Integrand size = 30, antiderivative size = 510 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\frac {b d^3 x \sqrt {a+b x^2}}{2 f^4}+\frac {(b e-a f) (d e-c f)^3 x \sqrt {a+b x^2}}{6 e f^4 \left (e+f x^2\right )^3}-\frac {(d e-c f)^2 (2 b e (10 d e-c f)-a f (13 d e+5 c f)) x \sqrt {a+b x^2}}{24 e^2 f^4 \left (e+f x^2\right )^2}+\frac {(d e-c f) \left (4 b^2 e^2 \left (26 d^2 e^2-7 c d e f-c^2 f^2\right )-2 a b e f \left (67 d^2 e^2+c d e f+4 c^2 f^2\right )+3 a^2 f^2 \left (11 d^2 e^2+8 c d e f+5 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{48 e^3 f^4 (b e-a f) \left (e+f x^2\right )}-\frac {\sqrt {b} d^2 (8 b d e-6 b c f-3 a d f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 f^5}+\frac {\left (16 b^3 d^2 e^5 (4 d e-3 c f)-24 a b^2 d^2 e^4 f (5 d e-3 c f)+6 a^2 b e f^2 \left (10 d^3 e^3-3 c d^2 e^2 f+c^3 f^3\right )-a^3 f^3 \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{16 e^{7/2} f^5 (b e-a f)^{3/2}} \] Output:
1/2*b*d^3*x*(b*x^2+a)^(1/2)/f^4+1/6*(-a*f+b*e)*(-c*f+d*e)^3*x*(b*x^2+a)^(1 /2)/e/f^4/(f*x^2+e)^3-1/24*(-c*f+d*e)^2*(2*b*e*(-c*f+10*d*e)-a*f*(5*c*f+13 *d*e))*x*(b*x^2+a)^(1/2)/e^2/f^4/(f*x^2+e)^2+1/48*(-c*f+d*e)*(4*b^2*e^2*(- c^2*f^2-7*c*d*e*f+26*d^2*e^2)-2*a*b*e*f*(4*c^2*f^2+c*d*e*f+67*d^2*e^2)+3*a ^2*f^2*(5*c^2*f^2+8*c*d*e*f+11*d^2*e^2))*x*(b*x^2+a)^(1/2)/e^3/f^4/(-a*f+b *e)/(f*x^2+e)-1/2*b^(1/2)*d^2*(-3*a*d*f-6*b*c*f+8*b*d*e)*arctanh(b^(1/2)*x /(b*x^2+a)^(1/2))/f^5+1/16*(16*b^3*d^2*e^5*(-3*c*f+4*d*e)-24*a*b^2*d^2*e^4 *f*(-3*c*f+5*d*e)+6*a^2*b*e*f^2*(c^3*f^3-3*c*d^2*e^2*f+10*d^3*e^3)-a^3*f^3 *(5*c^3*f^3+3*c^2*d*e*f^2+3*c*d^2*e^2*f+5*d^3*e^3))*arctanh((-a*f+b*e)^(1/ 2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(7/2)/f^5/(-a*f+b*e)^(3/2)
Time = 11.80 (sec) , antiderivative size = 456, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\frac {f x \sqrt {a+b x^2} \left (24 b d^3+\frac {8 (b e-a f) (d e-c f)^3}{e \left (e+f x^2\right )^3}-\frac {2 (d e-c f)^2 (2 b e (10 d e-c f)-a f (13 d e+5 c f))}{e^2 \left (e+f x^2\right )^2}+\frac {(d e-c f) \left (4 b^2 e^2 \left (26 d^2 e^2-7 c d e f-c^2 f^2\right )-2 a b e f \left (67 d^2 e^2+c d e f+4 c^2 f^2\right )+3 a^2 f^2 \left (11 d^2 e^2+8 c d e f+5 c^2 f^2\right )\right )}{e^3 (b e-a f) \left (e+f x^2\right )}\right )-\frac {3 \left (16 b^3 d^2 e^5 (4 d e-3 c f)-24 a b^2 d^2 e^4 f (5 d e-3 c f)+6 a^2 b e f^2 \left (10 d^3 e^3-3 c d^2 e^2 f+c^3 f^3\right )-a^3 f^3 \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{7/2} (-b e+a f)^{3/2}}-24 \sqrt {b} d^2 (8 b d e-6 b c f-3 a d f) \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )}{48 f^5} \] Input:
Integrate[((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2)^4,x]
Output:
(f*x*Sqrt[a + b*x^2]*(24*b*d^3 + (8*(b*e - a*f)*(d*e - c*f)^3)/(e*(e + f*x ^2)^3) - (2*(d*e - c*f)^2*(2*b*e*(10*d*e - c*f) - a*f*(13*d*e + 5*c*f)))/( e^2*(e + f*x^2)^2) + ((d*e - c*f)*(4*b^2*e^2*(26*d^2*e^2 - 7*c*d*e*f - c^2 *f^2) - 2*a*b*e*f*(67*d^2*e^2 + c*d*e*f + 4*c^2*f^2) + 3*a^2*f^2*(11*d^2*e ^2 + 8*c*d*e*f + 5*c^2*f^2)))/(e^3*(b*e - a*f)*(e + f*x^2))) - (3*(16*b^3* d^2*e^5*(4*d*e - 3*c*f) - 24*a*b^2*d^2*e^4*f*(5*d*e - 3*c*f) + 6*a^2*b*e*f ^2*(10*d^3*e^3 - 3*c*d^2*e^2*f + c^3*f^3) - a^3*f^3*(5*d^3*e^3 + 3*c*d^2*e ^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]* Sqrt[a + b*x^2])])/(e^(7/2)*(-(b*e) + a*f)^(3/2)) - 24*Sqrt[b]*d^2*(8*b*d* e - 6*b*c*f - 3*a*d*f)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(48*f^5)
Leaf count is larger than twice the leaf count of optimal. \(2602\) vs. \(2(510)=1020\).
Time = 2.45 (sec) , antiderivative size = 2602, normalized size of antiderivative = 5.10, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {425, 425, 425, 420, 299, 224, 219, 398, 224, 219, 291, 221, 425, 398, 224, 219, 291, 221, 402, 27, 291, 221, 402, 27, 291, 221, 402, 27, 291, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{\left (f x^2+e\right )^3}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{\left (f x^2+e\right )^4}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\int \frac {4 b (d e-c f) x^2+6 b c e-a d e-5 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\int \frac {4 b (d e-c f) x^2+6 b c e-a d e-5 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\int \frac {4 b (d e-c f) x^2+6 b c e-a d e-5 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\int \frac {4 b (d e-c f) x^2+6 b c e-a d e-5 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {3 f (d e+5 c f) a^2-2 b e (4 d e+17 c f) a+24 b^2 c e^2+2 b (2 b e (2 d e-5 c f)+a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {3 f (d e+5 c f) a^2-2 b e (4 d e+17 c f) a+24 b^2 c e^2+2 b (2 b e (2 d e-5 c f)+a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {3 f (d e+5 c f) a^2-2 b e (4 d e+17 c f) a+24 b^2 c e^2+2 b (2 b e (2 d e-5 c f)+a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {3 f (d e+5 c f) a^2-2 b e (4 d e+17 c f) a+24 b^2 c e^2+2 b (2 b e (2 d e-5 c f)+a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (4 b^2 (2 d e-11 c f) e^2+2 a b f (5 d e+22 c f) e-3 a^2 f^2 (d e+5 c f)\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {3 \left (-f^2 (d e+5 c f) a^3+2 b e f (2 d e+9 c f) a^2-8 b^2 e^2 (d e+3 c f) a+16 b^3 c e^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (4 b^2 (2 d e-11 c f) e^2+2 a b f (5 d e+22 c f) e-3 a^2 f^2 (d e+5 c f)\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {3 \left (-f^2 (d e+5 c f) a^3+2 b e f (2 d e+9 c f) a^2-8 b^2 e^2 (d e+3 c f) a+16 b^3 c e^3\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (4 b^2 (2 d e-11 c f) e^2+2 a b f (5 d e+22 c f) e-3 a^2 f^2 (d e+5 c f)\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {3 \left (-f^2 (d e+5 c f) a^3+2 b e f (2 d e+9 c f) a^2-8 b^2 e^2 (d e+3 c f) a+16 b^3 c e^3\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \sqrt {b x^2+a} x}{2 b}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (4 b^2 (2 d e-11 c f) e^2+2 a b f (5 d e+22 c f) e-3 a^2 f^2 (d e+5 c f)\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {3 \left (-f^2 (d e+5 c f) a^3+2 b e f (2 d e+9 c f) a^2-8 b^2 e^2 (d e+3 c f) a+16 b^3 c e^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\right )}{f}\) |
Input:
Int[((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2)^4,x]
Output:
(b*((b*((d*((d*((d*x*Sqrt[a + b*x^2])/(2*b) + ((2*b*c - a*d)*ArcTanh[(Sqrt [b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/f - ((d*e - c*f)*((d*ArcTanh[(Sqrt[ b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - ((d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f ]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f*Sqrt[b*e - a*f])))/f))/f - ((d *e - c*f)*((d*((d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - ((d* e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]* f*Sqrt[b*e - a*f])))/f - ((d*e - c*f)*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2* e*(b*e - a*f)*(e + f*x^2)) + ((2*b*c*e - a*(d*e + c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2))))/f))/ f))/f - ((b*e - a*f)*((d*((d*((d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sq rt[b]*f) - ((d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^ 2])])/(Sqrt[e]*f*Sqrt[b*e - a*f])))/f - ((d*e - c*f)*(((d*e - c*f)*x*Sqrt[ a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + ((2*b*c*e - a*(d*e + c*f))*Arc Tanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f )^(3/2))))/f))/f - ((d*e - c*f)*((d*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2*e* (b*e - a*f)*(e + f*x^2)) + ((2*b*c*e - a*(d*e + c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2))))/f - (( d*e - c*f)*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2 ) + (((2*b*e*(d*e - 3*c*f) + a*f*(d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b *e - a*f)*(e + f*x^2)) + ((8*b^2*c*e^2 - 4*a*b*e*(d*e + 2*c*f) + a^2*f*...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[d/b Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*x^2 )), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^2)^p*(c + d*x ^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt Q[p, 0] && GtQ[q, 0]
Time = 1.46 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.17
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (\left (f \,x^{2}+e \right )^{3} \left (\frac {5 a^{3} \left (c^{2} f^{2}-\frac {2}{5} c d e f +d^{2} e^{2}\right ) f^{3} \left (c f +d e \right ) \sqrt {b}}{48}+b^{\frac {3}{2}} \left (-\frac {4 b^{2} d^{3} e^{5}}{3}+\frac {5 \left (a d +\frac {2 b c}{5}\right ) d^{2} b f \,e^{4}}{2}-\frac {5 a \left (a d +\frac {6 b c}{5}\right ) d^{2} f^{2} e^{3}}{4}+\frac {3 a^{2} c \,d^{2} e^{2} f^{3}}{8}-\frac {a^{2} c^{3} f^{5}}{8}\right ) e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\frac {\sqrt {\left (a f -b e \right ) e}\, \left (-6 \left (-\frac {8 b d e}{3}+f \left (a d +2 b c \right )\right ) d^{2} \left (a f -b e \right ) b \left (f \,x^{2}+e \right )^{3} e^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\left (-\frac {11 a^{2} \left (c f -d e \right ) \left (\frac {5 e^{4} d^{2}}{11}+\frac {8 d \left (\frac {5 x^{2} d}{3}+c \right ) f \,e^{3}}{11}+f^{2} \left (\frac {64}{33} c d \,x^{2}+d^{2} x^{4}+c^{2}\right ) e^{2}+\frac {40 c \left (\frac {3 x^{2} d}{5}+c \right ) x^{2} f^{3} e}{33}+\frac {5 c^{2} f^{4} x^{4}}{11}\right ) f^{2} \sqrt {b}}{4}+\left (8 b \,d^{3} e^{6}-9 \left (-\frac {20}{9} b d \,x^{2}+\frac {2}{3} b c +a d \right ) d^{2} f \,e^{5}+\frac {9 \left (\frac {88 b d \,x^{4}}{27}+\left (-\frac {137 a d}{27}-\frac {10 b c}{3}\right ) x^{2}+a c \right ) d^{2} f^{2} e^{4}}{2}+\frac {23 d^{2} \left (\frac {4 b d \,x^{4}}{23}+\left (-\frac {103 a d}{69}-\frac {22 b c}{23}\right ) x^{2}+a c \right ) x^{2} f^{3} e^{3}}{2}+\frac {5 \left (-\frac {4 a \,d^{3} x^{6}}{5}+\left (\frac {22}{5} a c \,d^{2}+\frac {4}{5} b \,c^{2} d \right ) x^{4}+\left (\frac {7}{5} a \,c^{2} d +\frac {2}{5} c^{3} b \right ) x^{2}+c^{3} a \right ) f^{4} e^{2}}{2}+\frac {11 c^{2} \left (\left (-\frac {3 a d}{11}+\frac {2 b c}{11}\right ) x^{2}+a c \right ) x^{2} f^{5} e}{6}+\frac {2 a \,c^{3} f^{6} x^{4}}{3}\right ) b^{\frac {3}{2}} e \right ) \sqrt {b \,x^{2}+a}\, x f \right )}{12}\right )}{\sqrt {\left (a f -b e \right ) e}\, \sqrt {b}\, \left (a f -b e \right ) f^{5} \left (f \,x^{2}+e \right )^{3} e^{3}}\) | \(599\) |
| default | \(\text {Expression too large to display}\) | \(13094\) |
| risch | \(\text {Expression too large to display}\) | \(36543\) |
Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^4,x,method=_RETURNVERBOSE)
Output:
-3*((f*x^2+e)^3*(5/48*a^3*(c^2*f^2-2/5*c*d*e*f+d^2*e^2)*f^3*(c*f+d*e)*b^(1 /2)+b^(3/2)*(-4/3*b^2*d^3*e^5+5/2*(a*d+2/5*b*c)*d^2*b*f*e^4-5/4*a*(a*d+6/5 *b*c)*d^2*f^2*e^3+3/8*a^2*c*d^2*e^2*f^3-1/8*a^2*c^3*f^5)*e)*arctan(e*(b*x^ 2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+1/12*((a*f-b*e)*e)^(1/2)*(-6*(-8/3*b*d*e +f*(a*d+2*b*c))*d^2*(a*f-b*e)*b*(f*x^2+e)^3*e^3*arctanh((b*x^2+a)^(1/2)/x/ b^(1/2))+(-11/4*a^2*(c*f-d*e)*(5/11*e^4*d^2+8/11*d*(5/3*x^2*d+c)*f*e^3+f^2 *(64/33*c*d*x^2+d^2*x^4+c^2)*e^2+40/33*c*(3/5*x^2*d+c)*x^2*f^3*e+5/11*c^2* f^4*x^4)*f^2*b^(1/2)+(8*b*d^3*e^6-9*(-20/9*b*d*x^2+2/3*b*c+a*d)*d^2*f*e^5+ 9/2*(88/27*b*d*x^4+(-137/27*a*d-10/3*b*c)*x^2+a*c)*d^2*f^2*e^4+23/2*d^2*(4 /23*b*d*x^4+(-103/69*a*d-22/23*b*c)*x^2+a*c)*x^2*f^3*e^3+5/2*(-4/5*a*d^3*x ^6+(22/5*a*c*d^2+4/5*b*c^2*d)*x^4+(7/5*a*c^2*d+2/5*c^3*b)*x^2+c^3*a)*f^4*e ^2+11/6*c^2*((-3/11*a*d+2/11*b*c)*x^2+a*c)*x^2*f^5*e+2/3*a*c^3*f^6*x^4)*b^ (3/2)*e)*(b*x^2+a)^(1/2)*x*f))/((a*f-b*e)*e)^(1/2)/b^(1/2)/(a*f-b*e)/f^5/( f*x^2+e)^3/e^3
Leaf count of result is larger than twice the leaf count of optimal. 1892 vs. \(2 (474) = 948\).
Time = 112.69 (sec) , antiderivative size = 7661, normalized size of antiderivative = 15.02 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^4,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{3}}{\left (e + f x^{2}\right )^{4}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(d*x**2+c)**3/(f*x**2+e)**4,x)
Output:
Integral((a + b*x**2)**(3/2)*(c + d*x**2)**3/(e + f*x**2)**4, x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{3}}{{\left (f x^{2} + e\right )}^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^4,x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^3/(f*x^2 + e)^4, x)
Leaf count of result is larger than twice the leaf count of optimal. 3926 vs. \(2 (474) = 948\).
Time = 0.26 (sec) , antiderivative size = 3926, normalized size of antiderivative = 7.70 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^4,x, algorithm="giac")
Output:
1/2*sqrt(b*x^2 + a)*b*d^3*x/f^4 - 1/16*(64*b^(7/2)*d^3*e^6 - 48*b^(7/2)*c* d^2*e^5*f - 120*a*b^(5/2)*d^3*e^5*f + 72*a*b^(5/2)*c*d^2*e^4*f^2 + 60*a^2* b^(3/2)*d^3*e^4*f^2 - 18*a^2*b^(3/2)*c*d^2*e^3*f^3 - 5*a^3*sqrt(b)*d^3*e^3 *f^3 - 3*a^3*sqrt(b)*c*d^2*e^2*f^4 + 6*a^2*b^(3/2)*c^3*e*f^5 - 3*a^3*sqrt( b)*c^2*d*e*f^5 - 5*a^3*sqrt(b)*c^3*f^6)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^ 2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((b*e^4*f^5 - a*e^3*f ^6)*sqrt(-b^2*e^2 + a*b*e*f)) + 1/24*(288*(sqrt(b)*x - sqrt(b*x^2 + a))^10 *b^(7/2)*d^3*e^6*f^2 - 432*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*c*d^2* e^5*f^3 - 504*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*d^3*e^5*f^3 + 144 *(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*c^2*d*e^4*f^4 + 648*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*c*d^2*e^4*f^4 + 252*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*d^3*e^4*f^4 - 144*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a *b^(5/2)*c^2*d*e^3*f^5 - 234*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)* c*d^2*e^3*f^5 - 33*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*d^3*e^3*f^ 5 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*c*d^2*e^2*f^6 - 18*(sqr t(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*c^3*e*f^7 + 9*(sqrt(b)*x - sqrt(b *x^2 + a))^10*a^3*sqrt(b)*c^2*d*e*f^7 + 15*(sqrt(b)*x - sqrt(b*x^2 + a))^1 0*a^3*sqrt(b)*c^3*f^8 + 1920*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(9/2)*d^3*e ^7*f - 2592*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(9/2)*c*d^2*e^6*f^2 - 4368*( sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*d^3*e^6*f^2 + 576*(sqrt(b)*x -...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^3}{{\left (f\,x^2+e\right )}^4} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2)^4,x)
Output:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2)^4, x)
Time = 2.49 (sec) , antiderivative size = 12619, normalized size of antiderivative = 24.74 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^4,x)
Output:
( - 15*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b* x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c**3*e**3*f**7 - 45*sqr t(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sq rt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c**3*e**2*f**8*x**2 - 45*sqrt(e)* sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f) *sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c**3*e*f**9*x**4 - 15*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b) *x)/(sqrt(e)*sqrt(b)))*a**4*c**3*f**10*x**6 - 9*sqrt(e)*sqrt(a*f - b*e)*at an((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt( e)*sqrt(b)))*a**4*c**2*d*e**4*f**6 - 27*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt (a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt( b)))*a**4*c**2*d*e**3*f**7*x**2 - 27*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a* f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)) )*a**4*c**2*d*e**2*f**8*x**4 - 9*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a* *4*c**2*d*e*f**9*x**6 - 9*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c*d* *2*e**5*f**5 - 27*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)* sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c*d**2*e**4* f**6*x**2 - 27*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*...