Integrand size = 30, antiderivative size = 414 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=-\frac {(b e-a f) (d e-c f)^2 x \sqrt {a+b x^2}}{6 e f^3 \left (e+f x^2\right )^3}+\frac {(d e-c f) (2 b e (7 d e-c f)-a f (7 d e+5 c f)) x \sqrt {a+b x^2}}{24 e^2 f^3 \left (e+f x^2\right )^2}-\frac {\left (4 b^2 e^2 \left (11 d^2 e^2-4 c d e f-c^2 f^2\right )-4 a b e f \left (11 d^2 e^2-c d e f+2 c^2 f^2\right )+3 a^2 f^2 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{48 e^3 f^3 (b e-a f) \left (e+f x^2\right )}+\frac {b^{3/2} d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f^4}-\frac {\left (16 b^3 d^2 e^5-24 a b^2 d^2 e^4 f+6 a^2 b e f^2 \left (d^2 e^2-c^2 f^2\right )+a^3 f^3 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{16 e^{7/2} f^4 (b e-a f)^{3/2}} \] Output:
-1/6*(-a*f+b*e)*(-c*f+d*e)^2*x*(b*x^2+a)^(1/2)/e/f^3/(f*x^2+e)^3+1/24*(-c* f+d*e)*(2*b*e*(-c*f+7*d*e)-a*f*(5*c*f+7*d*e))*x*(b*x^2+a)^(1/2)/e^2/f^3/(f *x^2+e)^2-1/48*(4*b^2*e^2*(-c^2*f^2-4*c*d*e*f+11*d^2*e^2)-4*a*b*e*f*(2*c^2 *f^2-c*d*e*f+11*d^2*e^2)+3*a^2*f^2*(5*c^2*f^2+2*c*d*e*f+d^2*e^2))*x*(b*x^2 +a)^(1/2)/e^3/f^3/(-a*f+b*e)/(f*x^2+e)+b^(3/2)*d^2*arctanh(b^(1/2)*x/(b*x^ 2+a)^(1/2))/f^4-1/16*(16*b^3*d^2*e^5-24*a*b^2*d^2*e^4*f+6*a^2*b*e*f^2*(-c^ 2*f^2+d^2*e^2)+a^3*f^3*(5*c^2*f^2+2*c*d*e*f+d^2*e^2))*arctanh((-a*f+b*e)^( 1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(7/2)/f^4/(-a*f+b*e)^(3/2)
Time = 11.46 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\frac {\frac {f x \sqrt {a+b x^2} \left (-8 e^2 (b e-a f) (d e-c f)^2+2 e (d e-c f) (2 b e (7 d e-c f)-a f (7 d e+5 c f)) \left (e+f x^2\right )-\frac {\left (4 a b e f \left (-11 d^2 e^2+c d e f-2 c^2 f^2\right )+4 b^2 e^2 \left (11 d^2 e^2-4 c d e f-c^2 f^2\right )+3 a^2 f^2 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \left (e+f x^2\right )^2}{b e-a f}\right )}{e^3 \left (e+f x^2\right )^3}+\frac {3 \left (16 b^3 d^2 e^5-24 a b^2 d^2 e^4 f+6 a^2 b e f^2 \left (d^2 e^2-c^2 f^2\right )+a^3 f^3 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{7/2} (-b e+a f)^{3/2}}+48 b^{3/2} d^2 \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )}{48 f^4} \] Input:
Integrate[((a + b*x^2)^(3/2)*(c + d*x^2)^2)/(e + f*x^2)^4,x]
Output:
((f*x*Sqrt[a + b*x^2]*(-8*e^2*(b*e - a*f)*(d*e - c*f)^2 + 2*e*(d*e - c*f)* (2*b*e*(7*d*e - c*f) - a*f*(7*d*e + 5*c*f))*(e + f*x^2) - ((4*a*b*e*f*(-11 *d^2*e^2 + c*d*e*f - 2*c^2*f^2) + 4*b^2*e^2*(11*d^2*e^2 - 4*c*d*e*f - c^2* f^2) + 3*a^2*f^2*(d^2*e^2 + 2*c*d*e*f + 5*c^2*f^2))*(e + f*x^2)^2)/(b*e - a*f)))/(e^3*(e + f*x^2)^3) + (3*(16*b^3*d^2*e^5 - 24*a*b^2*d^2*e^4*f + 6*a ^2*b*e*f^2*(d^2*e^2 - c^2*f^2) + a^3*f^3*(d^2*e^2 + 2*c*d*e*f + 5*c^2*f^2) )*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(e^(7/2)*(-(b* e) + a*f)^(3/2)) + 48*b^(3/2)*d^2*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(48* f^4)
Leaf count is larger than twice the leaf count of optimal. \(1559\) vs. \(2(414)=828\).
Time = 2.05 (sec) , antiderivative size = 1559, normalized size of antiderivative = 3.77, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {425, 425, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^3}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{\left (f x^2+e\right )^4}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )^4}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\int \frac {4 b (d e-c f) x^2+6 b c e-a d e-5 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\int \frac {4 b (d e-c f) x^2+6 b c e-a d e-5 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e (b e-a f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \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 {(b e-a 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 {b \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 {(b e-a 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}\) |
Input:
Int[((a + b*x^2)^(3/2)*(c + d*x^2)^2)/(e + f*x^2)^4,x]
Output:
(b*((b*((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*S qrt[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 - ((b*e - a*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]*S qrt[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*(d*e + 3*c*f))*ArcTan h[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^( 3/2)))/(4*e*(b*e - a*f))))/f))/f))/f - ((b*e - a*f)*((b*((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*(d*e + 3*c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*S qrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*(b*e - a*f))))/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[((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[((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 = 1.14 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.12
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (\left (\frac {16 b^{3} d^{2} e^{5}}{5}-\frac {24 a \,b^{2} d^{2} e^{4} f}{5}+\frac {6 a^{2} b \,d^{2} e^{3} f^{2}}{5}+\frac {a^{3} d^{2} e^{2} f^{3}}{5}+\frac {2 a^{2} c \,f^{4} \left (a d -3 b c \right ) e}{5}+a^{3} c^{2} f^{5}\right ) \left (f \,x^{2}+e \right )^{3} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {11 \left (\frac {16 d^{2} e^{3} \left (f \,x^{2}+e \right )^{3} \left (a f \,b^{\frac {3}{2}}-b^{\frac {5}{2}} e \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{11}+\left (\frac {8 b^{2} d^{2} e^{6}}{11}-\frac {6 \left (-\frac {10 b \,x^{2}}{3}+a \right ) d^{2} b f \,e^{5}}{11}-\frac {d^{2} f^{2} \left (-\frac {44}{3} b^{2} x^{4}+\frac {46}{3} a b \,x^{2}+a^{2}\right ) e^{4}}{11}-\frac {2 f^{3} \left (\left (\frac {22}{3} a b \,d^{2}+\frac {8}{3} b^{2} c d \right ) x^{4}+\left (\frac {4}{3} a^{2} d^{2}+\frac {14}{3} a b c d +2 b^{2} c^{2}\right ) x^{2}+a c \left (a d +5 b c \right )\right ) e^{3}}{11}+f^{4} \left (\left (\frac {1}{11} a^{2} d^{2}-\frac {4}{33} b^{2} c^{2}+\frac {4}{33} a b c d \right ) x^{4}+\left (-\frac {2}{3} b \,c^{2} a +\frac {16}{33} a^{2} c d \right ) x^{2}+a^{2} c^{2}\right ) e^{2}+\frac {40 a \left (\left (\frac {3 a d}{20}-\frac {b c}{5}\right ) x^{2}+a c \right ) c \,x^{2} f^{5} e}{33}+\frac {5 a^{2} c^{2} f^{6} x^{4}}{11}\right ) \sqrt {b \,x^{2}+a}\, x f \right ) \sqrt {\left (a f -b e \right ) e}}{5}\right )}{16 \sqrt {\left (a f -b e \right ) e}\, \left (f \,x^{2}+e \right )^{3} \left (a f -b e \right ) e^{3} f^{4}}\) | \(463\) |
| default | \(\text {Expression too large to display}\) | \(12984\) |
Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^2/(f*x^2+e)^4,x,method=_RETURNVERBOSE)
Output:
-5/16*((16/5*b^3*d^2*e^5-24/5*a*b^2*d^2*e^4*f+6/5*a^2*b*d^2*e^3*f^2+1/5*a^ 3*d^2*e^2*f^3+2/5*a^2*c*f^4*(a*d-3*b*c)*e+a^3*c^2*f^5)*(f*x^2+e)^3*arctan( e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))-11/5*(16/11*d^2*e^3*(f*x^2+e)^3*( a*f*b^(3/2)-b^(5/2)*e)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+(8/11*b^2*d^2*e^ 6-6/11*(-10/3*b*x^2+a)*d^2*b*f*e^5-1/11*d^2*f^2*(-44/3*b^2*x^4+46/3*a*b*x^ 2+a^2)*e^4-2/11*f^3*((22/3*a*b*d^2+8/3*b^2*c*d)*x^4+(4/3*a^2*d^2+14/3*a*b* c*d+2*b^2*c^2)*x^2+a*c*(a*d+5*b*c))*e^3+f^4*((1/11*a^2*d^2-4/33*b^2*c^2+4/ 33*a*b*c*d)*x^4+(-2/3*b*c^2*a+16/33*a^2*c*d)*x^2+a^2*c^2)*e^2+40/33*a*((3/ 20*a*d-1/5*b*c)*x^2+a*c)*c*x^2*f^5*e+5/11*a^2*c^2*f^6*x^4)*(b*x^2+a)^(1/2) *x*f)*((a*f-b*e)*e)^(1/2))/((a*f-b*e)*e)^(1/2)/(f*x^2+e)^3/(a*f-b*e)/e^3/f ^4
Leaf count of result is larger than twice the leaf count of optimal. 1247 vs. \(2 (384) = 768\).
Time = 29.43 (sec) , antiderivative size = 5081, normalized size of antiderivative = 12.27 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^2/(f*x^2+e)^4,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{2}}{\left (e + f x^{2}\right )^{4}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(d*x**2+c)**2/(f*x**2+e)**4,x)
Output:
Integral((a + b*x**2)**(3/2)*(c + d*x**2)**2/(e + f*x**2)**4, x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{2}}{{\left (f x^{2} + e\right )}^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^2/(f*x^2+e)^4,x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^2/(f*x^2 + e)^4, x)
Leaf count of result is larger than twice the leaf count of optimal. 2836 vs. \(2 (384) = 768\).
Time = 0.22 (sec) , antiderivative size = 2836, normalized size of antiderivative = 6.85 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^2/(f*x^2+e)^4,x, algorithm="giac")
Output:
1/16*(16*b^(7/2)*d^2*e^5 - 24*a*b^(5/2)*d^2*e^4*f + 6*a^2*b^(3/2)*d^2*e^3* f^2 + a^3*sqrt(b)*d^2*e^2*f^3 - 6*a^2*b^(3/2)*c^2*e*f^4 + 2*a^3*sqrt(b)*c* d*e*f^4 + 5*a^3*sqrt(b)*c^2*f^5)*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^4 - a*e^3*f^5)*sqr t(-b^2*e^2 + a*b*e*f)) - 1/2*b^(3/2)*d^2*log((sqrt(b)*x - sqrt(b*x^2 + a)) ^2)/f^4 - 1/24*(144*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*d^2*e^5*f^2 - 96*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*c*d*e^4*f^3 - 216*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*d^2*e^4*f^3 + 96*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*c*d*e^3*f^4 + 78*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^ (3/2)*d^2*e^3*f^4 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*d^2*e^2 *f^5 + 18*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*c^2*e*f^6 - 6*(sqrt (b)*x - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*c*d*e*f^6 - 15*(sqrt(b)*x - sqrt(b *x^2 + a))^10*a^3*sqrt(b)*c^2*f^7 + 864*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^ (9/2)*d^2*e^6*f - 384*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(9/2)*c*d*e^5*f^2 - 1728*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*d^2*e^5*f^2 - 96*(sqrt(b) *x - sqrt(b*x^2 + a))^8*b^(9/2)*c^2*e^4*f^3 + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*c*d*e^4*f^3 + 1188*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b ^(5/2)*d^2*e^4*f^3 + 96*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*c^2*e^3* f^4 - 96*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*c*d*e^3*f^4 - 324*(sq rt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(3/2)*d^2*e^3*f^4 + 180*(sqrt(b)*x -...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^2}{{\left (f\,x^2+e\right )}^4} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^2)/(e + f*x^2)^4,x)
Output:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^2)/(e + f*x^2)^4, x)
Time = 1.07 (sec) , antiderivative size = 8163, normalized size of antiderivative = 19.72 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^2/(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**2*e**3*f**6 - 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**2*e**2*f**7*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**2*e*f**8*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**2*f**9*x**6 - 6*sqrt(e)*sqrt(a*f - b*e)*ata n((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*e**4*f**5 - 18*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*e**3*f**6*x**2 - 18*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*e**2*f**7*x**4 - 6*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqr t(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c*d*e*f **8*x**6 - 3*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*d**2*e**5*f**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*d**2*e**4*f**5*x**2 - 9*sqrt (e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - ...