Integrand size = 30, antiderivative size = 394 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\frac {d^2 (5 a d f-12 b (d e-c f)) x \sqrt {a+b x^2}}{8 f^4}+\frac {b d^3 x^3 \sqrt {a+b x^2}}{4 f^3}+\frac {(b e-a f) (d e-c f)^3 x \sqrt {a+b x^2}}{4 e f^4 \left (e+f x^2\right )^2}-\frac {(d e-c f)^2 (2 b e (7 d e-c f)-3 a f (3 d e+c f)) x \sqrt {a+b x^2}}{8 e^2 f^4 \left (e+f x^2\right )}+\frac {3 d \left (a^2 d^2 f^2-12 a b d f (d e-c f)+8 b^2 \left (2 d^2 e^2-3 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b} f^5}-\frac {3 (d e-c f) \left (8 b^2 d e^3 (2 d e-c f)-4 a b d e^2 f (5 d e-c f)+a^2 f^2 \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} f^5 \sqrt {b e-a f}} \] Output:
1/8*d^2*(5*a*d*f-12*b*(-c*f+d*e))*x*(b*x^2+a)^(1/2)/f^4+1/4*b*d^3*x^3*(b*x ^2+a)^(1/2)/f^3+1/4*(-a*f+b*e)*(-c*f+d*e)^3*x*(b*x^2+a)^(1/2)/e/f^4/(f*x^2 +e)^2-1/8*(-c*f+d*e)^2*(2*b*e*(-c*f+7*d*e)-3*a*f*(c*f+3*d*e))*x*(b*x^2+a)^ (1/2)/e^2/f^4/(f*x^2+e)+3/8*d*(a^2*d^2*f^2-12*a*b*d*f*(-c*f+d*e)+8*b^2*(c^ 2*f^2-3*c*d*e*f+2*d^2*e^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/f^5 -3/8*(-c*f+d*e)*(8*b^2*d*e^3*(-c*f+2*d*e)-4*a*b*d*e^2*f*(-c*f+5*d*e)+a^2*f ^2*(c^2*f^2+2*c*d*e*f+5*d^2*e^2))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^ 2+a)^(1/2))/e^(5/2)/f^5/(-a*f+b*e)^(1/2)
Time = 10.94 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\frac {f x \sqrt {a+b x^2} \left (d^2 (-12 b d e+12 b c f+5 a d f)+2 b d^3 f x^2+\frac {2 (b e-a f) (d e-c f)^3}{e \left (e+f x^2\right )^2}-\frac {(d e-c f)^2 (2 b e (7 d e-c f)-3 a f (3 d e+c f))}{e^2 \left (e+f x^2\right )}\right )-\frac {3 (d e-c f) \left (8 b^2 d e^3 (2 d e-c f)+4 a b d e^2 f (-5 d e+c f)+a^2 f^2 \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} \sqrt {-b e+a f}}+\frac {3 d \left (a^2 d^2 f^2-12 a b d f (d e-c f)+8 b^2 \left (2 d^2 e^2-3 c d e f+c^2 f^2\right )\right ) \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )}{\sqrt {b}}}{8 f^5} \] Input:
Integrate[((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2)^3,x]
Output:
(f*x*Sqrt[a + b*x^2]*(d^2*(-12*b*d*e + 12*b*c*f + 5*a*d*f) + 2*b*d^3*f*x^2 + (2*(b*e - a*f)*(d*e - c*f)^3)/(e*(e + f*x^2)^2) - ((d*e - c*f)^2*(2*b*e *(7*d*e - c*f) - 3*a*f*(3*d*e + c*f)))/(e^2*(e + f*x^2))) - (3*(d*e - c*f) *(8*b^2*d*e^3*(2*d*e - c*f) + 4*a*b*d*e^2*f*(-5*d*e + c*f) + a^2*f^2*(5*d^ 2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[ a + b*x^2])])/(e^(5/2)*Sqrt[-(b*e) + a*f]) + (3*d*(a^2*d^2*f^2 - 12*a*b*d* f*(d*e - c*f) + 8*b^2*(2*d^2*e^2 - 3*c*d*e*f + c^2*f^2))*Log[b*x + Sqrt[b] *Sqrt[a + b*x^2]])/Sqrt[b])/(8*f^5)
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{\left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{\left (f x^2+e\right )^3}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\int \frac {3 d (2 b c-a d) x^2+c (4 b c-a d)}{\sqrt {b x^2+a}}dx}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^3}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{2 b^{3/2}}+\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{2 b}}{4 b}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}-\frac {(d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {b x^2+a} \left (d x^2+c\right )}{4 b}+\frac {\frac {3 d (2 b c-a d) \sqrt {b x^2+a} x}{2 b}+\frac {\left (8 b^2 c^2-8 a b d c+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}}{4 b}\right )}{f}-\frac {(d e-c f) \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}\right )}{f}-\frac {(b e-a f) \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 \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}\right )}{f}-\frac {(b e-a f) \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 \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}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {b x^2+a} \left (d x^2+c\right )}{4 b}+\frac {\frac {3 d (2 b c-a d) \sqrt {b x^2+a} x}{2 b}+\frac {\left (8 b^2 c^2-8 a b d c+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}}{4 b}\right )}{f}-\frac {(d e-c f) \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}\right )}{f}-\frac {(b e-a f) \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 \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}\right )}{f}-\frac {(b e-a f) \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 \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}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {b x^2+a} \left (d x^2+c\right )}{4 b}+\frac {\frac {3 d (2 b c-a d) \sqrt {b x^2+a} x}{2 b}+\frac {\left (8 b^2 c^2-8 a b d c+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}}{4 b}\right )}{f}-\frac {(d e-c f) \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}\right )}{f}-\frac {(b e-a f) \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) \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}\right )}{f}-\frac {(b e-a f) \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) \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}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {b x^2+a} \left (d x^2+c\right )}{4 b}+\frac {\frac {3 d (2 b c-a d) \sqrt {b x^2+a} x}{2 b}+\frac {\left (8 b^2 c^2-8 a b d c+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}}{4 b}\right )}{f}-\frac {(d e-c f) \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}\right )}{f}-\frac {(b e-a f) \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) \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}\right )}{f}-\frac {(b e-a f) \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) \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}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {b x^2+a} \left (d x^2+c\right )}{4 b}+\frac {\frac {3 d (2 b c-a d) \sqrt {b x^2+a} x}{2 b}+\frac {\left (8 b^2 c^2-8 a b d c+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}}{4 b}\right )}{f}-\frac {(d e-c f) \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}\right )}{f}-\frac {(b e-a f) \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) \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}\right )}{f}-\frac {(b e-a f) \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) \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}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {b x^2+a} \left (d x^2+c\right )}{4 b}+\frac {\frac {3 d (2 b c-a d) \sqrt {b x^2+a} x}{2 b}+\frac {\left (8 b^2 c^2-8 a b d c+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}}{4 b}\right )}{f}-\frac {(d e-c f) \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}\right )}{f}-\frac {(b e-a f) \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}\right )}{f}-\frac {(b e-a f) \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}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {b x^2+a} \left (d x^2+c\right )}{4 b}+\frac {\frac {3 d (2 b c-a d) \sqrt {b x^2+a} x}{2 b}+\frac {\left (8 b^2 c^2-8 a b d c+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}}{4 b}\right )}{f}-\frac {(d e-c f) \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}\right )}{f}-\frac {(b e-a f) \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}\right )}{f}-\frac {(b e-a f) \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}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {b x^2+a} \left (d x^2+c\right )}{4 b}+\frac {\frac {3 d (2 b c-a d) \sqrt {b x^2+a} x}{2 b}+\frac {\left (8 b^2 c^2-8 a b d c+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}}{4 b}\right )}{f}-\frac {(d e-c f) \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}\right )}{f}-\frac {(b e-a f) \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}\right )}{f}-\frac {(b e-a f) \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}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {b x^2+a} \left (d x^2+c\right )}{4 b}+\frac {\frac {3 d (2 b c-a d) \sqrt {b x^2+a} x}{2 b}+\frac {\left (8 b^2 c^2-8 a b d c+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}}{4 b}\right )}{f}-\frac {(d e-c f) \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}\right )}{f}-\frac {(b e-a f) \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}\right )}{f}-\frac {(b e-a f) \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}\) |
Input:
Int[((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2)^3,x]
Output:
$Aborted
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
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[(((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 = 1.34 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.13
| method | result | size |
| pseudoelliptic | \(\frac {3 \left (c f -d e \right ) \left (-\frac {f^{2} a^{2} \left (c^{2} f^{2}+2 c d e f +5 d^{2} e^{2}\right ) \sqrt {b}}{8}+b^{\frac {3}{2}} d \left (-2 b d \,e^{2}+f \left (b c +\frac {5 a d}{2}\right ) e -\frac {a c \,f^{2}}{2}\right ) e^{2}\right ) \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\frac {\sqrt {\left (a f -b e \right ) e}\, \left (\frac {3 \left (16 b^{2} d^{2} e^{2}-12 b d f \left (a d +2 b c \right ) e +f^{2} \left (a^{2} d^{2}+12 a b c d +8 b^{2} c^{2}\right )\right ) d \left (f \,x^{2}+e \right )^{2} e^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{2}+\left (\frac {5 \left (\frac {12 e^{4} d^{3}}{5}-\frac {9 d^{2} \left (-\frac {19 x^{2} d}{9}+c \right ) f \,e^{3}}{5}-\frac {3 d \left (-\frac {5}{3} d^{2} x^{4}+5 c d \,x^{2}+c^{2}\right ) f^{2} e^{2}}{5}+c^{2} f^{3} \left (\frac {3 x^{2} d}{5}+c \right ) e +\frac {3 c^{3} f^{4} x^{2}}{5}\right ) a f \sqrt {b}}{2}+b^{\frac {3}{2}} \left (-12 e^{4} d^{3}+18 d^{2} f \left (-x^{2} d +c \right ) e^{3}-6 d \left (\frac {2}{3} d^{2} x^{4}-\frac {9}{2} c d \,x^{2}+c^{2}\right ) f^{2} e^{2}-9 d \,x^{2} f^{3} \left (-\frac {1}{9} d^{2} x^{4}-\frac {2}{3} c d \,x^{2}+c^{2}\right ) e +c^{3} f^{4} x^{2}\right ) e \right ) \sqrt {b \,x^{2}+a}\, x f \right )}{4}}{\sqrt {\left (a f -b e \right ) e}\, \sqrt {b}\, \left (f \,x^{2}+e \right )^{2} f^{5} e^{2}}\) | \(445\) |
| risch | \(\text {Expression too large to display}\) | \(2919\) |
| default | \(\text {Expression too large to display}\) | \(7197\) |
Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
3*((c*f-d*e)*(-1/8*f^2*a^2*(c^2*f^2+2*c*d*e*f+5*d^2*e^2)*b^(1/2)+b^(3/2)*d *(-2*b*d*e^2+f*(b*c+5/2*a*d)*e-1/2*a*c*f^2)*e^2)*(f*x^2+e)^2*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)*(3/2*(16*b^2*d ^2*e^2-12*b*d*f*(a*d+2*b*c)*e+f^2*(a^2*d^2+12*a*b*c*d+8*b^2*c^2))*d*(f*x^2 +e)^2*e^2*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+(5/2*(12/5*e^4*d^3-9/5*d^2*(- 19/9*x^2*d+c)*f*e^3-3/5*d*(-5/3*d^2*x^4+5*c*d*x^2+c^2)*f^2*e^2+c^2*f^3*(3/ 5*x^2*d+c)*e+3/5*c^3*f^4*x^2)*a*f*b^(1/2)+b^(3/2)*(-12*e^4*d^3+18*d^2*f*(- d*x^2+c)*e^3-6*d*(2/3*d^2*x^4-9/2*c*d*x^2+c^2)*f^2*e^2-9*d*x^2*f^3*(-1/9*d ^2*x^4-2/3*c*d*x^2+c^2)*e+c^3*f^4*x^2)*e)*(b*x^2+a)^(1/2)*x*f))/((a*f-b*e) *e)^(1/2)/b^(1/2)/(f*x^2+e)^2/f^5/e^2
Leaf count of result is larger than twice the leaf count of optimal. 1376 vs. \(2 (358) = 716\).
Time = 38.26 (sec) , antiderivative size = 5597, normalized size of antiderivative = 14.21 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{3}}{\left (e + f x^{2}\right )^{3}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(d*x**2+c)**3/(f*x**2+e)**3,x)
Output:
Integral((a + b*x**2)**(3/2)*(c + d*x**2)**3/(e + f*x**2)**3, x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{3}}{{\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^3/(f*x^2 + e)^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 1960 vs. \(2 (358) = 716\).
Time = 0.22 (sec) , antiderivative size = 1960, normalized size of antiderivative = 4.97 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="giac")
Output:
1/8*sqrt(b*x^2 + a)*(2*b*d^3*x^2/f^3 - (12*b^3*d^3*e*f^8 - 12*b^3*c*d^2*f^ 9 - 5*a*b^2*d^3*f^9)/(b^2*f^12))*x - 3/16*(16*b^2*d^3*e^2 - 24*b^2*c*d^2*e *f - 12*a*b*d^3*e*f + 8*b^2*c^2*d*f^2 + 12*a*b*c*d^2*f^2 + a^2*d^3*f^2)*lo g((sqrt(b)*x - sqrt(b*x^2 + a))^2)/(sqrt(b)*f^5) + 3/8*(16*b^(5/2)*d^3*e^5 - 24*b^(5/2)*c*d^2*e^4*f - 20*a*b^(3/2)*d^3*e^4*f + 8*b^(5/2)*c^2*d*e^3*f ^2 + 24*a*b^(3/2)*c*d^2*e^3*f^2 + 5*a^2*sqrt(b)*d^3*e^3*f^2 - 4*a*b^(3/2)* c^2*d*e^2*f^3 - 3*a^2*sqrt(b)*c*d^2*e^2*f^3 - a^2*sqrt(b)*c^2*d*e*f^4 - a^ 2*sqrt(b)*c^3*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))/(sqrt(-b^2*e^2 + a*b*e*f)*e^2*f^5) - 1/4*( 32*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*d^3*e^5*f - 72*(sqrt(b)*x - sqr t(b*x^2 + a))^6*b^(5/2)*c*d^2*e^4*f^2 - 36*(sqrt(b)*x - sqrt(b*x^2 + a))^6 *a*b^(3/2)*d^3*e^4*f^2 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c^2*d* e^3*f^3 + 72*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*d^2*e^3*f^3 + 9*( sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d^3*e^3*f^3 - 8*(sqrt(b)*x - sq rt(b*x^2 + a))^6*b^(5/2)*c^3*e^2*f^4 - 36*(sqrt(b)*x - sqrt(b*x^2 + a))^6* a*b^(3/2)*c^2*d*e^2*f^4 - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c *d^2*e^2*f^4 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c^2*d*e*f^5 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c^3*f^6 + 112*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d^3*e^6 - 240*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b ^(7/2)*c*d^2*e^5*f - 184*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*d^3*...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^3}{{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2)^3,x)
Output:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2)^3, x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{3}}{\left (f \,x^{2}+e \right )^{3}}d x \] Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^3,x)
Output:
int((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e)^3,x)