\(\int \frac {(a+b x^2)^{5/2}}{(c+d x^2) (e+f x^2)} \, dx\) [315]

Optimal result

Integrand size = 30, antiderivative size = 201 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {b^2 x \sqrt {a+b x^2}}{2 d f}+\frac {b^{3/2} (5 a d f-2 b (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^2 f^2}-\frac {(b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d^2 (d e-c f)}+\frac {(b e-a f)^{5/2} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^2 (d e-c f)} \] Output:

1/2*b^2*x*(b*x^2+a)^(1/2)/d/f+1/2*b^(3/2)*(5*a*d*f-2*b*(c*f+d*e))*arctanh( 
b^(1/2)*x/(b*x^2+a)^(1/2))/d^2/f^2-(-a*d+b*c)^(5/2)*arctanh((-a*d+b*c)^(1/ 
2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(1/2)/d^2/(-c*f+d*e)+(-a*f+b*e)^(5/2)*arct 
anh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(1/2)/f^2/(-c*f+d*e)
 

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {b^2 x \sqrt {a+b x^2}}{2 d f}+\frac {(-b c+a d)^{5/2} \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c} d^2 (-d e+c f)}+\frac {(-b e+a f)^{5/2} \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} f^2 (d e-c f)}+\frac {b^{3/2} (-5 a d f+2 b (d e+c f)) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^2 f^2} \] Input:

Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)*(e + f*x^2)),x]
 

Output:

(b^2*x*Sqrt[a + b*x^2])/(2*d*f) + ((-(b*c) + a*d)^(5/2)*ArcTan[(-(d*x*Sqrt 
[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(Sqrt[c]*Sqrt[-(b*c) + a*d])])/(Sqrt[c 
]*d^2*(-(d*e) + c*f)) + ((-(b*e) + a*f)^(5/2)*ArcTan[(-(f*x*Sqrt[a + b*x^2 
]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(Sqrt[e]*f^2*(d*e 
 - c*f)) + (b^(3/2)*(-5*a*d*f + 2*b*(d*e + c*f))*Log[-(Sqrt[b]*x) + Sqrt[a 
 + b*x^2]])/(2*d^2*f^2)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(429\) vs. \(2(201)=402\).

Time = 0.76 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.13, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {420, 318, 25, 398, 224, 219, 291, 221, 420, 301, 224, 219, 291, 221, 422, 301, 224, 219, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 420

\(\displaystyle \frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{f x^2+e}dx}{d}-\frac {(b c-a d) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\)

\(\Big \downarrow \) 318

\(\displaystyle \frac {b \left (\frac {\int -\frac {b (2 b e-3 a f) x^2+a (b e-2 a f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}+\frac {b x \sqrt {a+b x^2}}{2 f}\right )}{d}-\frac {(b c-a d) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\int \frac {b (2 b e-3 a f) x^2+a (b e-2 a f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}\right )}{d}-\frac {(b c-a d) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {b (2 b e-3 a f) \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {2 (b e-a f)^2 \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{d}-\frac {(b c-a d) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {b (2 b e-3 a f) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {2 (b e-a f)^2 \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{d}-\frac {(b c-a d) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^2 \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{d}-\frac {(b c-a d) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^2 \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}}{2 f}\right )}{d}-\frac {(b c-a d) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\)

\(\Big \downarrow \) 420

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{f x^2+e}dx}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 422

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{f x^2+e}dx}{d e-c f}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{d e-c f}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b e-3 a f)}{f}-\frac {2 (b e-a f)^{3/2} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{d}\right )}{d}\)

Input:

Int[(a + b*x^2)^(5/2)/((c + d*x^2)*(e + f*x^2)),x]
 

Output:

(b*((b*x*Sqrt[a + b*x^2])/(2*f) - ((Sqrt[b]*(2*b*e - 3*a*f)*ArcTanh[(Sqrt[ 
b]*x)/Sqrt[a + b*x^2]])/f - (2*(b*e - a*f)^(3/2)*ArcTanh[(Sqrt[b*e - a*f]* 
x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f))/(2*f)))/d - ((b*c - a*d)*((b*( 
(Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/f - (Sqrt[b*e - a*f]*ArcTan 
h[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f)))/d - ((b*c 
- a*d)*((d*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - (Sqrt[b*c - 
 a*d]*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d)) 
)/(d*e - c*f) - (f*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/f - (Sq 
rt[b*e - a*f]*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqr 
t[e]*f)))/(d*e - c*f)))/d))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 219

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])
 

rule 221

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]
 

rule 224

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]
 

rule 291

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]
 

rule 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 318

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]
 

rule 398

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]
 

rule 420

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]
 

rule 422

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[-d/(b*c - a*d)   Int[(c + d*x^2)^q*(e + f*x^2)^r, 
 x], x] + Simp[b/(b*c - a*d)   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] && LeQ[q, -1]
 

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(\frac {\left (a d -b c \right )^{3} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{d^{2} \left (c f -d e \right ) \sqrt {\left (a d -b c \right ) c}}-\frac {b^{2} \left (-\sqrt {b \,x^{2}+a}\, d f x -\frac {\left (5 a d f -2 b c f -2 b d e \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{\sqrt {b}}\right )}{2 d^{2} f^{2}}-\frac {\left (a f -b e \right )^{3} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{f^{2} \left (c f -d e \right ) \sqrt {\left (a f -b e \right ) e}}\)	\(193\)
risch	\(\frac {b^{2} x \sqrt {b \,x^{2}+a}}{2 d f}+\frac {\frac {b^{\frac {3}{2}} \left (5 a d f -2 b c f -2 b d e \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d f}-\frac {f^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {\frac {2 a d -2 b c}{d}+\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{\sqrt {-c d}\, \left (\sqrt {-c d}\, f +\sqrt {-e f}\, d \right ) \left (\sqrt {-c d}\, f -\sqrt {-e f}\, d \right ) \sqrt {\frac {a d -b c}{d}}}-\frac {d^{2} \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\left (\sqrt {-c d}\, f +\sqrt {-e f}\, d \right ) \left (\sqrt {-c d}\, f -\sqrt {-e f}\, d \right ) \sqrt {-e f}\, \sqrt {\frac {a f -b e}{f}}}+\frac {f^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {\frac {2 a d -2 b c}{d}-\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{\sqrt {-c d}\, \left (\sqrt {-c d}\, f +\sqrt {-e f}\, d \right ) \left (\sqrt {-c d}\, f -\sqrt {-e f}\, d \right ) \sqrt {\frac {a d -b c}{d}}}+\frac {d^{2} \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\left (\sqrt {-c d}\, f +\sqrt {-e f}\, d \right ) \left (\sqrt {-c d}\, f -\sqrt {-e f}\, d \right ) \sqrt {-e f}\, \sqrt {\frac {a f -b e}{f}}}}{2 d f}\)	\(980\)
default	\(\text {Expression too large to display}\)	\(4266\)

Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

(a*d-b*c)^3/d^2/(c*f-d*e)/((a*d-b*c)*c)^(1/2)*arctan(c*(b*x^2+a)^(1/2)/x/( 
(a*d-b*c)*c)^(1/2))-1/2*b^2*(-(b*x^2+a)^(1/2)*d*f*x-(5*a*d*f-2*b*c*f-2*b*d 
*e)/b^(1/2)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2)))/d^2/f^2-(a*f-b*e)^3/f^2/(c 
*f-d*e)/((a*f-b*e)*e)^(1/2)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2) 
)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)/(d*x**2+c)/(f*x**2+e),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)*(f*x^2 + e)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{\left (d\,x^2+c\right )\,\left (f\,x^2+e\right )} \,d x \] Input:

int((a + b*x^2)^(5/2)/((c + d*x^2)*(e + f*x^2)),x)
 

Output:

int((a + b*x^2)^(5/2)/((c + d*x^2)*(e + f*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 1.09 (sec) , antiderivative size = 924, normalized size of antiderivative = 4.60 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e),x)
 

Output:

(2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2 
) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**2*d**2*e*f**2 - 4*sqrt(c)*sqr 
t(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sq 
rt(b)*x)/(sqrt(c)*sqrt(b)))*a*b*c*d*e*f**2 + 2*sqrt(c)*sqrt(a*d - b*c)*ata 
n((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c 
)*sqrt(b)))*b**2*c**2*e*f**2 + 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - 
b*c) + sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a* 
*2*d**2*e*f**2 - 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d) 
*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*b*c*d*e*f**2 + 
 2*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2 
) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*b**2*c**2*e*f**2 - 2*sqrt(e)*sqr 
t(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sq 
rt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*c*d**2*f**2 + 4*sqrt(e)*sqrt(a*f - b*e)*a 
tan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt 
(e)*sqrt(b)))*a*b*c*d**2*e*f - 2*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)))*b* 
*2*c*d**2*e**2 - 2*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**2*c*d**2*f**2 
 + 4*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*b*c*d**2*e*f - 2*sqrt(e)*...