\(\int \frac {(c+d x^2)^3}{\sqrt {a+b x^2} (e+f x^2)^2} \, dx\) [329]

Optimal result

Integrand size = 30, antiderivative size = 216 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\frac {d^3 x \sqrt {a+b x^2}}{2 b f^2}+\frac {(d e-c f)^3 x \sqrt {a+b x^2}}{2 e f^2 (b e-a f) \left (e+f x^2\right )}-\frac {d^2 (4 b d e-6 b c f+a d f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2} f^3}+\frac {(d e-c f)^2 (2 b e (2 d e+c f)-a f (5 d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} f^3 (b e-a f)^{3/2}} \] Output:

1/2*d^3*x*(b*x^2+a)^(1/2)/b/f^2+1/2*(-c*f+d*e)^3*x*(b*x^2+a)^(1/2)/e/f^2/( 
-a*f+b*e)/(f*x^2+e)-1/2*d^2*(a*d*f-6*b*c*f+4*b*d*e)*arctanh(b^(1/2)*x/(b*x 
^2+a)^(1/2))/b^(3/2)/f^3+1/2*(-c*f+d*e)^2*(2*b*e*(c*f+2*d*e)-a*f*(c*f+5*d* 
e))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(3/2)/f^3/(-a*f+ 
b*e)^(3/2)
 

Mathematica [A] (verified)

Time = 1.55 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.19 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\frac {-\frac {f x \sqrt {a+b x^2} \left (a d^3 e f \left (e+f x^2\right )+b \left (3 c d^2 e^2 f-3 c^2 d e f^2+c^3 f^3-d^3 e^2 \left (2 e+f x^2\right )\right )\right )}{b e (b e-a f) \left (e+f x^2\right )}+\frac {(d e-c f)^2 (2 b e (2 d e+c f)-a f (5 d e+c f)) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2} (-b e+a f)^{3/2}}+\frac {d^2 (4 b d e-6 b c f+a d f) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{2 f^3} \] Input:

Integrate[(c + d*x^2)^3/(Sqrt[a + b*x^2]*(e + f*x^2)^2),x]
 

Output:

(-((f*x*Sqrt[a + b*x^2]*(a*d^3*e*f*(e + f*x^2) + b*(3*c*d^2*e^2*f - 3*c^2* 
d*e*f^2 + c^3*f^3 - d^3*e^2*(2*e + f*x^2))))/(b*e*(b*e - a*f)*(e + f*x^2)) 
) + ((d*e - c*f)^2*(2*b*e*(2*d*e + c*f) - a*f*(5*d*e + c*f))*ArcTan[(-(f*x 
*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(e 
^(3/2)*(-(b*e) + a*f)^(3/2)) + (d^2*(4*b*d*e - 6*b*c*f + a*d*f)*Log[-(Sqrt 
[b]*x) + Sqrt[a + b*x^2]])/b^(3/2))/(2*f^3)
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.91, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {425, 420, 299, 224, 219, 398, 224, 219, 291, 221, 425, 398, 224, 219, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

Int[(c + d*x^2)^3/(Sqrt[a + b*x^2]*(e + f*x^2)^2),x]
 

Output:

(d*((d*((d*x*Sqrt[a + b*x^2])/(2*b) + ((2*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/S 
qrt[a + b*x^2]])/(2*b^(3/2))))/f - ((d*e - c*f)*((d*ArcTanh[(Sqrt[b]*x)/Sq 
rt[a + b*x^2]])/(Sqrt[b]*f) - ((d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sq 
rt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f*Sqrt[b*e - a*f])))/f))/f - ((d*e - c*f 
)*((d*((d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - ((d*e - c*f) 
*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f*Sqrt[b 
*e - a*f])))/f - ((d*e - c*f)*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2*e*(b*e - 
 a*f)*(e + f*x^2)) + ((2*b*c*e - a*(d*e + c*f))*ArcTanh[(Sqrt[b*e - a*f]*x 
)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2))))/f))/f
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 299

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]
 

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 402

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]
 

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 425

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]
 

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.23

method	result	size
pseudoelliptic	\(\frac {\left (2 b d \,e^{2}+f \left (-\frac {5 a d}{2}+b c \right ) e -\frac {a c \,f^{2}}{2}\right ) \left (c f -d e \right )^{2} \left (f \,x^{2}+e \right ) b^{\frac {5}{2}} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\frac {\left (-d^{2} \left (a f -b e \right ) b \left (f \,x^{2}+e \right ) \left (4 b d e +f \left (a d -6 b c \right )\right ) e \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\sqrt {b \,x^{2}+a}\, x f \left (-2 b \,d^{3} e^{3}+\left (\left (-x^{2} d +3 c \right ) b +a d \right ) d^{2} f \,e^{2}+d \,f^{2} \left (a \,d^{2} x^{2}-3 b \,c^{2}\right ) e +b \,c^{3} f^{3}\right ) b^{\frac {3}{2}}\right ) \sqrt {\left (a f -b e \right ) e}}{2}}{\sqrt {\left (a f -b e \right ) e}\, b^{\frac {5}{2}} f^{3} \left (a f -b e \right ) e \left (f \,x^{2}+e \right )}\)	\(265\)
risch	\(\text {Expression too large to display}\)	\(1035\)
default	\(\text {Expression too large to display}\)	\(1055\)

Input:

int((d*x^2+c)^3/(b*x^2+a)^(1/2)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

((2*b*d*e^2+f*(-5/2*a*d+b*c)*e-1/2*a*c*f^2)*(c*f-d*e)^2*(f*x^2+e)*b^(5/2)* 
arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+1/2*(-d^2*(a*f-b*e)*b*(f*x 
^2+e)*(4*b*d*e+f*(a*d-6*b*c))*e*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+(b*x^2+ 
a)^(1/2)*x*f*(-2*b*d^3*e^3+((-d*x^2+3*c)*b+a*d)*d^2*f*e^2+d*f^2*(a*d^2*x^2 
-3*b*c^2)*e+b*c^3*f^3)*b^(3/2))*((a*f-b*e)*e)^(1/2))/((a*f-b*e)*e)^(1/2)/b 
^(5/2)/f^3/(a*f-b*e)/e/(f*x^2+e)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 747 vs. \(2 (188) = 376\).

Time = 22.76 (sec) , antiderivative size = 3081, normalized size of antiderivative = 14.26 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^3/(b*x^2+a)^(1/2)/(f*x^2+e)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {\left (c + d x^{2}\right )^{3}}{\sqrt {a + b x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:

integrate((d*x**2+c)**3/(b*x**2+a)**(1/2)/(f*x**2+e)**2,x)
 

Output:

Integral((c + d*x**2)**3/(sqrt(a + b*x**2)*(e + f*x**2)**2), x)
 

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3}}{\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:

integrate((d*x^2+c)^3/(b*x^2+a)^(1/2)/(f*x^2+e)^2,x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^3/(sqrt(b*x^2 + a)*(f*x^2 + e)^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (188) = 376\).

Time = 0.17 (sec) , antiderivative size = 652, normalized size of antiderivative = 3.02 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\frac {\sqrt {b x^{2} + a} d^{3} x}{2 \, b f^{2}} - \frac {{\left (4 \, b^{\frac {3}{2}} d^{3} e^{4} - 6 \, b^{\frac {3}{2}} c d^{2} e^{3} f - 5 \, a \sqrt {b} d^{3} e^{3} f + 9 \, a \sqrt {b} c d^{2} e^{2} f^{2} + 2 \, b^{\frac {3}{2}} c^{3} e f^{3} - 3 \, a \sqrt {b} c^{2} d e f^{3} - a \sqrt {b} c^{3} f^{4}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{2 \, {\left (b e^{2} f^{3} - a e f^{4}\right )} \sqrt {-b^{2} e^{2} + a b e f}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{2} d^{3} e^{4} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{2} c d^{2} e^{3} f - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b d^{3} e^{3} f + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{2} c^{2} d e^{2} f^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b c d^{2} e^{2} f^{2} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{2} c^{3} e f^{3} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b c^{2} d e f^{3} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b c^{3} f^{4} + a^{2} b d^{3} e^{3} f - 3 \, a^{2} b c d^{2} e^{2} f^{2} + 3 \, a^{2} b c^{2} d e f^{3} - a^{2} b c^{3} f^{4}}{{\left (b^{\frac {3}{2}} e^{2} f^{3} - a \sqrt {b} e f^{4}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )}} + \frac {{\left (4 \, b d^{3} e - 6 \, b c d^{2} f + a d^{3} f\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{4 \, b^{\frac {3}{2}} f^{3}} \] Input:

integrate((d*x^2+c)^3/(b*x^2+a)^(1/2)/(f*x^2+e)^2,x, algorithm="giac")
 

Output:

1/2*sqrt(b*x^2 + a)*d^3*x/(b*f^2) - 1/2*(4*b^(3/2)*d^3*e^4 - 6*b^(3/2)*c*d 
^2*e^3*f - 5*a*sqrt(b)*d^3*e^3*f + 9*a*sqrt(b)*c*d^2*e^2*f^2 + 2*b^(3/2)*c 
^3*e*f^3 - 3*a*sqrt(b)*c^2*d*e*f^3 - a*sqrt(b)*c^3*f^4)*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 
^2*f^3 - a*e*f^4)*sqrt(-b^2*e^2 + a*b*e*f)) + (2*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^2*b^2*d^3*e^4 - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^2*c*d^2*e^3*f - ( 
sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b*d^3*e^3*f + 6*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^2*b^2*c^2*d*e^2*f^2 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b*c*d^2*e^2 
*f^2 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^2*c^3*e*f^3 - 3*(sqrt(b)*x - sq 
rt(b*x^2 + a))^2*a*b*c^2*d*e*f^3 + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b*c^3 
*f^4 + a^2*b*d^3*e^3*f - 3*a^2*b*c*d^2*e^2*f^2 + 3*a^2*b*c^2*d*e*f^3 - a^2 
*b*c^3*f^4)/((b^(3/2)*e^2*f^3 - a*sqrt(b)*e*f^4)*((sqrt(b)*x - sqrt(b*x^2 
+ a))^4*f + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*e - 2*(sqrt(b)*x - sqrt(b* 
x^2 + a))^2*a*f + a^2*f)) + 1/4*(4*b*d^3*e - 6*b*c*d^2*f + a*d^3*f)*log((s 
qrt(b)*x - sqrt(b*x^2 + a))^2)/(b^(3/2)*f^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {{\left (d\,x^2+c\right )}^3}{\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:

int((c + d*x^2)^3/((a + b*x^2)^(1/2)*(e + f*x^2)^2),x)
 

Output:

int((c + d*x^2)^3/((a + b*x^2)^(1/2)*(e + f*x^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 2669, normalized size of antiderivative = 12.36 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^3/(b*x^2+a)^(1/2)/(f*x^2+e)^2,x)
 

Output:

( - 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**2*c**3*e*f**4 - sqrt(e)*sq 
rt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*s 
qrt(b)*x)/(sqrt(e)*sqrt(b)))*a*b**2*c**3*f**5*x**2 - 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*b**2*c**2*d*e**2*f**3 - 3*sqrt(e)*sqrt(a*f - b*e)*at 
an((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt( 
e)*sqrt(b)))*a*b**2*c**2*d*e*f**4*x**2 + 9*sqrt(e)*sqrt(a*f - b*e)*atan((s 
qrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sq 
rt(b)))*a*b**2*c*d**2*e**3*f**2 + 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*b**2*c*d**2*e**2*f**3*x**2 - 5*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**2*d**3*e**4*f - 5*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**2*d**3*e 
**3*f**2*x**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)))*b**3*c**3*e**2*f* 
*3 + 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**3*c**3*e*f**4*x**2 - 6*sq 
rt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) ...