\(\int \frac {(c+d x^2)^3}{\sqrt {a+b x^2} (e+f x^2)^3} \, dx\) [330]

Optimal result

Integrand size = 30, antiderivative size = 309 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx=\frac {(d e-c f)^3 x \sqrt {a+b x^2}}{4 e f^2 (b e-a f) \left (e+f x^2\right )^2}-\frac {3 (d e-c f)^2 (2 b e (d e+c f)-a f (3 d e+c f)) x \sqrt {a+b x^2}}{8 e^2 f^2 (b e-a f)^2 \left (e+f x^2\right )}+\frac {d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f^3}-\frac {(d e-c f) \left (8 b^2 e^2 \left (d^2 e^2+c d e f+c^2 f^2\right )+3 a^2 f^2 \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-4 a b e f \left (5 d^2 e^2+5 c d e f+2 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} f^3 (b e-a f)^{5/2}} \] Output:

1/4*(-c*f+d*e)^3*x*(b*x^2+a)^(1/2)/e/f^2/(-a*f+b*e)/(f*x^2+e)^2-3/8*(-c*f+ 
d*e)^2*(2*b*e*(c*f+d*e)-a*f*(c*f+3*d*e))*x*(b*x^2+a)^(1/2)/e^2/f^2/(-a*f+b 
*e)^2/(f*x^2+e)+d^3*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/f^3-1/8*(-c 
*f+d*e)*(8*b^2*e^2*(c^2*f^2+c*d*e*f+d^2*e^2)+3*a^2*f^2*(c^2*f^2+2*c*d*e*f+ 
5*d^2*e^2)-4*a*b*e*f*(2*c^2*f^2+5*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^3/(-a*f+b*e)^(5/2)
 

Mathematica [A] (verified)

Time = 11.49 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.32 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx=\frac {-\frac {12 d (d e-c f)^2 x \left (f \left (a+b x^2\right )-\frac {(2 b e-a f) \left (e+f x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{e \sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{e (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {(d e-c f)^3 x \left (e f \left (a+b x^2\right ) \left (2 b e \left (4 e+3 f x^2\right )-a f \left (5 e+3 f x^2\right )\right )-\frac {\left (8 b^2 e^2-8 a b e f+3 a^2 f^2\right ) \left (e+f x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{e^3 (b e-a f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}+\frac {8 d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}-\frac {24 d^2 (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f}}}{8 f^3} \] Input:

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

Output:

((-12*d*(d*e - c*f)^2*x*(f*(a + b*x^2) - ((2*b*e - a*f)*(e + f*x^2)*ArcTan 
h[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/(e*Sqrt[((b*e - a*f)*x^2)/(e*( 
a + b*x^2))])))/(e*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)) + ((d*e - c*f) 
^3*x*(e*f*(a + b*x^2)*(2*b*e*(4*e + 3*f*x^2) - a*f*(5*e + 3*f*x^2)) - ((8* 
b^2*e^2 - 8*a*b*e*f + 3*a^2*f^2)*(e + f*x^2)^2*ArcTanh[Sqrt[((b*e - a*f)*x 
^2)/(e*(a + b*x^2))]])/Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]))/(e^3*(b*e 
 - a*f)^2*Sqrt[a + b*x^2]*(e + f*x^2)^2) + (8*d^3*ArcTanh[(Sqrt[b]*x)/Sqrt 
[a + b*x^2]])/Sqrt[b] - (24*d^2*(d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(S 
qrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*Sqrt[b*e - a*f]))/(8*f^3)
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.92, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {425, 425, 398, 224, 219, 291, 221, 402, 27, 291, 221, 402, 27, 291, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\int \frac {2 b c e-a (d e+c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(2 b c e-a (c f+d e)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(2 b c e-a (c f+d e)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {(2 b c e-a (c f+d e)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(2 b c e-a (c f+d e)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\int \frac {f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (a f (3 c f+d e)+2 b e (d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\left (a^2 f (3 c f+d e)-4 a b e (2 c f+d e)+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (a f (3 c f+d e)+2 b e (d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\left (a^2 f (3 c f+d e)-4 a b e (2 c f+d e)+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (a f (3 c f+d e)+2 b e (d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}-\frac {(d e-c f) \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b c e-a (c f+d e))}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f (3 c f+d e)-4 a b e (2 c f+d e)+8 b^2 c e^2\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} (a f (3 c f+d e)+2 b e (d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}\)

Input:

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

Output:

(d*((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 - ((d 
*e - c*f)*((d*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2 
)) + ((2*b*c*e - a*(d*e + c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[ 
a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2))))/f - ((d*e - c*f)*(((d*e - c* 
f)*x*Sqrt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2) + (((2*b*e*(d*e - 3* 
c*f) + a*f*(d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) 
 + ((8*b^2*c*e^2 - 4*a*b*e*(d*e + 2*c*f) + a^2*f*(d*e + 3*c*f))*ArcTanh[(S 
qrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2) 
))/(4*e*(b*e - a*f))))/f))/f
 

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 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 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 = 1.13 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.21

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {b}\, \left (a f -b e \right ) \left (3 a^{2} c^{2} f^{4}+6 a^{2} c d e \,f^{3}+15 a^{2} d^{2} e^{2} f^{2}-8 a b \,c^{2} e \,f^{3}-20 a b c d \,e^{2} f^{2}-20 a b \,d^{2} e^{3} f +8 b^{2} c^{2} e^{2} f^{2}+8 b^{2} c d \,e^{3} f +8 b^{2} d^{2} e^{4}\right ) \left (c f -d e \right ) \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8}+\left (d^{3} e^{2} \left (f \,x^{2}+e \right )^{2} \left (a f -b e \right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\frac {\sqrt {b}\, \sqrt {b \,x^{2}+a}\, \left (a f -b e \right ) \left (3 a c \,f^{3} x^{2}+9 e \,f^{2} a d \,x^{2}-6 b c e \,f^{2} x^{2}-6 b d \,e^{2} f \,x^{2}+5 a c e \,f^{2}+7 a d \,e^{2} f -8 b c \,e^{2} f -4 b d \,e^{3}\right ) \left (c f -d e \right )^{2} x f}{8}\right ) \sqrt {\left (a f -b e \right ) e}}{\sqrt {\left (a f -b e \right ) e}\, \sqrt {b}\, f^{3} \left (f \,x^{2}+e \right )^{2} e^{2} \left (a f -b e \right )^{3}}\)	\(373\)
default	\(\text {Expression too large to display}\)	\(2087\)

Input:

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

Output:

(-1/8*b^(1/2)*(a*f-b*e)*(3*a^2*c^2*f^4+6*a^2*c*d*e*f^3+15*a^2*d^2*e^2*f^2- 
8*a*b*c^2*e*f^3-20*a*b*c*d*e^2*f^2-20*a*b*d^2*e^3*f+8*b^2*c^2*e^2*f^2+8*b^ 
2*c*d*e^3*f+8*b^2*d^2*e^4)*(c*f-d*e)*(f*x^2+e)^2*arctan(e*(b*x^2+a)^(1/2)/ 
x/((a*f-b*e)*e)^(1/2))+(d^3*e^2*(f*x^2+e)^2*(a*f-b*e)^3*arctanh((b*x^2+a)^ 
(1/2)/x/b^(1/2))+1/8*b^(1/2)*(b*x^2+a)^(1/2)*(a*f-b*e)*(3*a*c*f^3*x^2+9*a* 
d*e*f^2*x^2-6*b*c*e*f^2*x^2-6*b*d*e^2*f*x^2+5*a*c*e*f^2+7*a*d*e^2*f-8*b*c* 
e^2*f-4*b*d*e^3)*(c*f-d*e)^2*x*f)*((a*f-b*e)*e)^(1/2))/((a*f-b*e)*e)^(1/2) 
/b^(1/2)/f^3/(f*x^2+e)^2/e^2/(a*f-b*e)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1138 vs. \(2 (283) = 566\).

Time = 66.38 (sec) , antiderivative size = 4646, normalized size of antiderivative = 15.04 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1863 vs. \(2 (283) = 566\).

Time = 0.18 (sec) , antiderivative size = 1863, normalized size of antiderivative = 6.03 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(8*b^(5/2)*d^3*e^5 - 20*a*b^(3/2)*d^3*e^4*f + 15*a^2*sqrt(b)*d^3*e^3*f 
^2 - 8*b^(5/2)*c^3*e^2*f^3 + 12*a*b^(3/2)*c^2*d*e^2*f^3 - 9*a^2*sqrt(b)*c* 
d^2*e^2*f^3 + 8*a*b^(3/2)*c^3*e*f^4 - 3*a^2*sqrt(b)*c^2*d*e*f^4 - 3*a^2*sq 
rt(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))/((b^2*e^4*f^3 - 2*a*b*e^3*f^4 + a^2*e^2*f^5)*s 
qrt(-b^2*e^2 + a*b*e*f)) - 1/2*d^3*log((sqrt(b)*x - sqrt(b*x^2 + a))^2)/(s 
qrt(b)*f^3) - 1/4*(16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*d^3*e^5*f - 
24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c*d^2*e^4*f^2 - 28*(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*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 - sqrt(b*x^2 + a))^6*b^(5/2)*c^3*e^2*f^4 - 1 
2*(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 - 8*(sqrt(b)*x - sqrt(b*x^2 
 + a))^6*a*b^(3/2)*c^3*e*f^5 + 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 + 4 
8*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d^3*e^6 - 48*(sqrt(b)*x - sqrt(b 
*x^2 + a))^4*b^(7/2)*c*d^2*e^5*f - 120*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b 
^(5/2)*d^3*e^5*f - 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^2*d*e^4*f^ 
2 + 168*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c*d^2*e^4*f^2 + 90*(sqrt 
(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d^3*e^4*f^2 + 48*(sqrt(b)*x - sq...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 7632, normalized size of antiderivative = 24.70 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*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**3*b*c**3*e**2*f**6 - 12*sq 
rt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - s 
qrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*b*c**3*e*f**7*x**2 - 6*sqrt(e)*s 
qrt(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**3*b*c**3*f**8*x**4 - 6*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**3*b*c**2*d*e**3*f**5 - 12*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)/(sqr 
t(e)*sqrt(b)))*a**3*b*c**2*d*e**2*f**6*x**2 - 6*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**3*b*c**2*d*e*f**7*x**4 - 18*sqrt(e)*sqrt(a*f - b*e)*atan(( 
sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*s 
qrt(b)))*a**3*b*c*d**2*e**4*f**4 - 36*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**3*b*c*d**2*e**3*f**5*x**2 - 18*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a* 
f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)) 
)*a**3*b*c*d**2*e**2*f**6*x**4 + 30*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**3*b*d**3*e**5*f**3 + 60*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*...