\(\int \frac {(c+d x^2)^3}{(a+b x^2)^{5/2} (e+f x^2)} \, dx\) [373]

Optimal result

Integrand size = 30, antiderivative size = 206 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d)^3 x}{3 a b^2 (b e-a f) \left (a+b x^2\right )^{3/2}}+\frac {(b c-a d)^2 \left (2 b^2 c e-4 a^2 d f+a b (7 d e-5 c f)\right ) x}{3 a^2 b^2 (b e-a f)^2 \sqrt {a+b x^2}}+\frac {d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2} f}-\frac {(d e-c f)^3 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f (b e-a f)^{5/2}} \] Output:

1/3*(-a*d+b*c)^3*x/a/b^2/(-a*f+b*e)/(b*x^2+a)^(3/2)+1/3*(-a*d+b*c)^2*(2*b^ 
2*c*e-4*a^2*d*f+a*b*(-5*c*f+7*d*e))*x/a^2/b^2/(-a*f+b*e)^2/(b*x^2+a)^(1/2) 
+d^3*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)/f-(-c*f+d*e)^3*arctanh((-a 
*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(1/2)/f/(-a*f+b*e)^(5/2)
 

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.07 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d)^2 x \left (-3 a^3 d f+2 b^3 c e x^2+a b^2 \left (3 c e+7 d e x^2-5 c f x^2\right )+2 a^2 b \left (3 d e-3 c f-2 d f x^2\right )\right )}{3 a^2 b^2 (b e-a f)^2 \left (a+b x^2\right )^{3/2}}+\frac {(d e-c f)^3 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} f (-b e+a f)^{5/2}}-\frac {d^3 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{5/2} f} \] Input:

Integrate[(c + d*x^2)^3/((a + b*x^2)^(5/2)*(e + f*x^2)),x]
 

Output:

((b*c - a*d)^2*x*(-3*a^3*d*f + 2*b^3*c*e*x^2 + a*b^2*(3*c*e + 7*d*e*x^2 - 
5*c*f*x^2) + 2*a^2*b*(3*d*e - 3*c*f - 2*d*f*x^2)))/(3*a^2*b^2*(b*e - a*f)^ 
2*(a + b*x^2)^(3/2)) + ((d*e - c*f)^3*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqr 
t[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/(Sqrt[e]*f*(-(b*e) + a*f) 
^(5/2)) - (d^3*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(b^(5/2)*f)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(466\) vs. \(2(206)=412\).

Time = 0.89 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.26, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.633, Rules used = {419, 25, 401, 25, 27, 401, 27, 299, 224, 219, 420, 299, 224, 219, 398, 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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 419

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

\(\displaystyle \frac {\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {b \left (d x^2+c\right ) \left (c \left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right )-d \left (2 d f a^2-b (5 d e-c f) a+2 b^2 c e\right ) x^2\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a b}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {b \left (d x^2+c\right ) \left (c \left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right )-d \left (2 d f a^2-b (5 d e-c f) a+2 b^2 c e\right ) x^2\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a b}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (d x^2+c\right ) \left (c \left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right )-d \left (2 d f a^2-b (5 d e-c f) a+2 b^2 c e\right ) x^2\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {\int \frac {d \left (\left (6 d^2 f a^3-b d (15 d e-7 c f) a^2+2 b^2 c (4 d e-5 c f) a+4 b^3 c^2 e\right ) x^2+a c \left (2 d f a^2-b (5 d e-c f) a+2 b^2 c e\right )\right )}{\sqrt {b x^2+a}}dx}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \int \frac {\left (6 d^2 f a^3-b d (15 d e-7 c f) a^2+2 b^2 c (4 d e-5 c f) a+4 b^3 c^2 e\right ) x^2+a c \left (2 d f a^2-b (5 d e-c f) a+2 b^2 c e\right )}{\sqrt {b x^2+a}}dx}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\left (d x^2+c\right )^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 420

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \int \frac {d x^2+c}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {(2 b c-a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {x \left (c+d x^2\right ) (b c-a d) \left (-2 a^2 d f+a b (5 d e-5 c f)+2 b^2 c e\right )}{a b \sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (15 d e-7 c f)+2 a b^2 c (4 d e-5 c f)+4 b^3 c^2 e\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a^2 d^2 f-a b d (5 d e-c f)+b^2 \left (6 c d e-4 c^2 f\right )\right )}{2 b^{3/2}}\right )}{a b}}{3 a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d)}{2 b^{3/2}}+\frac {d x \sqrt {a+b x^2}}{2 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{f}\right )}{(b e-a f)^2}\)

Input:

Int[(c + d*x^2)^3/((a + b*x^2)^(5/2)*(e + f*x^2)),x]
 

Output:

(((b*c - a*d)*(b*e - a*f)*x*(c + d*x^2)^2)/(3*a*(a + b*x^2)^(3/2)) + (((b* 
c - a*d)*(2*b^2*c*e - 2*a^2*d*f + a*b*(5*d*e - 5*c*f))*x*(c + d*x^2))/(a*b 
*Sqrt[a + b*x^2]) - (d*(((4*b^3*c^2*e + 6*a^3*d^2*f - a^2*b*d*(15*d*e - 7* 
c*f) + 2*a*b^2*c*(4*d*e - 5*c*f))*x*Sqrt[a + b*x^2])/(2*b) - (3*a^2*(2*a^2 
*d^2*f - a*b*d*(5*d*e - c*f) + b^2*(6*c*d*e - 4*c^2*f))*ArcTanh[(Sqrt[b]*x 
)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/(a*b))/(3*a))/(b*e - a*f)^2 - (f*(d*e - 
c*f)*((d*((d*x*Sqrt[a + b*x^2])/(2*b) + ((2*b*c - a*d)*ArcTanh[(Sqrt[b]*x) 
/Sqrt[a + b*x^2]])/(2*b^(3/2))))/f - ((d*e - c*f)*((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))/(b*e - a*f)^2
 

Defintions of rubi rules used

rule 25

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

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 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 419

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[b*((b*e - a*f)/(b*c - a*d)^2)   Int[(c + d*x^2)^( 
q + 2)*((e + f*x^2)^(r - 1)/(a + b*x^2)), x], x] - Simp[1/(b*c - a*d)^2   I 
nt[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*( 
b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && Gt 
Q[r, 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]
 

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(\frac {-a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{\frac {9}{2}} \left (c f -d e \right )^{3} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} d^{3} \left (a f -b e \right )^{2} b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\left (a d -b c \right )^{2} b^{\frac {5}{2}} \left (a^{3} d f +2 \left (\left (\frac {2 f \,x^{2}}{3}-e \right ) d +c f \right ) b \,a^{2}-\left (\frac {7 d e \,x^{2}}{3}+\left (-\frac {5 f \,x^{2}}{3}+e \right ) c \right ) b^{2} a -\frac {2 b^{3} c e \,x^{2}}{3}\right ) x f \right )}{\sqrt {\left (a f -b e \right ) e}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{\frac {9}{2}} f \left (a f -b e \right )^{2} a^{2}}\)	\(237\)
default	\(\text {Expression too large to display}\)	\(1706\)

Input:

int((d*x^2+c)^3/(b*x^2+a)^(5/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 962 vs. \(2 (182) = 364\).

Time = 17.38 (sec) , antiderivative size = 3941, normalized size of antiderivative = 19.13 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+c)^3/(b*x^2+a)^(5/2)/(f*x^2+e),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)**3/(b*x**2+a)**(5/2)/(f*x**2+e),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x^2+c)^3/(b*x^2+a)^(5/2)/(f*x^2+e),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*x^2+c)^3/(b*x^2+a)^(5/2)/(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. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^{5/2}\,\left (f\,x^2+e\right )} \,d x \] Input:

int((c + d*x^2)^3/((a + b*x^2)^(5/2)*(e + f*x^2)),x)
 

Output:

int((c + d*x^2)^3/((a + b*x^2)^(5/2)*(e + f*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 3361, normalized size of antiderivative = 16.32 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^3/(b*x^2+a)^(5/2)/(f*x^2+e),x)
 

Output:

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