\(\int \frac {\sqrt {a+b x^2} (c+d x^2)^3}{e+f x^2} \, dx\) [276]

Optimal result

Integrand size = 30, antiderivative size = 312 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=-\frac {d \left (a^2 d^2 f^2+2 a b d f (d e-3 c f)-8 b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{16 b^2 f^3}-\frac {d^2 (6 b d e-18 b c f-a d f) x^3 \sqrt {a+b x^2}}{24 b f^2}+\frac {d^3 x^5 \sqrt {a+b x^2}}{6 f}+\frac {\left (a^3 d^3 f^3+2 a^2 b d^2 f^2 (d e-3 c f)-16 b^3 (d e-c f)^3+8 a b^2 d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{5/2} f^4}+\frac {\sqrt {b e-a f} (d e-c f)^3 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^4} \] Output:

-1/16*d*(a^2*d^2*f^2+2*a*b*d*f*(-3*c*f+d*e)-8*b^2*(3*c^2*f^2-3*c*d*e*f+d^2 
*e^2))*x*(b*x^2+a)^(1/2)/b^2/f^3-1/24*d^2*(-a*d*f-18*b*c*f+6*b*d*e)*x^3*(b 
*x^2+a)^(1/2)/b/f^2+1/6*d^3*x^5*(b*x^2+a)^(1/2)/f+1/16*(a^3*d^3*f^3+2*a^2* 
b*d^2*f^2*(-3*c*f+d*e)-16*b^3*(-c*f+d*e)^3+8*a*b^2*d*f*(3*c^2*f^2-3*c*d*e* 
f+d^2*e^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)/f^4+(-a*f+b*e)^(1/2 
)*(-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^4
 

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\frac {\frac {d f x \sqrt {a+b x^2} \left (-3 a^2 d^2 f^2+2 a b d f \left (-3 d e+9 c f+d f x^2\right )+4 b^2 \left (18 c^2 f^2+9 c d f \left (-2 e+f x^2\right )+d^2 \left (6 e^2-3 e f x^2+2 f^2 x^4\right )\right )\right )}{b^2}+\frac {48 \sqrt {-b e+a f} (d e-c f)^3 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e}}+\frac {3 \left (-a^3 d^3 f^3-2 a^2 b d^2 f^2 (d e-3 c f)+16 b^3 (d e-c f)^3-8 a b^2 d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{5/2}}}{48 f^4} \] Input:

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

Output:

((d*f*x*Sqrt[a + b*x^2]*(-3*a^2*d^2*f^2 + 2*a*b*d*f*(-3*d*e + 9*c*f + d*f* 
x^2) + 4*b^2*(18*c^2*f^2 + 9*c*d*f*(-2*e + f*x^2) + d^2*(6*e^2 - 3*e*f*x^2 
 + 2*f^2*x^4))))/b^2 + (48*Sqrt[-(b*e) + a*f]*(d*e - c*f)^3*ArcTan[(-(f*x* 
Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-(b*e) + a*f])])/Sqr 
t[e] + (3*(-(a^3*d^3*f^3) - 2*a^2*b*d^2*f^2*(d*e - 3*c*f) + 16*b^3*(d*e - 
c*f)^3 - 8*a*b^2*d*f*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*Log[-(Sqrt[b]*x) + 
 Sqrt[a + b*x^2]])/b^(5/2))/(48*f^4)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.27, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {420, 318, 299, 211, 224, 219, 420, 299, 211, 224, 219, 403, 25, 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.

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 318

\(\displaystyle \frac {d \left (\frac {\int \sqrt {b x^2+a} \left (d (8 b c-3 a d) x^2+c (6 b c-a d)\right )dx}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \int \sqrt {b x^2+a}dx}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\)

\(\Big \downarrow \) 420

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \sqrt {b x^2+a} \left (d x^2+c\right )dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(4 b c-a d) \int \sqrt {b x^2+a}dx}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(4 b c-a d) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(4 b c-a d) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {\int -\frac {(2 b d e-2 b c f-a d f) x^2+a (d e-2 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}+\frac {d x \sqrt {a+b x^2}}{2 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\int \frac {(2 b d e-2 b c f-a d f) x^2+a (d e-2 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {(-a d f-2 b c f+2 b d e) \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {(-a d f-2 b c f+2 b d e) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}}{2 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}\)

Input:

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

Output:

(d*((d*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(6*b) + ((d*(8*b*c - 3*a*d)*x*(a + 
 b*x^2)^(3/2))/(4*b) + (3*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*((x*Sqrt[a + b 
*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/(4*b))/( 
6*b)))/f - ((d*e - c*f)*((d*((d*x*(a + b*x^2)^(3/2))/(4*b) + ((4*b*c - a*d 
)*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqr 
t[b])))/(4*b)))/f - ((d*e - c*f)*((d*x*Sqrt[a + b*x^2])/(2*f) - (((2*b*d*e 
 - 2*b*c*f - a*d*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - (2 
*Sqrt[b*e - a*f]*(d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + 
 b*x^2])])/(Sqrt[e]*f))/(2*f)))/f))/f
 

Defintions of rubi rules used

rule 25

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

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 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 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

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 = 0.98 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {2 \left (-a f +b e \right ) \left (c f -d e \right )^{3} b^{\frac {9}{2}} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (\frac {\left (-16 b^{3} d^{3} e^{3}+8 b^{2} d^{2} f \left (a d +6 b c \right ) e^{2}+2 b d \,f^{2} \left (a^{2} d^{2}-12 a b c d -24 b^{2} c^{2}\right ) e +f^{3} \left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d +16 b^{3} c^{3}\right )\right ) b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{8}+\left (b^{2} d^{2} e^{2}-\frac {d b f \left (2 \left (x^{2} d +6 c \right ) b +a d \right ) e}{4}-\frac {\left (4 \left (-\frac {2}{3} d^{2} x^{4}-3 c d \,x^{2}-6 c^{2}\right ) b^{2}-6 a d \left (\frac {x^{2} d}{9}+c \right ) b +a^{2} d^{2}\right ) f^{2}}{8}\right ) d \sqrt {b \,x^{2}+a}\, x f \,b^{\frac {5}{2}}\right )}{2 \sqrt {\left (a f -b e \right ) e}\, b^{\frac {9}{2}} f^{4}}\)	\(311\)
risch	\(-\frac {x d \left (-8 f^{2} x^{4} b^{2} d^{2}-2 a b \,d^{2} f^{2} x^{2}-36 b^{2} c d \,f^{2} x^{2}+12 b^{2} d^{2} e f \,x^{2}+3 a^{2} d^{2} f^{2}-18 a b c d \,f^{2}+6 a b \,d^{2} e f -72 b^{2} c^{2} f^{2}+72 b^{2} c d e f -24 b^{2} d^{2} e^{2}\right ) \sqrt {b \,x^{2}+a}}{48 b^{2} f^{3}}+\frac {\frac {\left (a^{3} d^{3} f^{3}-6 a^{2} b c \,d^{2} f^{3}+2 a^{2} b \,d^{3} e \,f^{2}+24 a \,b^{2} c^{2} d \,f^{3}-24 a \,b^{2} c \,d^{2} e \,f^{2}+8 a \,b^{2} d^{3} e^{2} f +16 b^{3} c^{3} f^{3}-48 b^{3} c^{2} d e \,f^{2}+48 b^{3} c \,d^{2} e^{2} f -16 b^{3} d^{3} e^{3}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{f \sqrt {b}}-\frac {8 b^{2} \left (c^{3} a \,f^{4}-3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}-a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}-3 b c \,d^{2} e^{3} f +e^{4} b \,d^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}+\frac {8 b^{2} \left (c^{3} a \,f^{4}-3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}-a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}-3 b c \,d^{2} e^{3} f +e^{4} b \,d^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}}{16 b^{2} f^{3}}\)	\(772\)
default	\(\text {Expression too large to display}\)	\(1012\)

Input:

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

Output:

1/2*(2*(-a*f+b*e)*(c*f-d*e)^3*b^(9/2)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e 
)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(1/8*(-16*b^3*d^3*e^3+8*b^2*d^2*f*(a*d+6*b 
*c)*e^2+2*b*d*f^2*(a^2*d^2-12*a*b*c*d-24*b^2*c^2)*e+f^3*(a^3*d^3-6*a^2*b*c 
*d^2+24*a*b^2*c^2*d+16*b^3*c^3))*b^2*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+(b 
^2*d^2*e^2-1/4*d*b*f*(2*(d*x^2+6*c)*b+a*d)*e-1/8*(4*(-2/3*d^2*x^4-3*c*d*x^ 
2-6*c^2)*b^2-6*a*d*(1/9*x^2*d+c)*b+a^2*d^2)*f^2)*d*(b*x^2+a)^(1/2)*x*f*b^( 
5/2)))/((a*f-b*e)*e)^(1/2)/b^(9/2)/f^4
 

Fricas [A] (verification not implemented)

Time = 28.22 (sec) , antiderivative size = 1809, normalized size of antiderivative = 5.80 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/96*(3*(16*b^3*d^3*e^3 - 8*(6*b^3*c*d^2 + a*b^2*d^3)*e^2*f + 2*(24*b^3* 
c^2*d + 12*a*b^2*c*d^2 - a^2*b*d^3)*e*f^2 - (16*b^3*c^3 + 24*a*b^2*c^2*d - 
 6*a^2*b*c*d^2 + a^3*d^3)*f^3)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sq 
rt(b)*x - a) + 24*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b 
^3*c^3*f^3)*sqrt((b*e - a*f)/e)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 
 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 - 4*(a*e^2*x + (2*b*e^2 - a*e*f 
)*x^3)*sqrt(b*x^2 + a)*sqrt((b*e - a*f)/e))/(f^2*x^4 + 2*e*f*x^2 + e^2)) - 
 2*(8*b^3*d^3*f^3*x^5 - 2*(6*b^3*d^3*e*f^2 - (18*b^3*c*d^2 + a*b^2*d^3)*f^ 
3)*x^3 + 3*(8*b^3*d^3*e^2*f - 2*(12*b^3*c*d^2 + a*b^2*d^3)*e*f^2 + (24*b^3 
*c^2*d + 6*a*b^2*c*d^2 - a^2*b*d^3)*f^3)*x)*sqrt(b*x^2 + a))/(b^3*f^4), 1/ 
48*(3*(16*b^3*d^3*e^3 - 8*(6*b^3*c*d^2 + a*b^2*d^3)*e^2*f + 2*(24*b^3*c^2* 
d + 12*a*b^2*c*d^2 - a^2*b*d^3)*e*f^2 - (16*b^3*c^3 + 24*a*b^2*c^2*d - 6*a 
^2*b*c*d^2 + a^3*d^3)*f^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 1 
2*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*sqrt 
((b*e - a*f)/e)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*( 
4*a*b*e^2 - 3*a^2*e*f)*x^2 - 4*(a*e^2*x + (2*b*e^2 - a*e*f)*x^3)*sqrt(b*x^ 
2 + a)*sqrt((b*e - a*f)/e))/(f^2*x^4 + 2*e*f*x^2 + e^2)) + (8*b^3*d^3*f^3* 
x^5 - 2*(6*b^3*d^3*e*f^2 - (18*b^3*c*d^2 + a*b^2*d^3)*f^3)*x^3 + 3*(8*b^3* 
d^3*e^2*f - 2*(12*b^3*c*d^2 + a*b^2*d^3)*e*f^2 + (24*b^3*c^2*d + 6*a*b^2*c 
*d^2 - a^2*b*d^3)*f^3)*x)*sqrt(b*x^2 + a))/(b^3*f^4), -1/96*(48*(b^3*d^...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3/(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. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3/(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. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^3}{f\,x^2+e} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 1077, normalized size of antiderivative = 3.45 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^3}{e+f x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 48*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*f**3 + 144*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**2*d*e*f**2 - 144*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*d**2*e**2*f + 48*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*d**3*e**3 - 48*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*f**3 + 144*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*s 
qrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**3*c**2*d*e*f**2 
 - 144*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*d**2*e**2*f + 48*sqrt 
(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqr 
t(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**3*d**3*e**3 - 3*sqrt(a + b*x**2)*a** 
2*b*d**3*e*f**3*x + 18*sqrt(a + b*x**2)*a*b**2*c*d**2*e*f**3*x - 6*sqrt(a 
+ b*x**2)*a*b**2*d**3*e**2*f**2*x + 2*sqrt(a + b*x**2)*a*b**2*d**3*e*f**3* 
x**3 + 72*sqrt(a + b*x**2)*b**3*c**2*d*e*f**3*x - 72*sqrt(a + b*x**2)*b**3 
*c*d**2*e**2*f**2*x + 36*sqrt(a + b*x**2)*b**3*c*d**2*e*f**3*x**3 + 24*sqr 
t(a + b*x**2)*b**3*d**3*e**3*f*x - 12*sqrt(a + b*x**2)*b**3*d**3*e**2*f...