Optimal. Leaf size=172 \[ \frac{x \left (c+d x^2\right ) (b c-a d) (5 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}-\frac{d x \sqrt{a+b x^2} \left (-15 a^2 d^2+8 a b c d+4 b^2 c^2\right )}{6 a^2 b^3}+\frac{d^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{7/2}}+\frac{x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.368917, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \left (c+d x^2\right ) (b c-a d) (5 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}-\frac{d x \sqrt{a+b x^2} \left (-15 a^2 d^2+8 a b c d+4 b^2 c^2\right )}{6 a^2 b^3}+\frac{d^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{7/2}}+\frac{x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^3/(a + b*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 50.9247, size = 162, normalized size = 0.94 \[ - \frac{d^{2} \left (5 a d - 6 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{7}{2}}} - \frac{x \left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{3 a b \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{x \left (c + d x^{2}\right ) \left (a d - b c\right ) \left (5 a d + 2 b c\right )}{3 a^{2} b^{2} \sqrt{a + b x^{2}}} + \frac{d x \sqrt{a + b x^{2}} \left (15 a^{2} d^{2} - 8 a b c d - 4 b^{2} c^{2}\right )}{6 a^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**3/(b*x**2+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.156789, size = 125, normalized size = 0.73 \[ \frac{x \left (3 a^2 d^3 \left (a+b x^2\right )^2+2 \left (a+b x^2\right ) (b c-a d)^2 (7 a d+2 b c)+2 a (b c-a d)^3\right )}{6 a^2 b^3 \left (a+b x^2\right )^{3/2}}+\frac{d^2 (6 b c-5 a d) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^3/(a + b*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.012, size = 228, normalized size = 1.3 \[{\frac{{c}^{3}x}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,{c}^{3}x}{3\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{{d}^{3}{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,a{d}^{3}{x}^{3}}{6\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,a{d}^{3}x}{2\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,a{d}^{3}}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}-{\frac{c{d}^{2}{x}^{3}}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-3\,{\frac{c{d}^{2}x}{{b}^{2}\sqrt{b{x}^{2}+a}}}+3\,{\frac{c{d}^{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{5/2}}}-{\frac{{c}^{2}dx}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{{c}^{2}dx}{ab}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^3/(b*x^2+a)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/(b*x^2 + a)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.302988, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \, a^{2} b^{2} d^{3} x^{5} + 2 \,{\left (2 \, b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 10 \, a^{3} b d^{3}\right )} x^{3} + 3 \,{\left (2 \, a b^{3} c^{3} - 6 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (6 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} +{\left (6 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \,{\left (6 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{12 \,{\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )} \sqrt{b}}, \frac{{\left (3 \, a^{2} b^{2} d^{3} x^{5} + 2 \,{\left (2 \, b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 10 \, a^{3} b d^{3}\right )} x^{3} + 3 \,{\left (2 \, a b^{3} c^{3} - 6 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (6 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} +{\left (6 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \,{\left (6 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{6 \,{\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )} \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/(b*x^2 + a)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**3/(b*x**2+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.233084, size = 213, normalized size = 1.24 \[ \frac{{\left ({\left (\frac{3 \, d^{3} x^{2}}{b} + \frac{2 \,{\left (2 \, b^{6} c^{3} + 3 \, a b^{5} c^{2} d - 12 \, a^{2} b^{4} c d^{2} + 10 \, a^{3} b^{3} d^{3}\right )}}{a^{2} b^{5}}\right )} x^{2} + \frac{3 \,{\left (2 \, a b^{5} c^{3} - 6 \, a^{3} b^{3} c d^{2} + 5 \, a^{4} b^{2} d^{3}\right )}}{a^{2} b^{5}}\right )} x}{6 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{{\left (6 \, b c d^{2} - 5 \, a d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/(b*x^2 + a)^(5/2),x, algorithm="giac")
[Out]