Optimal. Leaf size=67 \[ \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{14 b^2}-\frac{a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b^2} \]
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Rubi [A] time = 0.0516458, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1111, 640, 609} \[ \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{14 b^2}-\frac{a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b^2} \]
Antiderivative was successfully verified.
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Rule 1111
Rule 640
Rule 609
Rubi steps
\begin{align*} \int x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^2\right )\\ &=\frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{14 b^2}-\frac{a \operatorname{Subst}\left (\int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^2\right )}{2 b}\\ &=-\frac{a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b^2}+\frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{14 b^2}\\ \end{align*}
Mathematica [A] time = 0.021543, size = 83, normalized size = 1.24 \[ \frac{x^4 \sqrt{\left (a+b x^2\right )^2} \left (84 a^2 b^3 x^6+105 a^3 b^2 x^4+70 a^4 b x^2+21 a^5+35 a b^4 x^8+6 b^5 x^{10}\right )}{84 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.165, size = 80, normalized size = 1.2 \begin{align*}{\frac{{x}^{4} \left ( 6\,{b}^{5}{x}^{10}+35\,a{b}^{4}{x}^{8}+84\,{a}^{2}{b}^{3}{x}^{6}+105\,{b}^{2}{a}^{3}{x}^{4}+70\,{a}^{4}b{x}^{2}+21\,{a}^{5} \right ) }{84\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43888, size = 130, normalized size = 1.94 \begin{align*} \frac{1}{14} \, b^{5} x^{14} + \frac{5}{12} \, a b^{4} x^{12} + a^{2} b^{3} x^{10} + \frac{5}{4} \, a^{3} b^{2} x^{8} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{4} \, a^{5} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17274, size = 90, normalized size = 1.34 \begin{align*} \frac{1}{84} \,{\left (6 \, b^{5} x^{14} + 35 \, a b^{4} x^{12} + 84 \, a^{2} b^{3} x^{10} + 105 \, a^{3} b^{2} x^{8} + 70 \, a^{4} b x^{6} + 21 \, a^{5} x^{4}\right )} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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