Optimal. Leaf size=36 \[ \frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b} \]
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Rubi [A] time = 0.0258815, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1107, 609} \[ \frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b} \]
Antiderivative was successfully verified.
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Rule 1107
Rule 609
Rubi steps
\begin{align*} \int x \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^2\right )\\ &=\frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b}\\ \end{align*}
Mathematica [A] time = 0.0138417, size = 27, normalized size = 0.75 \[ \frac{\left (a+b x^2\right ) \left (\left (a+b x^2\right )^2\right )^{5/2}}{12 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.166, size = 79, normalized size = 2.2 \begin{align*}{\frac{{x}^{2} \left ({b}^{5}{x}^{10}+6\,a{b}^{4}{x}^{8}+15\,{a}^{2}{b}^{3}{x}^{6}+20\,{a}^{3}{b}^{2}{x}^{4}+15\,{a}^{4}b{x}^{2}+6\,{a}^{5} \right ) }{12\, \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 = 1.45982, size = 132, normalized size = 3.67 \begin{align*} \frac{1}{12} \, b^{5} x^{12} + \frac{1}{2} \, a b^{4} x^{10} + \frac{5}{4} \, a^{2} b^{3} x^{8} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{5}{4} \, a^{4} b x^{4} + \frac{1}{2} \, a^{5} x^{2} \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 \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 [B] time = 1.17384, size = 89, normalized size = 2.47 \begin{align*} \frac{1}{12} \,{\left (b^{5} x^{12} + 6 \, a b^{4} x^{10} + 15 \, a^{2} b^{3} x^{8} + 20 \, a^{3} b^{2} x^{6} + 15 \, a^{4} b x^{4} + 6 \, a^{5} x^{2}\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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