Optimal. Leaf size=167 \[ \frac{b^3 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac{a b^2 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{a^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0432692, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1112, 270} \[ \frac{b^3 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac{a b^2 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{a^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 270
Rubi steps
\begin{align*} \int x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int x^4 \left (a b+b^2 x^2\right )^3 \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (a^3 b^3 x^4+3 a^2 b^4 x^6+3 a b^5 x^8+b^6 x^{10}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{a^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{a b^2 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{b^3 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0154241, size = 61, normalized size = 0.37 \[ \frac{x^5 \sqrt{\left (a+b x^2\right )^2} \left (495 a^2 b x^2+231 a^3+385 a b^2 x^4+105 b^3 x^6\right )}{1155 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.164, size = 58, normalized size = 0.4 \begin{align*}{\frac{{x}^{5} \left ( 105\,{b}^{3}{x}^{6}+385\,a{x}^{4}{b}^{2}+495\,{a}^{2}b{x}^{2}+231\,{a}^{3} \right ) }{1155\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00699, size = 47, normalized size = 0.28 \begin{align*} \frac{1}{11} \, b^{3} x^{11} + \frac{1}{3} \, a b^{2} x^{9} + \frac{3}{7} \, a^{2} b x^{7} + \frac{1}{5} \, a^{3} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50835, size = 82, normalized size = 0.49 \begin{align*} \frac{1}{11} \, b^{3} x^{11} + \frac{1}{3} \, a b^{2} x^{9} + \frac{3}{7} \, a^{2} b x^{7} + \frac{1}{5} \, a^{3} x^{5} \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^{4} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12588, size = 90, normalized size = 0.54 \begin{align*} \frac{1}{11} \, b^{3} x^{11} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{3} \, a b^{2} x^{9} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{3}{7} \, a^{2} b x^{7} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{5} \, a^{3} x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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