Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b x^2\right )^{7/3}}{14 b^3}+\frac{3 \left (a+b x^2\right )^{13/3}}{26 b^3}-\frac{3 a \left (a+b x^2\right )^{10/3}}{10 b^3} \]
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Rubi [A] time = 0.0377544, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{3 a^2 \left (a+b x^2\right )^{7/3}}{14 b^3}+\frac{3 \left (a+b x^2\right )^{13/3}}{26 b^3}-\frac{3 a \left (a+b x^2\right )^{10/3}}{10 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b x^2\right )^{4/3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^{4/3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{4/3}}{b^2}-\frac{2 a (a+b x)^{7/3}}{b^2}+\frac{(a+b x)^{10/3}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a^2 \left (a+b x^2\right )^{7/3}}{14 b^3}-\frac{3 a \left (a+b x^2\right )^{10/3}}{10 b^3}+\frac{3 \left (a+b x^2\right )^{13/3}}{26 b^3}\\ \end{align*}
Mathematica [A] time = 0.0184864, size = 39, normalized size = 0.66 \[ \frac{3 \left (a+b x^2\right )^{7/3} \left (9 a^2-21 a b x^2+35 b^2 x^4\right )}{910 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.6 \begin{align*}{\frac{105\,{b}^{2}{x}^{4}-63\,ab{x}^{2}+27\,{a}^{2}}{910\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.96662, size = 63, normalized size = 1.07 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}}}{26 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a}{10 \, b^{3}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{2}}{14 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67103, size = 128, normalized size = 2.17 \begin{align*} \frac{3 \,{\left (35 \, b^{4} x^{8} + 49 \, a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{4} - 3 \, a^{3} b x^{2} + 9 \, a^{4}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{910 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.7216, size = 112, normalized size = 1.9 \begin{align*} \begin{cases} \frac{27 a^{4} \sqrt [3]{a + b x^{2}}}{910 b^{3}} - \frac{9 a^{3} x^{2} \sqrt [3]{a + b x^{2}}}{910 b^{2}} + \frac{3 a^{2} x^{4} \sqrt [3]{a + b x^{2}}}{455 b} + \frac{21 a x^{6} \sqrt [3]{a + b x^{2}}}{130} + \frac{3 b x^{8} \sqrt [3]{a + b x^{2}}}{26} & \text{for}\: b \neq 0 \\\frac{a^{\frac{4}{3}} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.73441, size = 143, normalized size = 2.42 \begin{align*} \frac{3 \,{\left (\frac{13 \,{\left (14 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2}\right )} a}{b^{2}} + \frac{140 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{3}}{b^{2}}\right )}}{3640 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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