Optimal. Leaf size=86 \[ \frac{3 a^2 \left (a+b x^{3/2}\right )^{4/3}}{2 b^4}-\frac{2 a^3 \sqrt [3]{a+b x^{3/2}}}{b^4}+\frac{\left (a+b x^{3/2}\right )^{10/3}}{5 b^4}-\frac{6 a \left (a+b x^{3/2}\right )^{7/3}}{7 b^4} \]
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Rubi [A] time = 0.0408598, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac{3 a^2 \left (a+b x^{3/2}\right )^{4/3}}{2 b^4}-\frac{2 a^3 \sqrt [3]{a+b x^{3/2}}}{b^4}+\frac{\left (a+b x^{3/2}\right )^{10/3}}{5 b^4}-\frac{6 a \left (a+b x^{3/2}\right )^{7/3}}{7 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{2/3}} \, dx,x,x^{3/2}\right )\\ &=\frac{2}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{2/3}}+\frac{3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac{3 a (a+b x)^{4/3}}{b^3}+\frac{(a+b x)^{7/3}}{b^3}\right ) \, dx,x,x^{3/2}\right )\\ &=-\frac{2 a^3 \sqrt [3]{a+b x^{3/2}}}{b^4}+\frac{3 a^2 \left (a+b x^{3/2}\right )^{4/3}}{2 b^4}-\frac{6 a \left (a+b x^{3/2}\right )^{7/3}}{7 b^4}+\frac{\left (a+b x^{3/2}\right )^{10/3}}{5 b^4}\\ \end{align*}
Mathematica [A] time = 0.0238537, size = 56, normalized size = 0.65 \[ \frac{\sqrt [3]{a+b x^{3/2}} \left (27 a^2 b x^{3/2}-81 a^3-18 a b^2 x^3+14 b^3 x^{9/2}\right )}{70 b^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{{x}^{5} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961197, size = 86, normalized size = 1. \begin{align*} \frac{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{10}{3}}}{5 \, b^{4}} - \frac{6 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{7}{3}} a}{7 \, b^{4}} + \frac{3 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{4}{3}} a^{2}}{2 \, b^{4}} - \frac{2 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}} a^{3}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.70544, size = 126, normalized size = 1.47 \begin{align*} -\frac{{\left (18 \, a b^{2} x^{3} + 81 \, a^{3} -{\left (14 \, b^{3} x^{4} + 27 \, a^{2} b x\right )} \sqrt{x}\right )}{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}}}{70 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 42.394, size = 102, normalized size = 1.19 \begin{align*} \begin{cases} - \frac{81 a^{3} \sqrt [3]{a + b x^{\frac{3}{2}}}}{70 b^{4}} + \frac{27 a^{2} x^{\frac{3}{2}} \sqrt [3]{a + b x^{\frac{3}{2}}}}{70 b^{3}} - \frac{9 a x^{3} \sqrt [3]{a + b x^{\frac{3}{2}}}}{35 b^{2}} + \frac{x^{\frac{9}{2}} \sqrt [3]{a + b x^{\frac{3}{2}}}}{5 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10293, size = 77, normalized size = 0.9 \begin{align*} \frac{14 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{10}{3}} - 60 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{7}{3}} a + 105 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{4}{3}} a^{2} - 140 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}} a^{3}}{70 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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