Optimal. Leaf size=38 \[ \frac{2 a}{3 b^2 \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{3 b^2} \]
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Rubi [A] time = 0.0225487, antiderivative size = 38, 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} \[ \frac{2 a}{3 b^2 \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{3 b^2} \]
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\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{3/2}}+\frac{1}{b \sqrt{a+b x}}\right ) \, dx,x,x^3\right )\\ &=\frac{2 a}{3 b^2 \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.011241, size = 27, normalized size = 0.71 \[ \frac{2 \left (2 a+b x^3\right )}{3 b^2 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 24, normalized size = 0.6 \begin{align*}{\frac{2\,b{x}^{3}+4\,a}{3\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957168, size = 41, normalized size = 1.08 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a}}{3 \, b^{2}} + \frac{2 \, a}{3 \, \sqrt{b x^{3} + a} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40859, size = 72, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (b x^{3} + 2 \, a\right )} \sqrt{b x^{3} + a}}{3 \,{\left (b^{3} x^{3} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.11768, size = 46, normalized size = 1.21 \begin{align*} \begin{cases} \frac{4 a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 x^{3}}{3 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12257, size = 35, normalized size = 0.92 \begin{align*} \frac{2 \,{\left (\sqrt{b x^{3} + a} + \frac{a}{\sqrt{b x^{3} + a}}\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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