Optimal. Leaf size=38 \[ \frac{2 \left (a+b x^3\right )^{3/2}}{9 b^2}-\frac{2 a \sqrt{a+b x^3}}{3 b^2} \]
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Rubi [A] time = 0.0229955, 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} \[ \frac{2 \left (a+b x^3\right )^{3/2}}{9 b^2}-\frac{2 a \sqrt{a+b x^3}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 a \sqrt{a+b x^3}}{3 b^2}+\frac{2 \left (a+b x^3\right )^{3/2}}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.0132131, size = 27, normalized size = 0.71 \[ \frac{2 \left (b x^3-2 a\right ) \sqrt{a+b x^3}}{9 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-2\,b{x}^{3}+4\,a}{9\,{b}^{2}}\sqrt{b{x}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955715, size = 41, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b^{2}} - \frac{2 \, \sqrt{b x^{3} + a} a}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47303, size = 53, normalized size = 1.39 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a}{\left (b x^{3} - 2 \, a\right )}}{9 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.991634, size = 46, normalized size = 1.21 \begin{align*} \begin{cases} - \frac{4 a \sqrt{a + b x^{3}}}{9 b^{2}} + \frac{2 x^{3} \sqrt{a + b x^{3}}}{9 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14388, size = 36, normalized size = 0.95 \begin{align*} \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{3} + a} a\right )}}{9 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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