Optimal. Leaf size=59 \[ -\frac{2 a^2}{3 b^3 \sqrt{a+b x^3}}-\frac{4 a \sqrt{a+b x^3}}{3 b^3}+\frac{2 \left (a+b x^3\right )^{3/2}}{9 b^3} \]
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Rubi [A] time = 0.0342384, 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} \[ -\frac{2 a^2}{3 b^3 \sqrt{a+b x^3}}-\frac{4 a \sqrt{a+b x^3}}{3 b^3}+\frac{2 \left (a+b x^3\right )^{3/2}}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/2}}-\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 a^2}{3 b^3 \sqrt{a+b x^3}}-\frac{4 a \sqrt{a+b x^3}}{3 b^3}+\frac{2 \left (a+b x^3\right )^{3/2}}{9 b^3}\\ \end{align*}
Mathematica [A] time = 0.0170295, size = 38, normalized size = 0.64 \[ \frac{2 \left (-8 a^2-4 a b x^3+b^2 x^6\right )}{9 b^3 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, normalized size = 0.6 \begin{align*} -{\frac{-2\,{b}^{2}{x}^{6}+8\,{x}^{3}ab+16\,{a}^{2}}{9\,{b}^{3}}{\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 = 0.970605, size = 63, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b^{3}} - \frac{4 \, \sqrt{b x^{3} + a} a}{3 \, b^{3}} - \frac{2 \, a^{2}}{3 \, \sqrt{b x^{3} + a} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42227, size = 93, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (b^{2} x^{6} - 4 \, a b x^{3} - 8 \, a^{2}\right )} \sqrt{b x^{3} + a}}{9 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.22148, size = 70, normalized size = 1.19 \begin{align*} \begin{cases} - \frac{16 a^{2}}{9 b^{3} \sqrt{a + b x^{3}}} - \frac{8 a x^{3}}{9 b^{2} \sqrt{a + b x^{3}}} + \frac{2 x^{6}}{9 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{x^{9}}{9 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 = 1.13177, size = 55, normalized size = 0.93 \begin{align*} \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{b x^{3} + a} a - \frac{3 \, a^{2}}{\sqrt{b x^{3} + a}}\right )}}{9 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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