Optimal. Leaf size=59 \[ \frac{2 a^2 \sqrt{a+b x^3}}{3 b^3}+\frac{2 \left (a+b x^3\right )^{5/2}}{15 b^3}-\frac{4 a \left (a+b x^3\right )^{3/2}}{9 b^3} \]
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Rubi [A] time = 0.0338344, 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} \[ \frac{2 a^2 \sqrt{a+b x^3}}{3 b^3}+\frac{2 \left (a+b x^3\right )^{5/2}}{15 b^3}-\frac{4 a \left (a+b x^3\right )^{3/2}}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt{a+b x}}-\frac{2 a \sqrt{a+b x}}{b^2}+\frac{(a+b x)^{3/2}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{2 a^2 \sqrt{a+b x^3}}{3 b^3}-\frac{4 a \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac{2 \left (a+b x^3\right )^{5/2}}{15 b^3}\\ \end{align*}
Mathematica [A] time = 0.0183546, size = 39, normalized size = 0.66 \[ \frac{2 \sqrt{a+b x^3} \left (8 a^2-4 a b x^3+3 b^2 x^6\right )}{45 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 36, normalized size = 0.6 \begin{align*}{\frac{6\,{b}^{2}{x}^{6}-8\,{x}^{3}ab+16\,{a}^{2}}{45\,{b}^{3}}\sqrt{b{x}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963801, size = 63, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{15 \, b^{3}} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{9 \, b^{3}} + \frac{2 \, \sqrt{b x^{3} + a} a^{2}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43677, size = 78, normalized size = 1.32 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{6} - 4 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{b x^{3} + a}}{45 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.56726, size = 70, normalized size = 1.19 \begin{align*} \begin{cases} \frac{16 a^{2} \sqrt{a + b x^{3}}}{45 b^{3}} - \frac{8 a x^{3} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 x^{6} \sqrt{a + b x^{3}}}{15 b} & \text{for}\: b \neq 0 \\\frac{x^{9}}{9 \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 = 1.14184, size = 58, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x^{3} + a} a^{2}\right )}}{45 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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