Optimal. Leaf size=78 \[ \frac{2 a^3}{3 b^4 \sqrt{a+b x^3}}+\frac{2 a^2 \sqrt{a+b x^3}}{b^4}-\frac{2 a \left (a+b x^3\right )^{3/2}}{3 b^4}+\frac{2 \left (a+b x^3\right )^{5/2}}{15 b^4} \]
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Rubi [A] time = 0.0437219, antiderivative size = 78, 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} \[ \frac{2 a^3}{3 b^4 \sqrt{a+b x^3}}+\frac{2 a^2 \sqrt{a+b x^3}}{b^4}-\frac{2 a \left (a+b x^3\right )^{3/2}}{3 b^4}+\frac{2 \left (a+b x^3\right )^{5/2}}{15 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{3/2}}+\frac{3 a^2}{b^3 \sqrt{a+b x}}-\frac{3 a \sqrt{a+b x}}{b^3}+\frac{(a+b x)^{3/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=\frac{2 a^3}{3 b^4 \sqrt{a+b x^3}}+\frac{2 a^2 \sqrt{a+b x^3}}{b^4}-\frac{2 a \left (a+b x^3\right )^{3/2}}{3 b^4}+\frac{2 \left (a+b x^3\right )^{5/2}}{15 b^4}\\ \end{align*}
Mathematica [A] time = 0.0222087, size = 49, normalized size = 0.63 \[ \frac{2 \left (8 a^2 b x^3+16 a^3-2 a b^2 x^6+b^3 x^9\right )}{15 b^4 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 46, normalized size = 0.6 \begin{align*}{\frac{2\,{b}^{3}{x}^{9}-4\,a{b}^{2}{x}^{6}+16\,{a}^{2}b{x}^{3}+32\,{a}^{3}}{15\,{b}^{4}}{\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 = 0.953148, size = 86, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{15 \, b^{4}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{3 \, b^{4}} + \frac{2 \, \sqrt{b x^{3} + a} a^{2}}{b^{4}} + \frac{2 \, a^{3}}{3 \, \sqrt{b x^{3} + a} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47078, size = 117, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (b^{3} x^{9} - 2 \, a b^{2} x^{6} + 8 \, a^{2} b x^{3} + 16 \, a^{3}\right )} \sqrt{b x^{3} + a}}{15 \,{\left (b^{5} x^{3} + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.85958, size = 94, normalized size = 1.21 \begin{align*} \begin{cases} \frac{32 a^{3}}{15 b^{4} \sqrt{a + b x^{3}}} + \frac{16 a^{2} x^{3}}{15 b^{3} \sqrt{a + b x^{3}}} - \frac{4 a x^{6}}{15 b^{2} \sqrt{a + b x^{3}}} + \frac{2 x^{9}}{15 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 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 = 1.11318, size = 74, normalized size = 0.95 \begin{align*} \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x^{3} + a} a^{2} + \frac{5 \, a^{3}}{\sqrt{b x^{3} + a}}\right )}}{15 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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