Optimal. Leaf size=80 \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2}}{3 b^4}-\frac{2 a^3 \sqrt{a+b x^3}}{3 b^4}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^4}-\frac{2 a \left (a+b x^3\right )^{5/2}}{5 b^4} \]
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Rubi [A] time = 0.0442464, antiderivative size = 80, 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} \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2}}{3 b^4}-\frac{2 a^3 \sqrt{a+b x^3}}{3 b^4}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^4}-\frac{2 a \left (a+b x^3\right )^{5/2}}{5 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 \sqrt{a+b x}}+\frac{3 a^2 \sqrt{a+b x}}{b^3}-\frac{3 a (a+b x)^{3/2}}{b^3}+\frac{(a+b x)^{5/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 a^3 \sqrt{a+b x^3}}{3 b^4}+\frac{2 a^2 \left (a+b x^3\right )^{3/2}}{3 b^4}-\frac{2 a \left (a+b x^3\right )^{5/2}}{5 b^4}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^4}\\ \end{align*}
Mathematica [A] time = 0.0276513, size = 50, normalized size = 0.62 \[ \frac{2 \sqrt{a+b x^3} \left (8 a^2 b x^3-16 a^3-6 a b^2 x^6+5 b^3 x^9\right )}{105 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-10\,{b}^{3}{x}^{9}+12\,a{b}^{2}{x}^{6}-16\,{a}^{2}b{x}^{3}+32\,{a}^{3}}{105\,{b}^{4}}\sqrt{b{x}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00705, size = 86, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{21 \, b^{4}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{5 \, b^{4}} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{3 \, b^{4}} - \frac{2 \, \sqrt{b x^{3} + a} a^{3}}{3 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45942, size = 103, normalized size = 1.29 \begin{align*} \frac{2 \,{\left (5 \, b^{3} x^{9} - 6 \, a b^{2} x^{6} + 8 \, a^{2} b x^{3} - 16 \, a^{3}\right )} \sqrt{b x^{3} + a}}{105 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.4574, size = 94, normalized size = 1.18 \begin{align*} \begin{cases} - \frac{32 a^{3} \sqrt{a + b x^{3}}}{105 b^{4}} + \frac{16 a^{2} x^{3} \sqrt{a + b x^{3}}}{105 b^{3}} - \frac{4 a x^{6} \sqrt{a + b x^{3}}}{35 b^{2}} + \frac{2 x^{9} \sqrt{a + b x^{3}}}{21 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 \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 = 1.12384, size = 77, normalized size = 0.96 \begin{align*} \frac{2 \,{\left (5 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 21 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b x^{3} + a} a^{3}\right )}}{105 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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