Optimal. Leaf size=80 \[ \frac{2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}-\frac{2 a^3 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac{2 \left (a+b x^3\right )^{9/2}}{27 b^4}-\frac{2 a \left (a+b x^3\right )^{7/2}}{7 b^4} \]
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Rubi [A] time = 0.0447942, 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} \[ \frac{2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}-\frac{2 a^3 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac{2 \left (a+b x^3\right )^{9/2}}{27 b^4}-\frac{2 a \left (a+b x^3\right )^{7/2}}{7 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{11} \sqrt{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 \sqrt{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3 \sqrt{a+b x}}{b^3}+\frac{3 a^2 (a+b x)^{3/2}}{b^3}-\frac{3 a (a+b x)^{5/2}}{b^3}+\frac{(a+b x)^{7/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 a^3 \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac{2 a^2 \left (a+b x^3\right )^{5/2}}{5 b^4}-\frac{2 a \left (a+b x^3\right )^{7/2}}{7 b^4}+\frac{2 \left (a+b x^3\right )^{9/2}}{27 b^4}\\ \end{align*}
Mathematica [A] time = 0.0249678, size = 50, normalized size = 0.62 \[ \frac{2 \left (a+b x^3\right )^{3/2} \left (24 a^2 b x^3-16 a^3-30 a b^2 x^6+35 b^3 x^9\right )}{945 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-70\,{b}^{3}{x}^{9}+60\,a{b}^{2}{x}^{6}-48\,{a}^{2}b{x}^{3}+32\,{a}^{3}}{945\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01943, size = 86, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}}}{27 \, b^{4}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a}{7 \, b^{4}} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{4}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}}{9 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.498, size = 127, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (35 \, b^{4} x^{12} + 5 \, a b^{3} x^{9} - 6 \, a^{2} b^{2} x^{6} + 8 \, a^{3} b x^{3} - 16 \, a^{4}\right )} \sqrt{b x^{3} + a}}{945 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.74896, size = 114, normalized size = 1.42 \begin{align*} \begin{cases} - \frac{32 a^{4} \sqrt{a + b x^{3}}}{945 b^{4}} + \frac{16 a^{3} x^{3} \sqrt{a + b x^{3}}}{945 b^{3}} - \frac{4 a^{2} x^{6} \sqrt{a + b x^{3}}}{315 b^{2}} + \frac{2 a x^{9} \sqrt{a + b x^{3}}}{189 b} + \frac{2 x^{12} \sqrt{a + b x^{3}}}{27} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11911, size = 77, normalized size = 0.96 \begin{align*} \frac{2 \,{\left (35 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}\right )}}{945 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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