Optimal. Leaf size=78 \[ \frac{\left (a+b x^3\right )^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 b^2}-\frac{a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 b^2} \]
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Rubi [A] time = 0.050472, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \[ \frac{\left (a+b x^3\right )^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 b^2}-\frac{a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 b^2} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int x^5 \left (a b+b^2 x^3\right )^3 \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int x \left (a b+b^2 x\right )^3 \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \left (-\frac{a \left (a b+b^2 x\right )^3}{b}+\frac{\left (a b+b^2 x\right )^4}{b^2}\right ) \, dx,x,x^3\right )}{3 b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac{a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 b^2}+\frac{\left (a+b x^3\right )^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 b^2}\\ \end{align*}
Mathematica [A] time = 0.0190837, size = 61, normalized size = 0.78 \[ \frac{x^6 \sqrt{\left (a+b x^3\right )^2} \left (20 a^2 b x^3+10 a^3+15 a b^2 x^6+4 b^3 x^9\right )}{60 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 58, normalized size = 0.7 \begin{align*}{\frac{{x}^{6} \left ( 4\,{b}^{3}{x}^{9}+15\,a{b}^{2}{x}^{6}+20\,{a}^{2}b{x}^{3}+10\,{a}^{3} \right ) }{60\, \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65279, size = 84, normalized size = 1.08 \begin{align*} \frac{1}{15} \, b^{3} x^{15} + \frac{1}{4} \, a b^{2} x^{12} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{6} \, a^{3} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09632, size = 61, normalized size = 0.78 \begin{align*} \frac{1}{60} \,{\left (4 \, b^{3} x^{15} + 15 \, a b^{2} x^{12} + 20 \, a^{2} b x^{9} + 10 \, a^{3} x^{6}\right )} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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