Optimal. Leaf size=167 \[ \frac{b^3 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{3 a^2 b x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{a^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0426231, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 270} \[ \frac{b^3 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{3 a^2 b x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{a^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 270
Rubi steps
\begin{align*} \int x^7 \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^7 \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} \int \left (a^3 b^3 x^7+3 a^2 b^4 x^{10}+3 a b^5 x^{13}+b^6 x^{16}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac{a^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{3 a^2 b x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{b^3 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0176902, size = 61, normalized size = 0.37 \[ \frac{x^8 \sqrt{\left (a+b x^3\right )^2} \left (2856 a^2 b x^3+1309 a^3+2244 a b^2 x^6+616 b^3 x^9\right )}{10472 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 58, normalized size = 0.4 \begin{align*}{\frac{{x}^{8} \left ( 616\,{b}^{3}{x}^{9}+2244\,a{b}^{2}{x}^{6}+2856\,{a}^{2}b{x}^{3}+1309\,{a}^{3} \right ) }{10472\, \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 [A] time = 1.03374, size = 47, normalized size = 0.28 \begin{align*} \frac{1}{17} \, b^{3} x^{17} + \frac{3}{14} \, a b^{2} x^{14} + \frac{3}{11} \, a^{2} b x^{11} + \frac{1}{8} \, a^{3} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72384, size = 88, normalized size = 0.53 \begin{align*} \frac{1}{17} \, b^{3} x^{17} + \frac{3}{14} \, a b^{2} x^{14} + \frac{3}{11} \, a^{2} b x^{11} + \frac{1}{8} \, a^{3} x^{8} \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^{7} \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 = 1.09728, size = 90, normalized size = 0.54 \begin{align*} \frac{1}{17} \, b^{3} x^{17} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{3}{14} \, a b^{2} x^{14} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{3}{11} \, a^{2} b x^{11} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{8} \, a^{3} x^{8} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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