Optimal. Leaf size=162 \[ -\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{b^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Rubi [A] time = 0.0409398, antiderivative size = 162, 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} \[ -\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{b^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{\left (a b+b^2 x^3\right )^3}{x^9} \, 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 (b^6+\frac{a^3 b^3}{x^9}+\frac{3 a^2 b^4}{x^6}+\frac{3 a b^5}{x^3}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{b^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}\\ \end{align*}
Mathematica [A] time = 0.01548, size = 61, normalized size = 0.38 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (24 a^2 b x^3+5 a^3+60 a b^2 x^6-40 b^3 x^9\right )}{40 x^8 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 58, normalized size = 0.4 \begin{align*} -{\frac{-40\,{b}^{3}{x}^{9}+60\,a{b}^{2}{x}^{6}+24\,{a}^{2}b{x}^{3}+5\,{a}^{3}}{40\,{x}^{8} \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 = 1.02377, size = 50, normalized size = 0.31 \begin{align*} \frac{40 \, b^{3} x^{9} - 60 \, a b^{2} x^{6} - 24 \, a^{2} b x^{3} - 5 \, a^{3}}{40 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79179, size = 82, normalized size = 0.51 \begin{align*} \frac{40 \, b^{3} x^{9} - 60 \, a b^{2} x^{6} - 24 \, a^{2} b x^{3} - 5 \, a^{3}}{40 \, 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 \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{9}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1176, size = 90, normalized size = 0.56 \begin{align*} b^{3} x \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{60 \, a b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 24 \, a^{2} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 5 \, a^{3} \mathrm{sgn}\left (b x^{3} + a\right )}{40 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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