Optimal. Leaf size=163 \[ \frac{b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{3 a b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{3 a^2 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.042097, antiderivative size = 163, 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} \[ \frac{b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{3 a b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{3 a^2 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
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^3} \, 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^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 (3 a^2 b^4+\frac{a^3 b^3}{x^3}+3 a b^5 x^3+b^6 x^6\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}}{2 x^2 \left (a+b x^3\right )}+\frac{3 a^2 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{3 a b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0178124, size = 61, normalized size = 0.37 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (84 a^2 b x^3-14 a^3+21 a b^2 x^6+4 b^3 x^9\right )}{28 x^2 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 58, normalized size = 0.4 \begin{align*} -{\frac{-4\,{b}^{3}{x}^{9}-21\,a{b}^{2}{x}^{6}-84\,{a}^{2}b{x}^{3}+14\,{a}^{3}}{28\,{x}^{2} \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 = 1.05122, size = 50, normalized size = 0.31 \begin{align*} \frac{4 \, b^{3} x^{9} + 21 \, a b^{2} x^{6} + 84 \, a^{2} b x^{3} - 14 \, a^{3}}{28 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72203, size = 82, normalized size = 0.5 \begin{align*} \frac{4 \, b^{3} x^{9} + 21 \, a b^{2} x^{6} + 84 \, a^{2} b x^{3} - 14 \, a^{3}}{28 \, x^{2}} \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^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11464, size = 88, normalized size = 0.54 \begin{align*} \frac{1}{7} \, b^{3} x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{3}{4} \, a b^{2} x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) + 3 \, a^{2} b x \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{a^{3} \mathrm{sgn}\left (b x^{3} + a\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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