Optimal. Leaf size=167 \[ -\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0409204, 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} \[ -\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]
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^{15}} \, 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^{15}} \, 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 (\frac{a^3 b^3}{x^{15}}+\frac{3 a^2 b^4}{x^{12}}+\frac{3 a b^5}{x^9}+\frac{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}}{14 x^{14} \left (a+b x^3\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0159379, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (840 a^2 b x^3+220 a^3+1155 a b^2 x^6+616 b^3 x^9\right )}{3080 x^{14} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 58, normalized size = 0.4 \begin{align*} -{\frac{616\,{b}^{3}{x}^{9}+1155\,a{b}^{2}{x}^{6}+840\,{a}^{2}b{x}^{3}+220\,{a}^{3}}{3080\,{x}^{14} \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 = 0.989244, size = 50, normalized size = 0.3 \begin{align*} -\frac{616 \, b^{3} x^{9} + 1155 \, a b^{2} x^{6} + 840 \, a^{2} b x^{3} + 220 \, a^{3}}{3080 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61594, size = 96, normalized size = 0.57 \begin{align*} -\frac{616 \, b^{3} x^{9} + 1155 \, a b^{2} x^{6} + 840 \, a^{2} b x^{3} + 220 \, a^{3}}{3080 \, x^{14}} \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^{15}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11327, size = 93, normalized size = 0.56 \begin{align*} -\frac{616 \, b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 1155 \, a b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 840 \, a^{2} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 220 \, a^{3} \mathrm{sgn}\left (b x^{3} + a\right )}{3080 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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