Optimal. Leaf size=253 \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{b^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0623355, antiderivative size = 253, 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{b^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]
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 )^{5/2}}{x^{11}} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{\left (a b+b^2 x^3\right )^5}{x^{11}} \, dx}{b^4 \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^5 b^5}{x^{11}}+\frac{5 a^4 b^6}{x^8}+\frac{10 a^3 b^7}{x^5}+\frac{10 a^2 b^8}{x^2}+5 a b^9 x+b^{10} x^4\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{b^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0163053, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (700 a^2 b^3 x^9+175 a^3 b^2 x^6+50 a^4 b x^3+7 a^5-175 a b^4 x^{12}-14 b^5 x^{15}\right )}{70 x^{10} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-14\,{b}^{5}{x}^{15}-175\,a{b}^{4}{x}^{12}+700\,{a}^{2}{b}^{3}{x}^{9}+175\,{a}^{3}{b}^{2}{x}^{6}+50\,{a}^{4}b{x}^{3}+7\,{a}^{5}}{70\,{x}^{10} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06051, size = 80, normalized size = 0.32 \begin{align*} \frac{14 \, b^{5} x^{15} + 175 \, a b^{4} x^{12} - 700 \, a^{2} b^{3} x^{9} - 175 \, a^{3} b^{2} x^{6} - 50 \, a^{4} b x^{3} - 7 \, a^{5}}{70 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77641, size = 136, normalized size = 0.54 \begin{align*} \frac{14 \, b^{5} x^{15} + 175 \, a b^{4} x^{12} - 700 \, a^{2} b^{3} x^{9} - 175 \, a^{3} b^{2} x^{6} - 50 \, a^{4} b x^{3} - 7 \, a^{5}}{70 \, x^{10}} \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{5}{2}}}{x^{11}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11812, size = 146, normalized size = 0.58 \begin{align*} \frac{1}{5} \, b^{5} x^{5} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{2} \, a b^{4} x^{2} \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{700 \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 175 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 50 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 7 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{70 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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