Optimal. Leaf size=249 \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0593276, antiderivative size = 249, 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} \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]
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^5} \, 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^5} \, 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^5}+\frac{5 a^4 b^6}{x^2}+10 a^3 b^7 x+10 a^2 b^8 x^4+5 a b^9 x^7+b^{10} x^{10}\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}}{4 x^4 \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{b^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0250369, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (176 a^2 b^3 x^9+440 a^3 b^2 x^6-440 a^4 b x^3-22 a^5+55 a b^4 x^{12}+8 b^5 x^{15}\right )}{88 x^4 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-8\,{b}^{5}{x}^{15}-55\,a{b}^{4}{x}^{12}-176\,{a}^{2}{b}^{3}{x}^{9}-440\,{a}^{3}{b}^{2}{x}^{6}+440\,{a}^{4}b{x}^{3}+22\,{a}^{5}}{88\,{x}^{4} \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 = 1.09904, size = 80, normalized size = 0.32 \begin{align*} \frac{8 \, b^{5} x^{15} + 55 \, a b^{4} x^{12} + 176 \, a^{2} b^{3} x^{9} + 440 \, a^{3} b^{2} x^{6} - 440 \, a^{4} b x^{3} - 22 \, a^{5}}{88 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70535, size = 135, normalized size = 0.54 \begin{align*} \frac{8 \, b^{5} x^{15} + 55 \, a b^{4} x^{12} + 176 \, a^{2} b^{3} x^{9} + 440 \, a^{3} b^{2} x^{6} - 440 \, a^{4} b x^{3} - 22 \, a^{5}}{88 \, x^{4}} \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^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13103, size = 144, normalized size = 0.58 \begin{align*} \frac{1}{11} \, b^{5} x^{11} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{8} \, a b^{4} x^{8} \mathrm{sgn}\left (b x^{3} + a\right ) + 2 \, a^{2} b^{3} x^{5} \mathrm{sgn}\left (b x^{3} + a\right ) + 5 \, a^{3} b^{2} x^{2} \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{20 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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