Optimal. Leaf size=252 \[ \frac{b^5 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{5 a b^4 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Rubi [A] time = 0.0735974, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \[ \frac{b^5 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{5 a b^4 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^4} \, 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^4} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^5}{x^2} \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \left (10 a^3 b^7+\frac{a^5 b^5}{x^2}+\frac{5 a^4 b^6}{x}+10 a^2 b^8 x+5 a b^9 x^2+b^{10} x^3\right ) \, dx,x,x^3\right )}{3 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}}{3 x^3 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a b^4 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{b^5 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end{align*}
Mathematica [A] time = 0.0261356, size = 85, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (60 a^2 b^3 x^9+120 a^3 b^2 x^6+180 a^4 b x^3 \log (x)-12 a^5+20 a b^4 x^{12}+3 b^5 x^{15}\right )}{36 x^3 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 82, normalized size = 0.3 \begin{align*}{\frac{3\,{b}^{5}{x}^{15}+20\,a{b}^{4}{x}^{12}+60\,{a}^{2}{b}^{3}{x}^{9}+120\,{a}^{3}{b}^{2}{x}^{6}+180\,{a}^{4}b\ln \left ( x \right ){x}^{3}-12\,{a}^{5}}{36\, \left ( b{x}^{3}+a \right ) ^{5}{x}^{3}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81542, size = 143, normalized size = 0.57 \begin{align*} \frac{3 \, b^{5} x^{15} + 20 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 120 \, a^{3} b^{2} x^{6} + 180 \, a^{4} b x^{3} \log \left (x\right ) - 12 \, a^{5}}{36 \, x^{3}} \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^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1099, size = 167, normalized size = 0.66 \begin{align*} \frac{1}{12} \, b^{5} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{9} \, a b^{4} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{2} b^{3} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{3} \, a^{3} b^{2} x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{5 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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