Optimal. Leaf size=251 \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Rubi [A] time = 0.0686626, antiderivative size = 251, 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} \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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} \, 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} \, 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} \, 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 (5 a^4 b^6+\frac{a^5 b^5}{x}+10 a^3 b^7 x+10 a^2 b^8 x^2+5 a b^9 x^3+b^{10} x^4\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end{align*}
Mathematica [A] time = 0.0260967, size = 82, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (b x^3 \left (200 a^2 b^2 x^6+300 a^3 b x^3+300 a^4+75 a b^3 x^9+12 b^4 x^{12}\right )+180 a^5 \log (x)\right )}{180 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 79, normalized size = 0.3 \begin{align*}{\frac{12\,{b}^{5}{x}^{15}+75\,a{b}^{4}{x}^{12}+200\,{a}^{2}{b}^{3}{x}^{9}+300\,{a}^{3}{b}^{2}{x}^{6}+300\,{a}^{4}b{x}^{3}+180\,{a}^{5}\ln \left ( x \right ) }{180\, \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 [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 = 1.63602, size = 134, normalized size = 0.53 \begin{align*} \frac{1}{15} \, b^{5} x^{15} + \frac{5}{12} \, a b^{4} x^{12} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{5}{3} \, a^{4} b x^{3} + a^{5} \log \left (x\right ) \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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09832, size = 140, normalized size = 0.56 \begin{align*} \frac{1}{15} \, b^{5} x^{15} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{12} \, a b^{4} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{9} \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{5} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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