Optimal. Leaf size=68 \[ -\frac{30 a^3 b^2}{\sqrt [3]{x}}+10 a^2 b^3 \log (x)-\frac{15 a^4 b}{2 x^{2/3}}-\frac{a^5}{x}+15 a b^4 \sqrt [3]{x}+\frac{3}{2} b^5 x^{2/3} \]
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Rubi [A] time = 0.0319101, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{30 a^3 b^2}{\sqrt [3]{x}}+10 a^2 b^3 \log (x)-\frac{15 a^4 b}{2 x^{2/3}}-\frac{a^5}{x}+15 a b^4 \sqrt [3]{x}+\frac{3}{2} b^5 x^{2/3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^5}{x^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (5 a b^4+\frac{a^5}{x^4}+\frac{5 a^4 b}{x^3}+\frac{10 a^3 b^2}{x^2}+\frac{10 a^2 b^3}{x}+b^5 x\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^5}{x}-\frac{15 a^4 b}{2 x^{2/3}}-\frac{30 a^3 b^2}{\sqrt [3]{x}}+15 a b^4 \sqrt [3]{x}+\frac{3}{2} b^5 x^{2/3}+10 a^2 b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0240683, size = 68, normalized size = 1. \[ -\frac{30 a^3 b^2}{\sqrt [3]{x}}+10 a^2 b^3 \log (x)-\frac{15 a^4 b}{2 x^{2/3}}-\frac{a^5}{x}+15 a b^4 \sqrt [3]{x}+\frac{3}{2} b^5 x^{2/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{x}}-{\frac{15\,{a}^{4}b}{2}{x}^{-{\frac{2}{3}}}}-30\,{\frac{{a}^{3}{b}^{2}}{\sqrt [3]{x}}}+15\,a{b}^{4}\sqrt [3]{x}+{\frac{3\,{b}^{5}}{2}{x}^{{\frac{2}{3}}}}+10\,{a}^{2}{b}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01243, size = 80, normalized size = 1.18 \begin{align*} 10 \, a^{2} b^{3} \log \left (x\right ) + \frac{3}{2} \, b^{5} x^{\frac{2}{3}} + 15 \, a b^{4} x^{\frac{1}{3}} - \frac{60 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} + 2 \, a^{5}}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5051, size = 147, normalized size = 2.16 \begin{align*} \frac{60 \, a^{2} b^{3} x \log \left (x^{\frac{1}{3}}\right ) - 2 \, a^{5} + 3 \,{\left (b^{5} x - 20 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 15 \,{\left (2 \, a b^{4} x - a^{4} b\right )} x^{\frac{1}{3}}}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.891623, size = 66, normalized size = 0.97 \begin{align*} - \frac{a^{5}}{x} - \frac{15 a^{4} b}{2 x^{\frac{2}{3}}} - \frac{30 a^{3} b^{2}}{\sqrt [3]{x}} + 10 a^{2} b^{3} \log{\left (x \right )} + 15 a b^{4} \sqrt [3]{x} + \frac{3 b^{5} x^{\frac{2}{3}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15233, size = 81, normalized size = 1.19 \begin{align*} 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) + \frac{3}{2} \, b^{5} x^{\frac{2}{3}} + 15 \, a b^{4} x^{\frac{1}{3}} - \frac{60 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} + 2 \, a^{5}}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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