Optimal. Leaf size=28 \[ a^2 \log (x)-\frac {6 a b}{\sqrt [3]{x}}-\frac {3 b^2}{2 x^{2/3}} \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {263, 266, 43} \[ a^2 \log (x)-\frac {6 a b}{\sqrt [3]{x}}-\frac {3 b^2}{2 x^{2/3}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^2}{x} \, dx &=\int \frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{5/3}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {(b+a x)^2}{x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {b^2}{x^3}+\frac {2 a b}{x^2}+\frac {a^2}{x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 b^2}{2 x^{2/3}}-\frac {6 a b}{\sqrt [3]{x}}+a^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 1.00 \[ a^2 \log (x)-\frac {6 a b}{\sqrt [3]{x}}-\frac {3 b^2}{2 x^{2/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 31, normalized size = 1.11 \[ \frac {3 \, {\left (2 \, a^{2} x \log \left (x^{\frac {1}{3}}\right ) - 4 \, a b x^{\frac {2}{3}} - b^{2} x^{\frac {1}{3}}\right )}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 24, normalized size = 0.86 \[ a^{2} \log \left ({\left | x \right |}\right ) - \frac {3 \, {\left (4 \, a b x^{\frac {1}{3}} + b^{2}\right )}}{2 \, x^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 23, normalized size = 0.82 \[ a^{2} \ln \relax (x )-\frac {6 a b}{x^{\frac {1}{3}}}-\frac {3 b^{2}}{2 x^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 22, normalized size = 0.79 \[ a^{2} \log \relax (x) - \frac {6 \, a b}{x^{\frac {1}{3}}} - \frac {3 \, b^{2}}{2 \, x^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 28, normalized size = 1.00 \[ 3\,a^2\,\ln \left (x^{1/3}\right )-\frac {\frac {3\,b^2}{2}+6\,a\,b\,x^{1/3}}{x^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 27, normalized size = 0.96 \[ a^{2} \log {\relax (x )} - \frac {6 a b}{\sqrt [3]{x}} - \frac {3 b^{2}}{2 x^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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