Optimal. Leaf size=42 \[ \frac{3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{2/3}}{2 b} \]
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Rubi [A] time = 0.021423, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{2/3}}{2 b} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{a+b \sqrt [3]{x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2}{a+b x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{2/3}}{2 b}+\frac{3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0206839, size = 42, normalized size = 1. \[ \frac{3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{2/3}}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 79, normalized size = 1.9 \begin{align*}{\frac{{a}^{2}\ln \left ({b}^{3}x+{a}^{3} \right ) }{{b}^{3}}}+{\frac{3}{2\,b}{x}^{{\frac{2}{3}}}}+2\,{\frac{{a}^{2}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{3}}}-{\frac{{a}^{2}}{{b}^{3}}\ln \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) }-3\,{\frac{a\sqrt [3]{x}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949821, size = 59, normalized size = 1.4 \begin{align*} \frac{3 \, a^{2} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{3}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2}}{2 \, b^{3}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )} a}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45872, size = 89, normalized size = 2.12 \begin{align*} \frac{3 \,{\left (2 \, a^{2} \log \left (b x^{\frac{1}{3}} + a\right ) + b^{2} x^{\frac{2}{3}} - 2 \, a b x^{\frac{1}{3}}\right )}}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.23262, size = 42, normalized size = 1. \begin{align*} \begin{cases} \frac{3 a^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{b^{3}} - \frac{3 a \sqrt [3]{x}}{b^{2}} + \frac{3 x^{\frac{2}{3}}}{2 b} & \text{for}\: b \neq 0 \\\frac{x}{a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14518, size = 47, normalized size = 1.12 \begin{align*} \frac{3 \, a^{2} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3}} + \frac{3 \,{\left (b x^{\frac{2}{3}} - 2 \, a x^{\frac{1}{3}}\right )}}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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