Optimal. Leaf size=46 \[ -\frac{3 a^2}{b^3 \left (a+b \sqrt [3]{x}\right )}-\frac{6 a \log \left (a+b \sqrt [3]{x}\right )}{b^3}+\frac{3 \sqrt [3]{x}}{b^2} \]
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Rubi [A] time = 0.0259325, antiderivative size = 46, 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} \[ -\frac{3 a^2}{b^3 \left (a+b \sqrt [3]{x}\right )}-\frac{6 a \log \left (a+b \sqrt [3]{x}\right )}{b^3}+\frac{3 \sqrt [3]{x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt [3]{x}\right )^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b^2}+\frac{a^2}{b^2 (a+b x)^2}-\frac{2 a}{b^2 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^2}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac{3 \sqrt [3]{x}}{b^2}-\frac{6 a \log \left (a+b \sqrt [3]{x}\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0306636, size = 42, normalized size = 0.91 \[ \frac{3 \left (-\frac{a^2}{a+b \sqrt [3]{x}}-2 a \log \left (a+b \sqrt [3]{x}\right )+b \sqrt [3]{x}\right )}{b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 160, normalized size = 3.5 \begin{align*} -3\,{\frac{{a}^{4}}{ \left ({b}^{3}x+{a}^{3} \right ){b}^{3}}}+3\,{\frac{\sqrt [3]{x}}{{b}^{2}}}-{\frac{{a}^{2}}{{b}^{2}}\sqrt [3]{x} \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{-1}}+2\,{\frac{{a}^{3}}{{b}^{3} \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) }}+2\,{\frac{a\ln \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) }{{b}^{3}}}-4\,{\frac{a\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{3}}}-2\,{\frac{{a}^{2}}{{b}^{3} \left ( a+b\sqrt [3]{x} \right ) }}-2\,{\frac{a\ln \left ({b}^{3}x+{a}^{3} \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981196, size = 59, normalized size = 1.28 \begin{align*} -\frac{6 \, a \log \left (b x^{\frac{1}{3}} + a\right )}{b^{3}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}}{b^{3}} - \frac{3 \, a^{2}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48487, size = 153, normalized size = 3.33 \begin{align*} -\frac{3 \,{\left (a^{2} b^{2} x^{\frac{2}{3}} + a^{4} + 2 \,{\left (a b^{3} x + a^{4}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) -{\left (b^{4} x + 2 \, a^{3} b\right )} x^{\frac{1}{3}}\right )}}{b^{6} x + a^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.414846, size = 109, normalized size = 2.37 \begin{align*} \begin{cases} - \frac{6 a^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a b^{3} + b^{4} \sqrt [3]{x}} - \frac{6 a^{2}}{a b^{3} + b^{4} \sqrt [3]{x}} - \frac{6 a b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a b^{3} + b^{4} \sqrt [3]{x}} + \frac{3 b^{2} x^{\frac{2}{3}}}{a b^{3} + b^{4} \sqrt [3]{x}} & \text{for}\: b \neq 0 \\\frac{x}{a^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20783, size = 55, normalized size = 1.2 \begin{align*} -\frac{6 \, a \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3}} + \frac{3 \, x^{\frac{1}{3}}}{b^{2}} - \frac{3 \, a^{2}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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