Optimal. Leaf size=54 \[ -\frac{3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac{6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]
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Rubi [A] time = 0.0299247, antiderivative size = 54, 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} \[ -\frac{3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac{6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]
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 )^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^3}-\frac{2 a}{b^2 (a+b x)^2}+\frac{1}{b^2 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac{6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0364597, size = 45, normalized size = 0.83 \[ \frac{3 \left (\frac{a \left (3 a+4 b \sqrt [3]{x}\right )}{\left (a+b \sqrt [3]{x}\right )^2}+2 \log \left (a+b \sqrt [3]{x}\right )\right )}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 237, normalized size = 4.4 \begin{align*} -{\frac{9\,{a}^{6}}{2\, \left ({b}^{3}x+{a}^{3} \right ) ^{2}{b}^{3}}}+9\,{\frac{{a}^{3}}{{b}^{3} \left ({b}^{3}x+{a}^{3} \right ) }}+{\frac{\ln \left ({b}^{3}x+{a}^{3} \right ) }{{b}^{3}}}-{\frac{13\,{a}^{2}}{2\,b}{x}^{{\frac{2}{3}}} \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{-2}}+5\,{\frac{{a}^{3}\sqrt [3]{x}}{{b}^{2} \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{2}}}-3\,{\frac{{a}^{4}}{{b}^{3} \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{2}}}-{\frac{1}{{b}^{3}}\ln \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) }+2\,{\frac{\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{3}}}-{\frac{{a}^{2}}{{b}^{3}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}+2\,{\frac{ax}{ \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{2}}}+4\,{\frac{a}{{b}^{3} \left ( a+b\sqrt [3]{x} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977779, size = 62, normalized size = 1.15 \begin{align*} \frac{3 \, \log \left (b x^{\frac{1}{3}} + a\right )}{b^{3}} + \frac{6 \, a}{{\left (b x^{\frac{1}{3}} + a\right )} b^{3}} - \frac{3 \, a^{2}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44315, size = 243, normalized size = 4.5 \begin{align*} \frac{3 \,{\left (6 \, a^{3} b^{3} x + 3 \, a^{6} + 2 \,{\left (b^{6} x^{2} + 2 \, a^{3} b^{3} x + a^{6}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) +{\left (4 \, a b^{5} x + a^{4} b^{2}\right )} x^{\frac{2}{3}} -{\left (5 \, a^{2} b^{4} x + 2 \, a^{5} b\right )} x^{\frac{1}{3}}\right )}}{2 \,{\left (b^{9} x^{2} + 2 \, a^{3} b^{6} x + a^{6} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.819298, size = 228, normalized size = 4.22 \begin{align*} \begin{cases} \frac{6 a^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} + \frac{9 a^{2}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} + \frac{12 a b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} + \frac{12 a b \sqrt [3]{x}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} + \frac{6 b^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} & \text{for}\: b \neq 0 \\\frac{x}{a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13608, size = 59, normalized size = 1.09 \begin{align*} \frac{3 \, \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3}} + \frac{3 \,{\left (4 \, a x^{\frac{1}{3}} + \frac{3 \, a^{2}}{b}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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