Optimal. Leaf size=104 \[ -\frac{3 b^4}{2 a^5 x^{2/3}}-\frac{3 b^2}{4 a^3 x^{4/3}}+\frac{3 b^5}{a^6 \sqrt [3]{x}}+\frac{b^3}{a^4 x}-\frac{3 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}+\frac{3 b}{5 a^2 x^{5/3}}-\frac{1}{2 a x^2} \]
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Rubi [A] time = 0.0537924, antiderivative size = 104, 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, 44} \[ -\frac{3 b^4}{2 a^5 x^{2/3}}-\frac{3 b^2}{4 a^3 x^{4/3}}+\frac{3 b^5}{a^6 \sqrt [3]{x}}+\frac{b^3}{a^4 x}-\frac{3 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}+\frac{3 b}{5 a^2 x^{5/3}}-\frac{1}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt [3]{x}\right ) x^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^7 (a+b x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{a x^7}-\frac{b}{a^2 x^6}+\frac{b^2}{a^3 x^5}-\frac{b^3}{a^4 x^4}+\frac{b^4}{a^5 x^3}-\frac{b^5}{a^6 x^2}+\frac{b^6}{a^7 x}-\frac{b^7}{a^7 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{1}{2 a x^2}+\frac{3 b}{5 a^2 x^{5/3}}-\frac{3 b^2}{4 a^3 x^{4/3}}+\frac{b^3}{a^4 x}-\frac{3 b^4}{2 a^5 x^{2/3}}+\frac{3 b^5}{a^6 \sqrt [3]{x}}-\frac{3 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}\\ \end{align*}
Mathematica [A] time = 0.0619384, size = 95, normalized size = 0.91 \[ \frac{\frac{a \left (-15 a^3 b^2 x^{2/3}+20 a^2 b^3 x+12 a^4 b \sqrt [3]{x}-10 a^5-30 a b^4 x^{4/3}+60 b^5 x^{5/3}\right )}{x^2}-60 b^6 \log \left (a+b \sqrt [3]{x}\right )+20 b^6 \log (x)}{20 a^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 87, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,a{x}^{2}}}+{\frac{3\,b}{5\,{a}^{2}}{x}^{-{\frac{5}{3}}}}-{\frac{3\,{b}^{2}}{4\,{a}^{3}}{x}^{-{\frac{4}{3}}}}+{\frac{{b}^{3}}{{a}^{4}x}}-{\frac{3\,{b}^{4}}{2\,{a}^{5}}{x}^{-{\frac{2}{3}}}}+3\,{\frac{{b}^{5}}{{a}^{6}\sqrt [3]{x}}}-3\,{\frac{{b}^{6}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{7}}}+{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99467, size = 116, normalized size = 1.12 \begin{align*} -\frac{3 \, b^{6} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{7}} + \frac{b^{6} \log \left (x\right )}{a^{7}} + \frac{60 \, b^{5} x^{\frac{5}{3}} - 30 \, a b^{4} x^{\frac{4}{3}} + 20 \, a^{2} b^{3} x - 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 12 \, a^{4} b x^{\frac{1}{3}} - 10 \, a^{5}}{20 \, a^{6} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4483, size = 230, normalized size = 2.21 \begin{align*} -\frac{60 \, b^{6} x^{2} \log \left (b x^{\frac{1}{3}} + a\right ) - 60 \, b^{6} x^{2} \log \left (x^{\frac{1}{3}}\right ) - 20 \, a^{3} b^{3} x + 10 \, a^{6} - 15 \,{\left (4 \, a b^{5} x - a^{4} b^{2}\right )} x^{\frac{2}{3}} + 6 \,{\left (5 \, a^{2} b^{4} x - 2 \, a^{5} b\right )} x^{\frac{1}{3}}}{20 \, a^{7} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.5118, size = 129, normalized size = 1.24 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{7 b x^{\frac{7}{3}}} & \text{for}\: a = 0 \\- \frac{1}{2 a x^{2}} & \text{for}\: b = 0 \\- \frac{1}{2 a x^{2}} + \frac{3 b}{5 a^{2} x^{\frac{5}{3}}} - \frac{3 b^{2}}{4 a^{3} x^{\frac{4}{3}}} + \frac{b^{3}}{a^{4} x} - \frac{3 b^{4}}{2 a^{5} x^{\frac{2}{3}}} + \frac{3 b^{5}}{a^{6} \sqrt [3]{x}} + \frac{b^{6} \log{\left (x \right )}}{a^{7}} - \frac{3 b^{6} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{7}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17873, size = 123, normalized size = 1.18 \begin{align*} -\frac{3 \, b^{6} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{7}} + \frac{b^{6} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{3}} - 30 \, a^{2} b^{4} x^{\frac{4}{3}} + 20 \, a^{3} b^{3} x - 15 \, a^{4} b^{2} x^{\frac{2}{3}} + 12 \, a^{5} b x^{\frac{1}{3}} - 10 \, a^{6}}{20 \, a^{7} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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