Optimal. Leaf size=104 \[ -\frac {3 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^7}+\frac {b^6 \log (x)}{a^7}+\frac {3 b^5}{a^6 \sqrt [3]{x}}-\frac {3 b^4}{2 a^5 x^{2/3}}+\frac {b^3}{a^4 x}-\frac {3 b^2}{4 a^3 x^{4/3}}+\frac {3 b}{5 a^2 x^{5/3}}-\frac {1}{2 a x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, 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 44
Rule 266
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*}
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Mathematica [A] time = 0.07, size = 95, normalized size = 0.91 \[ \frac {\frac {a \left (-10 a^5+12 a^4 b \sqrt [3]{x}-15 a^3 b^2 x^{2/3}+20 a^2 b^3 x-30 a b^4 x^{4/3}+60 b^5 x^{5/3}\right )}{x^2}-60 b^6 \log \left (a+b \sqrt [3]{x}\right )+20 b^6 \log (x)}{20 a^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 93, normalized size = 0.89 \[ -\frac {60 \, b^{6} x^{2} \log \left (b x^{\frac {1}{3}} + a\right ) - 60 \, b^{6} x^{2} \log \left (x^{\frac {1}{3}}\right ) - 20 \, a^{3} b^{3} x + 10 \, a^{6} - 15 \, {\left (4 \, a b^{5} x - a^{4} b^{2}\right )} x^{\frac {2}{3}} + 6 \, {\left (5 \, a^{2} b^{4} x - 2 \, a^{5} b\right )} x^{\frac {1}{3}}}{20 \, a^{7} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 91, normalized size = 0.88 \[ -\frac {3 \, b^{6} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{7}} + \frac {b^{6} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{\frac {5}{3}} - 30 \, a^{2} b^{4} x^{\frac {4}{3}} + 20 \, a^{3} b^{3} x - 15 \, a^{4} b^{2} x^{\frac {2}{3}} + 12 \, a^{5} b x^{\frac {1}{3}} - 10 \, a^{6}}{20 \, a^{7} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 87, normalized size = 0.84 \[ \frac {b^{6} \ln \relax (x )}{a^{7}}-\frac {3 b^{6} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{a^{7}}+\frac {3 b^{5}}{a^{6} x^{\frac {1}{3}}}-\frac {3 b^{4}}{2 a^{5} x^{\frac {2}{3}}}+\frac {b^{3}}{a^{4} x}-\frac {3 b^{2}}{4 a^{3} x^{\frac {4}{3}}}+\frac {3 b}{5 a^{2} x^{\frac {5}{3}}}-\frac {1}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 86, normalized size = 0.83 \[ -\frac {3 \, b^{6} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{7}} + \frac {b^{6} \log \relax (x)}{a^{7}} + \frac {60 \, b^{5} x^{\frac {5}{3}} - 30 \, a b^{4} x^{\frac {4}{3}} + 20 \, a^{2} b^{3} x - 15 \, a^{3} b^{2} x^{\frac {2}{3}} + 12 \, a^{4} b x^{\frac {1}{3}} - 10 \, a^{5}}{20 \, a^{6} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 83, normalized size = 0.80 \[ -\frac {\frac {1}{2\,a}-\frac {3\,b\,x^{1/3}}{5\,a^2}-\frac {b^3\,x}{a^4}+\frac {3\,b^2\,x^{2/3}}{4\,a^3}+\frac {3\,b^4\,x^{4/3}}{2\,a^5}-\frac {3\,b^5\,x^{5/3}}{a^6}}{x^2}-\frac {6\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.70, size = 129, normalized size = 1.24 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{7 b x^{\frac {7}{3}}} & \text {for}\: a = 0 \\- \frac {1}{2 a x^{2}} & \text {for}\: b = 0 \\- \frac {1}{2 a x^{2}} + \frac {3 b}{5 a^{2} x^{\frac {5}{3}}} - \frac {3 b^{2}}{4 a^{3} x^{\frac {4}{3}}} + \frac {b^{3}}{a^{4} x} - \frac {3 b^{4}}{2 a^{5} x^{\frac {2}{3}}} + \frac {3 b^{5}}{a^{6} \sqrt [3]{x}} + \frac {b^{6} \log {\relax (x )}}{a^{7}} - \frac {3 b^{6} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a^{7}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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