Optimal. Leaf size=60 \[ -\frac {3 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^4}-\frac {b^3 \log (x)}{a^4}+\frac {3 b^2 \sqrt [3]{x}}{a^3}-\frac {3 b x^{2/3}}{2 a^2}+\frac {x}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, 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, 44} \[ \frac {3 b^2 \sqrt [3]{x}}{a^3}-\frac {3 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^4}-\frac {b^3 \log (x)}{a^4}-\frac {3 b x^{2/3}}{2 a^2}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 44
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{a+\frac {b}{\sqrt [3]{x}}} \, dx &=-\left (3 \operatorname {Subst}\left (\int \frac {1}{x^4 (a+b x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname {Subst}\left (\int \left (\frac {1}{a x^4}-\frac {b}{a^2 x^3}+\frac {b^2}{a^3 x^2}-\frac {b^3}{a^4 x}+\frac {b^4}{a^4 (a+b x)}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=\frac {3 b^2 \sqrt [3]{x}}{a^3}-\frac {3 b x^{2/3}}{2 a^2}+\frac {x}{a}-\frac {3 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^4}-\frac {b^3 \log (x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.83 \[ -\frac {3 b^3 \log \left (a \sqrt [3]{x}+b\right )}{a^4}+\frac {3 b^2 \sqrt [3]{x}}{a^3}-\frac {3 b x^{2/3}}{2 a^2}+\frac {x}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 43, normalized size = 0.72 \[ \frac {2 \, a^{3} x - 6 \, b^{3} \log \left (a x^{\frac {1}{3}} + b\right ) - 3 \, a^{2} b x^{\frac {2}{3}} + 6 \, a b^{2} x^{\frac {1}{3}}}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 45, normalized size = 0.75 \[ -\frac {3 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{4}} + \frac {2 \, a^{2} x - 3 \, a b x^{\frac {2}{3}} + 6 \, b^{2} x^{\frac {1}{3}}}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 43, normalized size = 0.72 \[ \frac {x}{a}-\frac {3 b^{3} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{a^{4}}-\frac {3 b \,x^{\frac {2}{3}}}{2 a^{2}}+\frac {3 b^{2} x^{\frac {1}{3}}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 54, normalized size = 0.90 \[ -\frac {3 \, b^{3} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{4}} - \frac {b^{3} \log \relax (x)}{a^{4}} + \frac {{\left (2 \, a^{2} - \frac {3 \, a b}{x^{\frac {1}{3}}} + \frac {6 \, b^{2}}{x^{\frac {2}{3}}}\right )} x}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 42, normalized size = 0.70 \[ \frac {x}{a}-\frac {3\,b\,x^{2/3}}{2\,a^2}-\frac {3\,b^3\,\ln \left (b+a\,x^{1/3}\right )}{a^4}+\frac {3\,b^2\,x^{1/3}}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 58, normalized size = 0.97 \[ \begin {cases} \frac {x}{a} - \frac {3 b x^{\frac {2}{3}}}{2 a^{2}} + \frac {3 b^{2} \sqrt [3]{x}}{a^{3}} - \frac {3 b^{3} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{4}} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {4}{3}}}{4 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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