Optimal. Leaf size=51 \[ -\frac {3 a^2 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac {a^2 \log (x)}{b^3}+\frac {3 a}{b^2 \sqrt [3]{x}}-\frac {3}{2 b x^{2/3}} \]
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Rubi [A] time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {263, 266, 44} \[ -\frac {3 a^2 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac {a^2 \log (x)}{b^3}+\frac {3 a}{b^2 \sqrt [3]{x}}-\frac {3}{2 b x^{2/3}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^2} \, dx &=\int \frac {1}{\left (b+a \sqrt [3]{x}\right ) x^{5/3}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {1}{x^3 (b+a x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{b x^3}-\frac {a}{b^2 x^2}+\frac {a^2}{b^3 x}-\frac {a^3}{b^3 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3}{2 b x^{2/3}}+\frac {3 a}{b^2 \sqrt [3]{x}}-\frac {3 a^2 \log \left (b+a \sqrt [3]{x}\right )}{b^3}+\frac {a^2 \log (x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.94 \[ \frac {-6 a^2 \log \left (a \sqrt [3]{x}+b\right )+2 a^2 \log (x)-\frac {3 b \left (b-2 a \sqrt [3]{x}\right )}{x^{2/3}}}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 47, normalized size = 0.92 \[ -\frac {3 \, {\left (2 \, a^{2} x \log \left (a x^{\frac {1}{3}} + b\right ) - 2 \, a^{2} x \log \left (x^{\frac {1}{3}}\right ) - 2 \, a b x^{\frac {2}{3}} + b^{2} x^{\frac {1}{3}}\right )}}{2 \, b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 49, normalized size = 0.96 \[ -\frac {3 \, a^{2} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{3}} + \frac {a^{2} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {3 \, {\left (2 \, a b x^{\frac {1}{3}} - b^{2}\right )}}{2 \, b^{3} x^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 0.86 \[ \frac {a^{2} \ln \relax (x )}{b^{3}}-\frac {3 a^{2} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{b^{3}}+\frac {3 a}{b^{2} x^{\frac {1}{3}}}-\frac {3}{2 b \,x^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 44, normalized size = 0.86 \[ -\frac {3 \, a^{2} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{3}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2}}{2 \, b^{3}} + \frac {6 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 41, normalized size = 0.80 \[ -\frac {\frac {3}{2\,b}-\frac {3\,a\,x^{1/3}}{b^2}}{x^{2/3}}-\frac {6\,a^2\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.82, size = 73, normalized size = 1.43 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {2}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{a x} & \text {for}\: b = 0 \\- \frac {3}{2 b x^{\frac {2}{3}}} & \text {for}\: a = 0 \\\frac {a^{2} \log {\relax (x )}}{b^{3}} - \frac {3 a^{2} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{b^{3}} + \frac {3 a}{b^{2} \sqrt [3]{x}} - \frac {3}{2 b x^{\frac {2}{3}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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