Optimal. Leaf size=52 \[ \frac {6 a \log \left (a \sqrt [3]{x}+b\right )}{b^3}-\frac {2 a \log (x)}{b^3}-\frac {3 a}{b^2 \left (a \sqrt [3]{x}+b\right )}-\frac {3}{b^2 \sqrt [3]{x}} \]
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Rubi [A] time = 0.03, antiderivative size = 52, 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} \[ -\frac {3 a}{b^2 \left (a \sqrt [3]{x}+b\right )}+\frac {6 a \log \left (a \sqrt [3]{x}+b\right )}{b^3}-\frac {2 a \log (x)}{b^3}-\frac {3}{b^2 \sqrt [3]{x}} \]
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 )^2 x^2} \, dx &=\int \frac {1}{\left (b+a \sqrt [3]{x}\right )^2 x^{4/3}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {1}{x^2 (b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{b^2 x^2}-\frac {2 a}{b^3 x}+\frac {a^2}{b^2 (b+a x)^2}+\frac {2 a^2}{b^3 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a}{b^2 \left (b+a \sqrt [3]{x}\right )}-\frac {3}{b^2 \sqrt [3]{x}}+\frac {6 a \log \left (b+a \sqrt [3]{x}\right )}{b^3}-\frac {2 a \log (x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 46, normalized size = 0.88 \[ \frac {-\frac {3 a b}{a \sqrt [3]{x}+b}+6 a \log \left (a \sqrt [3]{x}+b\right )-2 a \log (x)-\frac {3 b}{\sqrt [3]{x}}}{b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 98, normalized size = 1.88 \[ \frac {3 \, {\left (a^{2} b^{2} x^{\frac {4}{3}} - a b^{3} x + 2 \, {\left (a^{4} x^{2} + a b^{3} x\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 2 \, {\left (a^{4} x^{2} + a b^{3} x\right )} \log \left (x^{\frac {1}{3}}\right ) - {\left (2 \, a^{3} b x + b^{4}\right )} x^{\frac {2}{3}}\right )}}{a^{3} b^{3} x^{2} + b^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 51, normalized size = 0.98 \[ \frac {6 \, a \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{3}} - \frac {2 \, a \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {3 \, {\left (2 \, a x^{\frac {1}{3}} + b\right )}}{{\left (a x^{\frac {2}{3}} + b x^{\frac {1}{3}}\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.90 \[ -\frac {3 a}{\left (a \,x^{\frac {1}{3}}+b \right ) b^{2}}-\frac {2 a \ln \relax (x )}{b^{3}}+\frac {6 a \ln \left (a \,x^{\frac {1}{3}}+b \right )}{b^{3}}-\frac {3}{b^{2} x^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 44, normalized size = 0.85 \[ \frac {6 \, a \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{3}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}}{b^{3}} + \frac {3 \, a^{2}}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 49, normalized size = 0.94 \[ \frac {12\,a\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^3}-\frac {\frac {3}{b}+\frac {6\,a\,x^{1/3}}{b^2}}{a\,x^{2/3}+b\,x^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.79, size = 211, normalized size = 4.06 \[ \begin {cases} \frac {\tilde {\infty }}{\sqrt [3]{x}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{a^{2} x} & \text {for}\: b = 0 \\- \frac {3}{b^{2} \sqrt [3]{x}} & \text {for}\: a = 0 \\- \frac {2 a^{2} x^{2} \log {\relax (x )}}{a b^{3} x^{2} + b^{4} x^{\frac {5}{3}}} + \frac {6 a^{2} x^{2} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a b^{3} x^{2} + b^{4} x^{\frac {5}{3}}} - \frac {2 a b x^{\frac {5}{3}} \log {\relax (x )}}{a b^{3} x^{2} + b^{4} x^{\frac {5}{3}}} + \frac {6 a b x^{\frac {5}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a b^{3} x^{2} + b^{4} x^{\frac {5}{3}}} - \frac {6 a b x^{\frac {5}{3}}}{a b^{3} x^{2} + b^{4} x^{\frac {5}{3}}} - \frac {3 b^{2} x^{\frac {4}{3}}}{a b^{3} x^{2} + b^{4} x^{\frac {5}{3}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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