Optimal. Leaf size=56 \[ \frac {3}{2 b \left (b+a \sqrt [3]{x}\right )^2}+\frac {3}{b^2 \left (b+a \sqrt [3]{x}\right )}-\frac {3 \log \left (b+a \sqrt [3]{x}\right )}{b^3}+\frac {\log (x)}{b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 56, 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 = {269, 272, 46}
\begin {gather*} -\frac {3 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac {3}{b^2 \left (a \sqrt [3]{x}+b\right )}+\frac {3}{2 b \left (a \sqrt [3]{x}+b\right )^2}+\frac {\log (x)}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx &=\int \frac {1}{\left (b+a \sqrt [3]{x}\right )^3 x} \, dx\\ &=3 \text {Subst}\left (\int \frac {1}{x (b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {1}{b^3 x}-\frac {a}{b (b+a x)^3}-\frac {a}{b^2 (b+a x)^2}-\frac {a}{b^3 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{2 b \left (b+a \sqrt [3]{x}\right )^2}+\frac {3}{b^2 \left (b+a \sqrt [3]{x}\right )}-\frac {3 \log \left (b+a \sqrt [3]{x}\right )}{b^3}+\frac {\log (x)}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 51, normalized size = 0.91 \begin {gather*} \frac {3 \left (\frac {b \left (3 b+2 a \sqrt [3]{x}\right )}{\left (b+a \sqrt [3]{x}\right )^2}-2 \log \left (b+a \sqrt [3]{x}\right )+\frac {2 \log (x)}{3}\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 49, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {3}{2 b \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {3}{b^{2} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {3 \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{3}}+\frac {\ln \left (x \right )}{b^{3}}\) | \(49\) |
default | \(\frac {3}{2 b \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {3}{b^{2} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {3 \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{3}}+\frac {\ln \left (x \right )}{b^{3}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 46, normalized size = 0.82 \begin {gather*} -\frac {3 \, \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{3}} - \frac {6 \, a}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{3}} + \frac {3 \, a^{2}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (48) = 96\).
time = 0.39, size = 129, normalized size = 2.30 \begin {gather*} \frac {3 \, {\left (3 \, b^{6} - 2 \, {\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 2 \, {\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (x^{\frac {1}{3}}\right ) + {\left (2 \, a^{5} b x + 5 \, a^{2} b^{4}\right )} x^{\frac {2}{3}} - {\left (a^{4} b^{2} x + 4 \, a b^{5}\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 406 vs.
\(2 (49) = 98\).
time = 1.34, size = 406, normalized size = 7.25 \begin {gather*} \begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\left (x \right )}}{b^{3}} & \text {for}\: a = 0 \\- \frac {1}{a^{3} x} & \text {for}\: b = 0 \\\frac {2 a^{2} x^{\frac {7}{3}} \log {\left (x \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {6 a^{2} x^{\frac {7}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {4 a b x^{2} \log {\left (x \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {12 a b x^{2} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {6 a b x^{2}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {2 b^{2} x^{\frac {5}{3}} \log {\left (x \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {6 b^{2} x^{\frac {5}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {9 b^{2} x^{\frac {5}{3}}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.37, size = 49, normalized size = 0.88 \begin {gather*} -\frac {3 \, \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{3}} + \frac {\log \left ({\left | x \right |}\right )}{b^{3}} + \frac {3 \, {\left (2 \, a b x^{\frac {1}{3}} + 3 \, b^{2}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 54, normalized size = 0.96 \begin {gather*} \frac {\frac {9}{2\,b}+\frac {3\,a\,x^{1/3}}{b^2}}{b^2+a^2\,x^{2/3}+2\,a\,b\,x^{1/3}}-\frac {6\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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