Optimal. Leaf size=39 \[ -\frac {a^3}{x}-\frac {9 a^2 b}{2 x^{2/3}}-\frac {9 a b^2}{\sqrt [3]{x}}+b^3 \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 39, 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 = {272, 45}
\begin {gather*} -\frac {a^3}{x}-\frac {9 a^2 b}{2 x^{2/3}}-\frac {9 a b^2}{\sqrt [3]{x}}+b^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt [3]{x}\right )^3}{x^2} \, dx &=3 \text {Subst}\left (\int \frac {(a+b x)^3}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {a^3}{x^4}+\frac {3 a^2 b}{x^3}+\frac {3 a b^2}{x^2}+\frac {b^3}{x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {a^3}{x}-\frac {9 a^2 b}{2 x^{2/3}}-\frac {9 a b^2}{\sqrt [3]{x}}+b^3 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 40, normalized size = 1.03 \begin {gather*} -\frac {a \left (2 a^2+9 a b \sqrt [3]{x}+18 b^2 x^{2/3}\right )}{2 x}+b^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 34, normalized size = 0.87
method | result | size |
derivativedivides | \(-\frac {a^{3}}{x}-\frac {9 a^{2} b}{2 x^{\frac {2}{3}}}-\frac {9 a \,b^{2}}{x^{\frac {1}{3}}}+b^{3} \ln \left (x \right )\) | \(34\) |
default | \(-\frac {a^{3}}{x}-\frac {9 a^{2} b}{2 x^{\frac {2}{3}}}-\frac {9 a \,b^{2}}{x^{\frac {1}{3}}}+b^{3} \ln \left (x \right )\) | \(34\) |
trager | \(\frac {a^{3} \left (x -1\right )}{x}-\frac {9 a^{2} b}{2 x^{\frac {2}{3}}}-\frac {9 a \,b^{2}}{x^{\frac {1}{3}}}-b^{3} \ln \left (\frac {1}{x}\right )\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 36, normalized size = 0.92 \begin {gather*} b^{3} \log \left (x\right ) - \frac {18 \, a b^{2} x^{\frac {2}{3}} + 9 \, a^{2} b x^{\frac {1}{3}} + 2 \, a^{3}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 39, normalized size = 1.00 \begin {gather*} \frac {6 \, b^{3} x \log \left (x^{\frac {1}{3}}\right ) - 18 \, a b^{2} x^{\frac {2}{3}} - 9 \, a^{2} b x^{\frac {1}{3}} - 2 \, a^{3}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 36, normalized size = 0.92 \begin {gather*} - \frac {a^{3}}{x} - \frac {9 a^{2} b}{2 x^{\frac {2}{3}}} - \frac {9 a b^{2}}{\sqrt [3]{x}} + b^{3} \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.91, size = 37, normalized size = 0.95 \begin {gather*} b^{3} \log \left ({\left | x \right |}\right ) - \frac {18 \, a b^{2} x^{\frac {2}{3}} + 9 \, a^{2} b x^{\frac {1}{3}} + 2 \, a^{3}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 37, normalized size = 0.95 \begin {gather*} 3\,b^3\,\ln \left (x^{1/3}\right )-\frac {a^3+\frac {9\,a^2\,b\,x^{1/3}}{2}+9\,a\,b^2\,x^{2/3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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