Optimal. Leaf size=27 \[ -\frac {a^2}{3 x^3}+\frac {b^2 x^3}{3}+2 a b \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} -\frac {a^2}{3 x^3}+2 a b \log (x)+\frac {b^2 x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^2}{x^4} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^2}{x^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^2}{3 x^3}+\frac {b^2 x^3}{3}+2 a b \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {a^2}{3 x^3}+\frac {b^2 x^3}{3}+2 a b \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 24, normalized size = 0.89
method | result | size |
default | \(-\frac {a^{2}}{3 x^{3}}+\frac {b^{2} x^{3}}{3}+2 a b \ln \left (x \right )\) | \(24\) |
risch | \(-\frac {a^{2}}{3 x^{3}}+\frac {b^{2} x^{3}}{3}+2 a b \ln \left (x \right )\) | \(24\) |
norman | \(\frac {-\frac {a^{2}}{3}+\frac {b^{2} x^{6}}{3}}{x^{3}}+2 a b \ln \left (x \right )\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 25, normalized size = 0.93 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} + \frac {2}{3} \, a b \log \left (x^{3}\right ) - \frac {a^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 27, normalized size = 1.00 \begin {gather*} \frac {b^{2} x^{6} + 6 \, a b x^{3} \log \left (x\right ) - a^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 24, normalized size = 0.89 \begin {gather*} - \frac {a^{2}}{3 x^{3}} + 2 a b \log {\left (x \right )} + \frac {b^{2} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.17, size = 32, normalized size = 1.19 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac {2 \, a b x^{3} + a^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 23, normalized size = 0.85 \begin {gather*} \frac {b^2\,x^3}{3}-\frac {a^2}{3\,x^3}+2\,a\,b\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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