Optimal. Leaf size=40 \[ -\frac {a^3}{6 x^6}+\frac {3 a^2 b}{4 x^4}-\frac {3 a b^2}{2 x^2}-b^3 \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {272, 45}
\begin {gather*} -\frac {a^3}{6 x^6}+\frac {3 a^2 b}{4 x^4}-\frac {3 a b^2}{2 x^2}-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 x^2\right )^3}{x^7} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a-b x)^3}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \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,x^2\right )\\ &=-\frac {a^3}{6 x^6}+\frac {3 a^2 b}{4 x^4}-\frac {3 a b^2}{2 x^2}-b^3 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 40, normalized size = 1.00 \begin {gather*} -\frac {a^3}{6 x^6}+\frac {3 a^2 b}{4 x^4}-\frac {3 a b^2}{2 x^2}-b^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 35, normalized size = 0.88
method | result | size |
default | \(-\frac {a^{3}}{6 x^{6}}+\frac {3 a^{2} b}{4 x^{4}}-\frac {3 a \,b^{2}}{2 x^{2}}-b^{3} \ln \left (x \right )\) | \(35\) |
norman | \(\frac {-\frac {1}{6} a^{3}-\frac {3}{2} a \,b^{2} x^{4}+\frac {3}{4} a^{2} b \,x^{2}}{x^{6}}-b^{3} \ln \left (x \right )\) | \(37\) |
risch | \(\frac {-\frac {1}{6} a^{3}-\frac {3}{2} a \,b^{2} x^{4}+\frac {3}{4} a^{2} b \,x^{2}}{x^{6}}-b^{3} \ln \left (x \right )\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.72, size = 39, normalized size = 0.98 \begin {gather*} -\frac {1}{2} \, b^{3} \log \left (x^{2}\right ) - \frac {18 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} + 2 \, a^{3}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 39, normalized size = 0.98 \begin {gather*} -\frac {12 \, b^{3} x^{6} \log \left (x\right ) + 18 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} + 2 \, a^{3}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 37, normalized size = 0.92 \begin {gather*} - b^{3} \log {\left (x \right )} - \frac {2 a^{3} - 9 a^{2} b x^{2} + 18 a b^{2} x^{4}}{12 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.58, size = 47, normalized size = 1.18 \begin {gather*} -\frac {1}{2} \, b^{3} \log \left (x^{2}\right ) + \frac {11 \, b^{3} x^{6} - 18 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 37, normalized size = 0.92 \begin {gather*} -b^3\,\ln \left (x\right )-\frac {\frac {a^3}{6}-\frac {3\,a^2\,b\,x^2}{4}+\frac {3\,a\,b^2\,x^4}{2}}{x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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