Optimal. Leaf size=50 \[ -\frac {a^4}{8 x^8}-\frac {2 a^3 b}{3 x^6}-\frac {3 a^2 b^2}{2 x^4}-\frac {2 a b^3}{x^2}+b^4 \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 45}
\begin {gather*} -\frac {a^4}{8 x^8}-\frac {2 a^3 b}{3 x^6}-\frac {3 a^2 b^2}{2 x^4}-\frac {2 a b^3}{x^2}+b^4 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^9} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^9} \, dx}{b^4}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^4}{x^5} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^4 b^4}{x^5}+\frac {4 a^3 b^5}{x^4}+\frac {6 a^2 b^6}{x^3}+\frac {4 a b^7}{x^2}+\frac {b^8}{x}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac {a^4}{8 x^8}-\frac {2 a^3 b}{3 x^6}-\frac {3 a^2 b^2}{2 x^4}-\frac {2 a b^3}{x^2}+b^4 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 50, normalized size = 1.00 \begin {gather*} -\frac {a^4}{8 x^8}-\frac {2 a^3 b}{3 x^6}-\frac {3 a^2 b^2}{2 x^4}-\frac {2 a b^3}{x^2}+b^4 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 45, normalized size = 0.90
method | result | size |
default | \(-\frac {a^{4}}{8 x^{8}}-\frac {2 a^{3} b}{3 x^{6}}-\frac {3 a^{2} b^{2}}{2 x^{4}}-\frac {2 a \,b^{3}}{x^{2}}+b^{4} \ln \left (x \right )\) | \(45\) |
norman | \(\frac {-\frac {1}{8} a^{4}-2 a \,b^{3} x^{6}-\frac {3}{2} a^{2} b^{2} x^{4}-\frac {2}{3} a^{3} b \,x^{2}}{x^{8}}+b^{4} \ln \left (x \right )\) | \(47\) |
risch | \(\frac {-\frac {1}{8} a^{4}-2 a \,b^{3} x^{6}-\frac {3}{2} a^{2} b^{2} x^{4}-\frac {2}{3} a^{3} b \,x^{2}}{x^{8}}+b^{4} \ln \left (x \right )\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, b^{4} \log \left (x^{2}\right ) - \frac {48 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 16 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 50, normalized size = 1.00 \begin {gather*} \frac {24 \, b^{4} x^{8} \log \left (x\right ) - 48 \, a b^{3} x^{6} - 36 \, a^{2} b^{2} x^{4} - 16 \, a^{3} b x^{2} - 3 \, a^{4}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 49, normalized size = 0.98 \begin {gather*} b^{4} \log {\left (x \right )} + \frac {- 3 a^{4} - 16 a^{3} b x^{2} - 36 a^{2} b^{2} x^{4} - 48 a b^{3} x^{6}}{24 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.89, size = 58, normalized size = 1.16 \begin {gather*} \frac {1}{2} \, b^{4} \log \left (x^{2}\right ) - \frac {25 \, b^{4} x^{8} + 48 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 16 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 47, normalized size = 0.94 \begin {gather*} b^4\,\ln \left (x\right )-\frac {\frac {a^4}{8}+\frac {2\,a^3\,b\,x^2}{3}+\frac {3\,a^2\,b^2\,x^4}{2}+2\,a\,b^3\,x^6}{x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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