Optimal. Leaf size=49 \[ -\frac {1}{4 a x^4}+\frac {b}{2 a^2 x^2}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log \left (a+b x^2\right )}{2 a^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, 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 = {1598, 272, 46}
\begin {gather*} -\frac {b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac {b^2 \log (x)}{a^3}+\frac {b}{2 a^2 x^2}-\frac {1}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a x+b x^3\right )} \, dx &=\int \frac {1}{x^5 \left (a+b x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a x^4}+\frac {b}{2 a^2 x^2}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log \left (a+b x^2\right )}{2 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 1.00 \begin {gather*} -\frac {1}{4 a x^4}+\frac {b}{2 a^2 x^2}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log \left (a+b x^2\right )}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 44, normalized size = 0.90
method | result | size |
default | \(-\frac {1}{4 a \,x^{4}}+\frac {b}{2 a^{2} x^{2}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{3}}\) | \(44\) |
norman | \(\frac {-\frac {1}{4 a}+\frac {b \,x^{2}}{2 a^{2}}}{x^{4}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{3}}\) | \(46\) |
risch | \(\frac {-\frac {1}{4 a}+\frac {b \,x^{2}}{2 a^{2}}}{x^{4}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{3}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 44, normalized size = 0.90 \begin {gather*} -\frac {b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {b^{2} \log \left (x\right )}{a^{3}} + \frac {2 \, b x^{2} - a}{4 \, a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.45, size = 45, normalized size = 0.92 \begin {gather*} -\frac {2 \, b^{2} x^{4} \log \left (b x^{2} + a\right ) - 4 \, b^{2} x^{4} \log \left (x\right ) - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 42, normalized size = 0.86 \begin {gather*} \frac {- a + 2 b x^{2}}{4 a^{2} x^{4}} + \frac {b^{2} \log {\left (x \right )}}{a^{3}} - \frac {b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.89, size = 57, normalized size = 1.16 \begin {gather*} \frac {b^{2} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3}} - \frac {3 \, b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 46, normalized size = 0.94 \begin {gather*} \frac {b^2\,\ln \left (x\right )}{a^3}-\frac {b^2\,\ln \left (b\,x^2+a\right )}{2\,a^3}-\frac {\frac {1}{4\,a}-\frac {b\,x^2}{2\,a^2}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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