Optimal. Leaf size=69 \[ -\frac {1}{2 a^3 x^2}+\frac {b}{4 a^2 \left (a-b x^2\right )^2}+\frac {b}{a^3 \left (a-b x^2\right )}+\frac {3 b \log (x)}{a^4}-\frac {3 b \log \left (a-b x^2\right )}{2 a^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 69, 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, 46}
\begin {gather*} -\frac {3 b \log \left (a-b x^2\right )}{2 a^4}+\frac {3 b \log (x)}{a^4}+\frac {b}{a^3 \left (a-b x^2\right )}-\frac {1}{2 a^3 x^2}+\frac {b}{4 a^2 \left (a-b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a-b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^3 x^2}+\frac {3 b}{a^4 x}+\frac {b^2}{a^2 (a-b x)^3}+\frac {2 b^2}{a^3 (a-b x)^2}+\frac {3 b^2}{a^4 (a-b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^3 x^2}+\frac {b}{4 a^2 \left (a-b x^2\right )^2}+\frac {b}{a^3 \left (a-b x^2\right )}+\frac {3 b \log (x)}{a^4}-\frac {3 b \log \left (a-b x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 60, normalized size = 0.87 \begin {gather*} \frac {\frac {a \left (-2 a^2+9 a b x^2-6 b^2 x^4\right )}{\left (a x-b x^3\right )^2}+12 b \log (x)-6 b \log \left (a-b x^2\right )}{4 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 75, normalized size = 1.09
method | result | size |
risch | \(\frac {-\frac {3 b^{2} x^{4}}{2 a^{3}}+\frac {9 b \,x^{2}}{4 a^{2}}-\frac {1}{2 a}}{x^{2} \left (-b \,x^{2}+a \right )^{2}}+\frac {3 b \ln \left (x \right )}{a^{4}}-\frac {3 b \ln \left (-b \,x^{2}+a \right )}{2 a^{4}}\) | \(65\) |
norman | \(\frac {-\frac {1}{2 a}+\frac {3 b^{2} x^{4}}{a^{3}}-\frac {9 b^{3} x^{6}}{4 a^{4}}}{x^{2} \left (-b \,x^{2}+a \right )^{2}}+\frac {3 b \ln \left (x \right )}{a^{4}}-\frac {3 b \ln \left (-b \,x^{2}+a \right )}{2 a^{4}}\) | \(67\) |
default | \(\frac {b^{2} \left (-\frac {3 \ln \left (-b \,x^{2}+a \right )}{b}+\frac {2 a}{b \left (-b \,x^{2}+a \right )}+\frac {a^{2}}{2 b \left (-b \,x^{2}+a \right )^{2}}\right )}{2 a^{4}}-\frac {1}{2 a^{3} x^{2}}+\frac {3 b \ln \left (x \right )}{a^{4}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 79, normalized size = 1.14 \begin {gather*} -\frac {6 \, b^{2} x^{4} - 9 \, a b x^{2} + 2 \, a^{2}}{4 \, {\left (a^{3} b^{2} x^{6} - 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} - \frac {3 \, b \log \left (b x^{2} - a\right )}{2 \, a^{4}} + \frac {3 \, b \log \left (x^{2}\right )}{2 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.45, size = 121, normalized size = 1.75 \begin {gather*} -\frac {6 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} + 2 \, a^{3} + 6 \, {\left (b^{3} x^{6} - 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (b x^{2} - a\right ) - 12 \, {\left (b^{3} x^{6} - 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{4} b^{2} x^{6} - 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 78, normalized size = 1.13 \begin {gather*} - \frac {2 a^{2} - 9 a b x^{2} + 6 b^{2} x^{4}}{4 a^{5} x^{2} - 8 a^{4} b x^{4} + 4 a^{3} b^{2} x^{6}} + \frac {3 b \log {\left (x \right )}}{a^{4}} - \frac {3 b \log {\left (- \frac {a}{b} + x^{2} \right )}}{2 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.72, size = 84, normalized size = 1.22 \begin {gather*} \frac {3 \, b \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {3 \, b \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{4}} + \frac {9 \, b^{3} x^{4} - 22 \, a b^{2} x^{2} + 14 \, a^{2} b}{4 \, {\left (b x^{2} - a\right )}^{2} a^{4}} - \frac {3 \, b x^{2} + a}{2 \, a^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.61, size = 76, normalized size = 1.10 \begin {gather*} \frac {3\,b\,\ln \left (x\right )}{a^4}-\frac {3\,b\,\ln \left (a-b\,x^2\right )}{2\,a^4}-\frac {\frac {1}{2\,a}-\frac {9\,b\,x^2}{4\,a^2}+\frac {3\,b^2\,x^4}{2\,a^3}}{a^2\,x^2-2\,a\,b\,x^4+b^2\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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