Optimal. Leaf size=106 \[ -\frac {5 b \log \left (a-b x^2\right )}{2 a^6}+\frac {5 b \log (x)}{a^6}+\frac {2 b}{a^5 \left (a-b x^2\right )}-\frac {1}{2 a^5 x^2}+\frac {3 b}{4 a^4 \left (a-b x^2\right )^2}+\frac {b}{3 a^3 \left (a-b x^2\right )^3}+\frac {b}{8 a^2 \left (a-b x^2\right )^4} \]
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Rubi [A] time = 0.09, antiderivative size = 106, 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 = {266, 44} \[ \frac {2 b}{a^5 \left (a-b x^2\right )}+\frac {3 b}{4 a^4 \left (a-b x^2\right )^2}+\frac {b}{3 a^3 \left (a-b x^2\right )^3}+\frac {b}{8 a^2 \left (a-b x^2\right )^4}-\frac {5 b \log \left (a-b x^2\right )}{2 a^6}+\frac {5 b \log (x)}{a^6}-\frac {1}{2 a^5 x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a-b x)^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^5 x^2}+\frac {5 b}{a^6 x}+\frac {b^2}{a^2 (a-b x)^5}+\frac {2 b^2}{a^3 (a-b x)^4}+\frac {3 b^2}{a^4 (a-b x)^3}+\frac {4 b^2}{a^5 (a-b x)^2}+\frac {5 b^2}{a^6 (a-b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^5 x^2}+\frac {b}{8 a^2 \left (a-b x^2\right )^4}+\frac {b}{3 a^3 \left (a-b x^2\right )^3}+\frac {3 b}{4 a^4 \left (a-b x^2\right )^2}+\frac {2 b}{a^5 \left (a-b x^2\right )}+\frac {5 b \log (x)}{a^6}-\frac {5 b \log \left (a-b x^2\right )}{2 a^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.78 \[ \frac {\frac {a \left (-12 a^4+125 a^3 b x^2-260 a^2 b^2 x^4+210 a b^3 x^6-60 b^4 x^8\right )}{x^2 \left (a-b x^2\right )^4}-60 b \log \left (a-b x^2\right )+120 b \log (x)}{24 a^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 209, normalized size = 1.97 \[ -\frac {60 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} + 260 \, a^{3} b^{2} x^{4} - 125 \, a^{4} b x^{2} + 12 \, a^{5} + 60 \, {\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (b x^{2} - a\right ) - 120 \, {\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \relax (x)}{24 \, {\left (a^{6} b^{4} x^{10} - 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} - 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 106, normalized size = 1.00 \[ \frac {5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} - \frac {5 \, b \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{6}} - \frac {5 \, b x^{2} + a}{2 \, a^{6} x^{2}} + \frac {125 \, b^{5} x^{8} - 548 \, a b^{4} x^{6} + 912 \, a^{2} b^{3} x^{4} - 688 \, a^{3} b^{2} x^{2} + 202 \, a^{4} b}{24 \, {\left (b x^{2} - a\right )}^{4} a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 102, normalized size = 0.96 \[ \frac {b}{8 \left (b \,x^{2}-a \right )^{4} a^{2}}-\frac {b}{3 \left (b \,x^{2}-a \right )^{3} a^{3}}+\frac {3 b}{4 \left (b \,x^{2}-a \right )^{2} a^{4}}-\frac {2 b}{\left (b \,x^{2}-a \right ) a^{5}}+\frac {5 b \ln \relax (x )}{a^{6}}-\frac {5 b \ln \left (b \,x^{2}-a \right )}{2 a^{6}}-\frac {1}{2 a^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 123, normalized size = 1.16 \[ -\frac {60 \, b^{4} x^{8} - 210 \, a b^{3} x^{6} + 260 \, a^{2} b^{2} x^{4} - 125 \, a^{3} b x^{2} + 12 \, a^{4}}{24 \, {\left (a^{5} b^{4} x^{10} - 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} - 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} - \frac {5 \, b \log \left (b x^{2} - a\right )}{2 \, a^{6}} + \frac {5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.35, size = 120, normalized size = 1.13 \[ \frac {5\,b\,\ln \relax (x)}{a^6}-\frac {5\,b\,\ln \left (a-b\,x^2\right )}{2\,a^6}-\frac {\frac {1}{2\,a}-\frac {125\,b\,x^2}{24\,a^2}+\frac {65\,b^2\,x^4}{6\,a^3}-\frac {35\,b^3\,x^6}{4\,a^4}+\frac {5\,b^4\,x^8}{2\,a^5}}{a^4\,x^2-4\,a^3\,b\,x^4+6\,a^2\,b^2\,x^6-4\,a\,b^3\,x^8+b^4\,x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.76, size = 126, normalized size = 1.19 \[ - \frac {12 a^{4} - 125 a^{3} b x^{2} + 260 a^{2} b^{2} x^{4} - 210 a b^{3} x^{6} + 60 b^{4} x^{8}}{24 a^{9} x^{2} - 96 a^{8} b x^{4} + 144 a^{7} b^{2} x^{6} - 96 a^{6} b^{3} x^{8} + 24 a^{5} b^{4} x^{10}} + \frac {5 b \log {\relax (x )}}{a^{6}} - \frac {5 b \log {\left (- \frac {a}{b} + x^{2} \right )}}{2 a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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