Optimal. Leaf size=95 \[ \frac {5 b^3 \log \left (a+b x^2\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6}-\frac {2 b^3}{a^5 \left (a+b x^2\right )}-\frac {3 b^2}{a^5 x^2}-\frac {b^3}{4 a^4 \left (a+b x^2\right )^2}+\frac {3 b}{4 a^4 x^4}-\frac {1}{6 a^3 x^6} \]
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Rubi [A] time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac {2 b^3}{a^5 \left (a+b x^2\right )}-\frac {b^3}{4 a^4 \left (a+b x^2\right )^2}-\frac {3 b^2}{a^5 x^2}+\frac {5 b^3 \log \left (a+b x^2\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6}+\frac {3 b}{4 a^4 x^4}-\frac {1}{6 a^3 x^6} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 (a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^4}-\frac {3 b}{a^4 x^3}+\frac {6 b^2}{a^5 x^2}-\frac {10 b^3}{a^6 x}+\frac {b^4}{a^4 (a+b x)^3}+\frac {4 b^4}{a^5 (a+b x)^2}+\frac {10 b^4}{a^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{6 a^3 x^6}+\frac {3 b}{4 a^4 x^4}-\frac {3 b^2}{a^5 x^2}-\frac {b^3}{4 a^4 \left (a+b x^2\right )^2}-\frac {2 b^3}{a^5 \left (a+b x^2\right )}-\frac {10 b^3 \log (x)}{a^6}+\frac {5 b^3 \log \left (a+b x^2\right )}{a^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 85, normalized size = 0.89 \[ -\frac {\frac {a \left (2 a^4-5 a^3 b x^2+20 a^2 b^2 x^4+90 a b^3 x^6+60 b^4 x^8\right )}{x^6 \left (a+b x^2\right )^2}-60 b^3 \log \left (a+b x^2\right )+120 b^3 \log (x)}{12 a^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 145, normalized size = 1.53 \[ -\frac {60 \, a b^{4} x^{8} + 90 \, a^{2} b^{3} x^{6} + 20 \, a^{3} b^{2} x^{4} - 5 \, a^{4} b x^{2} + 2 \, a^{5} - 60 \, {\left (b^{5} x^{10} + 2 \, a b^{4} x^{8} + a^{2} b^{3} x^{6}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{5} x^{10} + 2 \, a b^{4} x^{8} + a^{2} b^{3} x^{6}\right )} \log \relax (x)}{12 \, {\left (a^{6} b^{2} x^{10} + 2 \, a^{7} b x^{8} + a^{8} x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 110, normalized size = 1.16 \[ -\frac {5 \, b^{3} \log \left (x^{2}\right )}{a^{6}} + \frac {5 \, b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{6}} - \frac {30 \, b^{5} x^{4} + 68 \, a b^{4} x^{2} + 39 \, a^{2} b^{3}}{4 \, {\left (b x^{2} + a\right )}^{2} a^{6}} + \frac {110 \, b^{3} x^{6} - 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{6} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.95 \[ -\frac {b^{3}}{4 \left (b \,x^{2}+a \right )^{2} a^{4}}-\frac {2 b^{3}}{\left (b \,x^{2}+a \right ) a^{5}}-\frac {10 b^{3} \ln \relax (x )}{a^{6}}+\frac {5 b^{3} \ln \left (b \,x^{2}+a \right )}{a^{6}}-\frac {3 b^{2}}{a^{5} x^{2}}+\frac {3 b}{4 a^{4} x^{4}}-\frac {1}{6 a^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 103, normalized size = 1.08 \[ -\frac {60 \, b^{4} x^{8} + 90 \, a b^{3} x^{6} + 20 \, a^{2} b^{2} x^{4} - 5 \, a^{3} b x^{2} + 2 \, a^{4}}{12 \, {\left (a^{5} b^{2} x^{10} + 2 \, a^{6} b x^{8} + a^{7} x^{6}\right )}} + \frac {5 \, b^{3} \log \left (b x^{2} + a\right )}{a^{6}} - \frac {5 \, b^{3} \log \left (x^{2}\right )}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.69, size = 101, normalized size = 1.06 \[ \frac {5\,b^3\,\ln \left (b\,x^2+a\right )}{a^6}-\frac {\frac {1}{6\,a}-\frac {5\,b\,x^2}{12\,a^2}+\frac {5\,b^2\,x^4}{3\,a^3}+\frac {15\,b^3\,x^6}{2\,a^4}+\frac {5\,b^4\,x^8}{a^5}}{a^2\,x^6+2\,a\,b\,x^8+b^2\,x^{10}}-\frac {10\,b^3\,\ln \relax (x)}{a^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 104, normalized size = 1.09 \[ \frac {- 2 a^{4} + 5 a^{3} b x^{2} - 20 a^{2} b^{2} x^{4} - 90 a b^{3} x^{6} - 60 b^{4} x^{8}}{12 a^{7} x^{6} + 24 a^{6} b x^{8} + 12 a^{5} b^{2} x^{10}} - \frac {10 b^{3} \log {\relax (x )}}{a^{6}} + \frac {5 b^{3} \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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