Optimal. Leaf size=140 \[ -\frac {21 b^2 \log \left (a+b x^2\right )}{2 a^8}+\frac {21 b^2 \log (x)}{a^8}+\frac {15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac {3 b}{a^7 x^2}+\frac {5 b^2}{2 a^6 \left (a+b x^2\right )^2}-\frac {1}{4 a^6 x^4}+\frac {b^2}{a^5 \left (a+b x^2\right )^3}+\frac {3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac {b^2}{10 a^3 \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.14, antiderivative size = 140, 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, 266, 44} \begin {gather*} \frac {15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac {5 b^2}{2 a^6 \left (a+b x^2\right )^2}+\frac {b^2}{a^5 \left (a+b x^2\right )^3}+\frac {3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac {b^2}{10 a^3 \left (a+b x^2\right )^5}-\frac {21 b^2 \log \left (a+b x^2\right )}{2 a^8}+\frac {21 b^2 \log (x)}{a^8}+\frac {3 b}{a^7 x^2}-\frac {1}{4 a^6 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{x^5 \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{2} b^6 \operatorname {Subst}\left (\int \frac {1}{x^3 \left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^6 \operatorname {Subst}\left (\int \left (\frac {1}{a^6 b^6 x^3}-\frac {6}{a^7 b^5 x^2}+\frac {21}{a^8 b^4 x}-\frac {1}{a^3 b^3 (a+b x)^6}-\frac {3}{a^4 b^3 (a+b x)^5}-\frac {6}{a^5 b^3 (a+b x)^4}-\frac {10}{a^6 b^3 (a+b x)^3}-\frac {15}{a^7 b^3 (a+b x)^2}-\frac {21}{a^8 b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a^6 x^4}+\frac {3 b}{a^7 x^2}+\frac {b^2}{10 a^3 \left (a+b x^2\right )^5}+\frac {3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac {b^2}{a^5 \left (a+b x^2\right )^3}+\frac {5 b^2}{2 a^6 \left (a+b x^2\right )^2}+\frac {15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac {21 b^2 \log (x)}{a^8}-\frac {21 b^2 \log \left (a+b x^2\right )}{2 a^8}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 107, normalized size = 0.76 \begin {gather*} \frac {\frac {a \left (-10 a^6+70 a^5 b x^2+959 a^4 b^2 x^4+2695 a^3 b^3 x^6+3290 a^2 b^4 x^8+1890 a b^5 x^{10}+420 b^6 x^{12}\right )}{x^4 \left (a+b x^2\right )^5}-420 b^2 \log \left (a+b x^2\right )+840 b^2 \log (x)}{40 a^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.89, size = 266, normalized size = 1.90 \begin {gather*} \frac {420 \, a b^{6} x^{12} + 1890 \, a^{2} b^{5} x^{10} + 3290 \, a^{3} b^{4} x^{8} + 2695 \, a^{4} b^{3} x^{6} + 959 \, a^{5} b^{2} x^{4} + 70 \, a^{6} b x^{2} - 10 \, a^{7} - 420 \, {\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 840 \, {\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \relax (x)}{40 \, {\left (a^{8} b^{5} x^{14} + 5 \, a^{9} b^{4} x^{12} + 10 \, a^{10} b^{3} x^{10} + 10 \, a^{11} b^{2} x^{8} + 5 \, a^{12} b x^{6} + a^{13} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 130, normalized size = 0.93 \begin {gather*} \frac {21 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{8}} - \frac {21 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{8}} - \frac {63 \, b^{2} x^{4} - 12 \, a b x^{2} + a^{2}}{4 \, a^{8} x^{4}} + \frac {959 \, b^{7} x^{10} + 5095 \, a b^{6} x^{8} + 10890 \, a^{2} b^{5} x^{6} + 11730 \, a^{3} b^{4} x^{4} + 6390 \, a^{4} b^{3} x^{2} + 1418 \, a^{5} b^{2}}{40 \, {\left (b x^{2} + a\right )}^{5} a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 129, normalized size = 0.92 \begin {gather*} \frac {b^{2}}{10 \left (b \,x^{2}+a \right )^{5} a^{3}}+\frac {3 b^{2}}{8 \left (b \,x^{2}+a \right )^{4} a^{4}}+\frac {b^{2}}{\left (b \,x^{2}+a \right )^{3} a^{5}}+\frac {5 b^{2}}{2 \left (b \,x^{2}+a \right )^{2} a^{6}}+\frac {15 b^{2}}{2 \left (b \,x^{2}+a \right ) a^{7}}+\frac {21 b^{2} \ln \relax (x )}{a^{8}}-\frac {21 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{8}}+\frac {3 b}{a^{7} x^{2}}-\frac {1}{4 a^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 158, normalized size = 1.13 \begin {gather*} \frac {420 \, b^{6} x^{12} + 1890 \, a b^{5} x^{10} + 3290 \, a^{2} b^{4} x^{8} + 2695 \, a^{3} b^{3} x^{6} + 959 \, a^{4} b^{2} x^{4} + 70 \, a^{5} b x^{2} - 10 \, a^{6}}{40 \, {\left (a^{7} b^{5} x^{14} + 5 \, a^{8} b^{4} x^{12} + 10 \, a^{9} b^{3} x^{10} + 10 \, a^{10} b^{2} x^{8} + 5 \, a^{11} b x^{6} + a^{12} x^{4}\right )}} - \frac {21 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{8}} + \frac {21 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.91, size = 155, normalized size = 1.11 \begin {gather*} \frac {\frac {7\,b\,x^2}{4\,a^2}-\frac {1}{4\,a}+\frac {959\,b^2\,x^4}{40\,a^3}+\frac {539\,b^3\,x^6}{8\,a^4}+\frac {329\,b^4\,x^8}{4\,a^5}+\frac {189\,b^5\,x^{10}}{4\,a^6}+\frac {21\,b^6\,x^{12}}{2\,a^7}}{a^5\,x^4+5\,a^4\,b\,x^6+10\,a^3\,b^2\,x^8+10\,a^2\,b^3\,x^{10}+5\,a\,b^4\,x^{12}+b^5\,x^{14}}-\frac {21\,b^2\,\ln \left (b\,x^2+a\right )}{2\,a^8}+\frac {21\,b^2\,\ln \relax (x)}{a^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.96, size = 165, normalized size = 1.18 \begin {gather*} \frac {- 10 a^{6} + 70 a^{5} b x^{2} + 959 a^{4} b^{2} x^{4} + 2695 a^{3} b^{3} x^{6} + 3290 a^{2} b^{4} x^{8} + 1890 a b^{5} x^{10} + 420 b^{6} x^{12}}{40 a^{12} x^{4} + 200 a^{11} b x^{6} + 400 a^{10} b^{2} x^{8} + 400 a^{9} b^{3} x^{10} + 200 a^{8} b^{4} x^{12} + 40 a^{7} b^{5} x^{14}} + \frac {21 b^{2} \log {\relax (x )}}{a^{8}} - \frac {21 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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