Optimal. Leaf size=95 \[ \frac {5 c^3 \log \left (b+c x^2\right )}{b^6}-\frac {10 c^3 \log (x)}{b^6}-\frac {2 c^3}{b^5 \left (b+c x^2\right )}-\frac {3 c^2}{b^5 x^2}-\frac {c^3}{4 b^4 \left (b+c x^2\right )^2}+\frac {3 c}{4 b^4 x^4}-\frac {1}{6 b^3 x^6} \]
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Rubi [A] time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 266, 44} \[ -\frac {2 c^3}{b^5 \left (b+c x^2\right )}-\frac {c^3}{4 b^4 \left (b+c x^2\right )^2}-\frac {3 c^2}{b^5 x^2}+\frac {5 c^3 \log \left (b+c x^2\right )}{b^6}-\frac {10 c^3 \log (x)}{b^6}+\frac {3 c}{4 b^4 x^4}-\frac {1}{6 b^3 x^6} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {1}{x \left (b x^2+c x^4\right )^3} \, dx &=\int \frac {1}{x^7 \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b^3 x^4}-\frac {3 c}{b^4 x^3}+\frac {6 c^2}{b^5 x^2}-\frac {10 c^3}{b^6 x}+\frac {c^4}{b^4 (b+c x)^3}+\frac {4 c^4}{b^5 (b+c x)^2}+\frac {10 c^4}{b^6 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{6 b^3 x^6}+\frac {3 c}{4 b^4 x^4}-\frac {3 c^2}{b^5 x^2}-\frac {c^3}{4 b^4 \left (b+c x^2\right )^2}-\frac {2 c^3}{b^5 \left (b+c x^2\right )}-\frac {10 c^3 \log (x)}{b^6}+\frac {5 c^3 \log \left (b+c x^2\right )}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 85, normalized size = 0.89 \[ -\frac {\frac {b \left (2 b^4-5 b^3 c x^2+20 b^2 c^2 x^4+90 b c^3 x^6+60 c^4 x^8\right )}{x^6 \left (b+c x^2\right )^2}-60 c^3 \log \left (b+c x^2\right )+120 c^3 \log (x)}{12 b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 145, normalized size = 1.53 \[ -\frac {60 \, b c^{4} x^{8} + 90 \, b^{2} c^{3} x^{6} + 20 \, b^{3} c^{2} x^{4} - 5 \, b^{4} c x^{2} + 2 \, b^{5} - 60 \, {\left (c^{5} x^{10} + 2 \, b c^{4} x^{8} + b^{2} c^{3} x^{6}\right )} \log \left (c x^{2} + b\right ) + 120 \, {\left (c^{5} x^{10} + 2 \, b c^{4} x^{8} + b^{2} c^{3} x^{6}\right )} \log \relax (x)}{12 \, {\left (b^{6} c^{2} x^{10} + 2 \, b^{7} c x^{8} + b^{8} x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 110, normalized size = 1.16 \[ -\frac {5 \, c^{3} \log \left (x^{2}\right )}{b^{6}} + \frac {5 \, c^{3} \log \left ({\left | c x^{2} + b \right |}\right )}{b^{6}} - \frac {30 \, c^{5} x^{4} + 68 \, b c^{4} x^{2} + 39 \, b^{2} c^{3}}{4 \, {\left (c x^{2} + b\right )}^{2} b^{6}} + \frac {110 \, c^{3} x^{6} - 36 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} - 2 \, b^{3}}{12 \, b^{6} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 90, normalized size = 0.95 \[ -\frac {c^{3}}{4 \left (c \,x^{2}+b \right )^{2} b^{4}}-\frac {2 c^{3}}{\left (c \,x^{2}+b \right ) b^{5}}-\frac {10 c^{3} \ln \relax (x )}{b^{6}}+\frac {5 c^{3} \ln \left (c \,x^{2}+b \right )}{b^{6}}-\frac {3 c^{2}}{b^{5} x^{2}}+\frac {3 c}{4 b^{4} x^{4}}-\frac {1}{6 b^{3} x^{6}} \]
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 \[ -\frac {60 \, c^{4} x^{8} + 90 \, b c^{3} x^{6} + 20 \, b^{2} c^{2} x^{4} - 5 \, b^{3} c x^{2} + 2 \, b^{4}}{12 \, {\left (b^{5} c^{2} x^{10} + 2 \, b^{6} c x^{8} + b^{7} x^{6}\right )}} + \frac {5 \, c^{3} \log \left (c x^{2} + b\right )}{b^{6}} - \frac {5 \, c^{3} \log \left (x^{2}\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 101, normalized size = 1.06 \[ \frac {5\,c^3\,\ln \left (c\,x^2+b\right )}{b^6}-\frac {\frac {1}{6\,b}-\frac {5\,c\,x^2}{12\,b^2}+\frac {5\,c^2\,x^4}{3\,b^3}+\frac {15\,c^3\,x^6}{2\,b^4}+\frac {5\,c^4\,x^8}{b^5}}{b^2\,x^6+2\,b\,c\,x^8+c^2\,x^{10}}-\frac {10\,c^3\,\ln \relax (x)}{b^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 104, normalized size = 1.09 \[ \frac {- 2 b^{4} + 5 b^{3} c x^{2} - 20 b^{2} c^{2} x^{4} - 90 b c^{3} x^{6} - 60 c^{4} x^{8}}{12 b^{7} x^{6} + 24 b^{6} c x^{8} + 12 b^{5} c^{2} x^{10}} - \frac {10 c^{3} \log {\relax (x )}}{b^{6}} + \frac {5 c^{3} \log {\left (\frac {b}{c} + x^{2} \right )}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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