Optimal. Leaf size=95 \[ -\frac {1}{6 b^3 x^6}+\frac {3 c}{4 b^4 x^4}-\frac {3 c^2}{b^5 x^2}-\frac {c^3}{4 b^4 \left (b+c x^2\right )^2}-\frac {2 c^3}{b^5 \left (b+c x^2\right )}-\frac {10 c^3 \log (x)}{b^6}+\frac {5 c^3 \log \left (b+c x^2\right )}{b^6} \]
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Rubi [A]
time = 0.05, 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 = {1598, 272, 46}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 1598
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} \text {Subst}\left (\int \frac {1}{x^4 (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 = 0.05, size = 85, normalized size = 0.89 \begin {gather*} -\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}+120 c^3 \log (x)-60 c^3 \log \left (b+c x^2\right )}{12 b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 94, normalized size = 0.99
method | result | size |
norman | \(\frac {-\frac {1}{6 b}+\frac {5 c \,x^{2}}{12 b^{2}}-\frac {5 c^{2} x^{4}}{3 b^{3}}+\frac {10 c^{4} x^{8}}{b^{5}}+\frac {15 c^{5} x^{10}}{2 b^{6}}}{x^{6} \left (c \,x^{2}+b \right )^{2}}-\frac {10 c^{3} \ln \left (x \right )}{b^{6}}+\frac {5 c^{3} \ln \left (c \,x^{2}+b \right )}{b^{6}}\) | \(89\) |
risch | \(\frac {-\frac {5 c^{4} x^{8}}{b^{5}}-\frac {15 c^{3} x^{6}}{2 b^{4}}-\frac {5 c^{2} x^{4}}{3 b^{3}}+\frac {5 c \,x^{2}}{12 b^{2}}-\frac {1}{6 b}}{x^{6} \left (c \,x^{2}+b \right )^{2}}-\frac {10 c^{3} \ln \left (x \right )}{b^{6}}+\frac {5 c^{3} \ln \left (-c \,x^{2}-b \right )}{b^{6}}\) | \(92\) |
default | \(\frac {c^{4} \left (\frac {10 \ln \left (c \,x^{2}+b \right )}{c}-\frac {b^{2}}{2 c \left (c \,x^{2}+b \right )^{2}}-\frac {4 b}{c \left (c \,x^{2}+b \right )}\right )}{2 b^{6}}-\frac {1}{6 b^{3} x^{6}}-\frac {10 c^{3} \ln \left (x \right )}{b^{6}}-\frac {3 c^{2}}{b^{5} x^{2}}+\frac {3 c}{4 b^{4} x^{4}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 103, normalized size = 1.08 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 145, normalized size = 1.53 \begin {gather*} -\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 \left (x\right )}{12 \, {\left (b^{6} c^{2} x^{10} + 2 \, b^{7} c x^{8} + b^{8} x^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 104, normalized size = 1.09 \begin {gather*} \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 {\left (x \right )}}{b^{6}} + \frac {5 c^{3} \log {\left (\frac {b}{c} + x^{2} \right )}}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.43, size = 110, normalized size = 1.16 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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 \left (x\right )}{b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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