Optimal. Leaf size=57 \[ -\frac {b x^2}{c^3}+\frac {x^4}{4 c^2}+\frac {b^3}{2 c^4 \left (b+c x^2\right )}+\frac {3 b^2 \log \left (b+c x^2\right )}{2 c^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 57, 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, 45}
\begin {gather*} \frac {b^3}{2 c^4 \left (b+c x^2\right )}+\frac {3 b^2 \log \left (b+c x^2\right )}{2 c^4}-\frac {b x^2}{c^3}+\frac {x^4}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^7}{\left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {2 b}{c^3}+\frac {x}{c^2}-\frac {b^3}{c^3 (b+c x)^2}+\frac {3 b^2}{c^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b x^2}{c^3}+\frac {x^4}{4 c^2}+\frac {b^3}{2 c^4 \left (b+c x^2\right )}+\frac {3 b^2 \log \left (b+c x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 0.86 \begin {gather*} \frac {-4 b c x^2+c^2 x^4+\frac {2 b^3}{b+c x^2}+6 b^2 \log \left (b+c x^2\right )}{4 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 55, normalized size = 0.96
method | result | size |
default | \(\frac {\left (-c \,x^{2}+2 b \right )^{2}}{4 c^{4}}+\frac {b^{2} \left (\frac {3 \ln \left (c \,x^{2}+b \right )}{c}+\frac {b}{c \left (c \,x^{2}+b \right )}\right )}{2 c^{3}}\) | \(55\) |
risch | \(\frac {x^{4}}{4 c^{2}}-\frac {b \,x^{2}}{c^{3}}+\frac {b^{2}}{c^{4}}+\frac {b^{3}}{2 c^{4} \left (c \,x^{2}+b \right )}+\frac {3 b^{2} \ln \left (c \,x^{2}+b \right )}{2 c^{4}}\) | \(59\) |
norman | \(\frac {\frac {x^{9}}{4 c}-\frac {3 b \,x^{7}}{4 c^{2}}+\frac {3 b^{3} x^{3}}{2 c^{4}}}{x^{3} \left (c \,x^{2}+b \right )}+\frac {3 b^{2} \ln \left (c \,x^{2}+b \right )}{2 c^{4}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 54, normalized size = 0.95 \begin {gather*} \frac {b^{3}}{2 \, {\left (c^{5} x^{2} + b c^{4}\right )}} + \frac {3 \, b^{2} \log \left (c x^{2} + b\right )}{2 \, c^{4}} + \frac {c x^{4} - 4 \, b x^{2}}{4 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 70, normalized size = 1.23 \begin {gather*} \frac {c^{3} x^{6} - 3 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} + 2 \, b^{3} + 6 \, {\left (b^{2} c x^{2} + b^{3}\right )} \log \left (c x^{2} + b\right )}{4 \, {\left (c^{5} x^{2} + b c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 53, normalized size = 0.93 \begin {gather*} \frac {b^{3}}{2 b c^{4} + 2 c^{5} x^{2}} + \frac {3 b^{2} \log {\left (b + c x^{2} \right )}}{2 c^{4}} - \frac {b x^{2}}{c^{3}} + \frac {x^{4}}{4 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.75, size = 67, normalized size = 1.18 \begin {gather*} \frac {3 \, b^{2} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} + \frac {c^{2} x^{4} - 4 \, b c x^{2}}{4 \, c^{4}} - \frac {3 \, b^{2} c x^{2} + 2 \, b^{3}}{2 \, {\left (c x^{2} + b\right )} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.14, size = 57, normalized size = 1.00 \begin {gather*} \frac {x^4}{4\,c^2}+\frac {b^3}{2\,c\,\left (c^4\,x^2+b\,c^3\right )}-\frac {b\,x^2}{c^3}+\frac {3\,b^2\,\ln \left (c\,x^2+b\right )}{2\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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