Optimal. Leaf size=44 \[ \frac {x^2}{2 c^2}-\frac {b^2}{2 c^3 \left (b+c x^2\right )}-\frac {b \log \left (b+c x^2\right )}{c^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 44, 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^2}{2 c^3 \left (b+c x^2\right )}-\frac {b \log \left (b+c x^2\right )}{c^3}+\frac {x^2}{2 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^9}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^5}{\left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{c^2}+\frac {b^2}{c^2 (b+c x)^2}-\frac {2 b}{c^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 c^2}-\frac {b^2}{2 c^3 \left (b+c x^2\right )}-\frac {b \log \left (b+c x^2\right )}{c^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 0.86 \begin {gather*} \frac {c x^2-\frac {b^2}{b+c x^2}-2 b \log \left (b+c x^2\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 44, normalized size = 1.00
method | result | size |
risch | \(\frac {x^{2}}{2 c^{2}}-\frac {b^{2}}{2 c^{3} \left (c \,x^{2}+b \right )}-\frac {b \ln \left (c \,x^{2}+b \right )}{c^{3}}\) | \(41\) |
default | \(\frac {x^{2}}{2 c^{2}}-\frac {b \left (\frac {2 \ln \left (c \,x^{2}+b \right )}{c}+\frac {b}{c \left (c \,x^{2}+b \right )}\right )}{2 c^{2}}\) | \(44\) |
norman | \(\frac {\frac {x^{7}}{2 c}-\frac {b^{2} x^{3}}{c^{3}}}{x^{3} \left (c \,x^{2}+b \right )}-\frac {b \ln \left (c \,x^{2}+b \right )}{c^{3}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 43, normalized size = 0.98 \begin {gather*} -\frac {b^{2}}{2 \, {\left (c^{4} x^{2} + b c^{3}\right )}} + \frac {x^{2}}{2 \, c^{2}} - \frac {b \log \left (c x^{2} + b\right )}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 56, normalized size = 1.27 \begin {gather*} \frac {c^{2} x^{4} + b c x^{2} - b^{2} - 2 \, {\left (b c x^{2} + b^{2}\right )} \log \left (c x^{2} + b\right )}{2 \, {\left (c^{4} x^{2} + b c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 39, normalized size = 0.89 \begin {gather*} - \frac {b^{2}}{2 b c^{3} + 2 c^{4} x^{2}} - \frac {b \log {\left (b + c x^{2} \right )}}{c^{3}} + \frac {x^{2}}{2 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.07, size = 49, normalized size = 1.11 \begin {gather*} \frac {x^{2}}{2 \, c^{2}} - \frac {b \log \left ({\left | c x^{2} + b \right |}\right )}{c^{3}} + \frac {2 \, b c x^{2} + b^{2}}{2 \, {\left (c x^{2} + b\right )} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 45, normalized size = 1.02 \begin {gather*} \frac {x^2}{2\,c^2}-\frac {b^2}{2\,\left (c^4\,x^2+b\,c^3\right )}-\frac {b\,\ln \left (c\,x^2+b\right )}{c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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