Optimal. Leaf size=68 \[ -\frac {b B-A c}{4 b c \left (b+c x^2\right )^2}+\frac {A}{2 b^2 \left (b+c x^2\right )}+\frac {A \log (x)}{b^3}-\frac {A \log \left (b+c x^2\right )}{2 b^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 68, 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 = {1598, 457, 78}
\begin {gather*} -\frac {A \log \left (b+c x^2\right )}{2 b^3}+\frac {A \log (x)}{b^3}+\frac {A}{2 b^2 \left (b+c x^2\right )}-\frac {b B-A c}{4 b c \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^5 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {A+B x^2}{x \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{b^3 x}+\frac {b B-A c}{b (b+c x)^3}-\frac {A c}{b^2 (b+c x)^2}-\frac {A c}{b^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b B-A c}{4 b c \left (b+c x^2\right )^2}+\frac {A}{2 b^2 \left (b+c x^2\right )}+\frac {A \log (x)}{b^3}-\frac {A \log \left (b+c x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 59, normalized size = 0.87 \begin {gather*} \frac {\frac {b \left (-b^2 B+3 A b c+2 A c^2 x^2\right )}{c \left (b+c x^2\right )^2}+4 A \log (x)-2 A \log \left (b+c x^2\right )}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 63, normalized size = 0.93
method | result | size |
risch | \(\frac {\frac {A c \,x^{2}}{2 b^{2}}+\frac {3 A c -B b}{4 b c}}{\left (c \,x^{2}+b \right )^{2}}+\frac {A \ln \left (x \right )}{b^{3}}-\frac {A \ln \left (c \,x^{2}+b \right )}{2 b^{3}}\) | \(61\) |
default | \(-\frac {A \ln \left (c \,x^{2}+b \right )-\frac {b^{2} \left (A c -B b \right )}{2 c \left (c \,x^{2}+b \right )^{2}}-\frac {A b}{c \,x^{2}+b}}{2 b^{3}}+\frac {A \ln \left (x \right )}{b^{3}}\) | \(63\) |
norman | \(\frac {-\frac {\left (2 A c -B b \right ) x^{7}}{2 b^{2}}-\frac {c \left (3 A c -B b \right ) x^{9}}{4 b^{3}}}{x^{5} \left (c \,x^{2}+b \right )^{2}}+\frac {A \ln \left (x \right )}{b^{3}}-\frac {A \ln \left (c \,x^{2}+b \right )}{2 b^{3}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 77, normalized size = 1.13 \begin {gather*} \frac {2 \, A c^{2} x^{2} - B b^{2} + 3 \, A b c}{4 \, {\left (b^{2} c^{3} x^{4} + 2 \, b^{3} c^{2} x^{2} + b^{4} c\right )}} - \frac {A \log \left (c x^{2} + b\right )}{2 \, b^{3}} + \frac {A \log \left (x^{2}\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.68, size = 119, normalized size = 1.75 \begin {gather*} \frac {2 \, A b c^{2} x^{2} - B b^{3} + 3 \, A b^{2} c - 2 \, {\left (A c^{3} x^{4} + 2 \, A b c^{2} x^{2} + A b^{2} c\right )} \log \left (c x^{2} + b\right ) + 4 \, {\left (A c^{3} x^{4} + 2 \, A b c^{2} x^{2} + A b^{2} c\right )} \log \left (x\right )}{4 \, {\left (b^{3} c^{3} x^{4} + 2 \, b^{4} c^{2} x^{2} + b^{5} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 75, normalized size = 1.10 \begin {gather*} \frac {A \log {\left (x \right )}}{b^{3}} - \frac {A \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{3}} + \frac {3 A b c + 2 A c^{2} x^{2} - B b^{2}}{4 b^{4} c + 8 b^{3} c^{2} x^{2} + 4 b^{2} c^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 76, normalized size = 1.12 \begin {gather*} \frac {A \log \left (x^{2}\right )}{2 \, b^{3}} - \frac {A \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{3}} + \frac {3 \, A c^{3} x^{4} + 8 \, A b c^{2} x^{2} - B b^{3} + 6 \, A b^{2} c}{4 \, {\left (c x^{2} + b\right )}^{2} b^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 71, normalized size = 1.04 \begin {gather*} \frac {\frac {3\,A\,c-B\,b}{4\,b\,c}+\frac {A\,c\,x^2}{2\,b^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}-\frac {A\,\ln \left (c\,x^2+b\right )}{2\,b^3}+\frac {A\,\ln \left (x\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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