Optimal. Leaf size=61 \[ \frac {B x^2}{2 c^2}-\frac {b (b B-A c)}{2 c^3 \left (b+c x^2\right )}-\frac {(2 b B-A c) \log \left (b+c x^2\right )}{2 c^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 61, 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 {b (b B-A c)}{2 c^3 \left (b+c x^2\right )}-\frac {(2 b B-A c) \log \left (b+c x^2\right )}{2 c^3}+\frac {B x^2}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^7 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^3 \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x (A+B x)}{(b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {B}{c^2}+\frac {b (b B-A c)}{c^2 (b+c x)^2}+\frac {-2 b B+A c}{c^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {B x^2}{2 c^2}-\frac {b (b B-A c)}{2 c^3 \left (b+c x^2\right )}-\frac {(2 b B-A c) \log \left (b+c x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.82 \begin {gather*} \frac {B c x^2+\frac {b (-b B+A c)}{b+c x^2}+(-2 b B+A c) \log \left (b+c x^2\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 59, normalized size = 0.97
method | result | size |
default | \(\frac {B \,x^{2}}{2 c^{2}}+\frac {\frac {\left (A c -2 B b \right ) \ln \left (c \,x^{2}+b \right )}{c}+\frac {b \left (A c -B b \right )}{c \left (c \,x^{2}+b \right )}}{2 c^{2}}\) | \(59\) |
norman | \(\frac {\frac {B \,x^{7}}{2 c}+\frac {b \left (A c -2 B b \right ) x^{3}}{2 c^{3}}}{x^{3} \left (c \,x^{2}+b \right )}+\frac {\left (A c -2 B b \right ) \ln \left (c \,x^{2}+b \right )}{2 c^{3}}\) | \(63\) |
risch | \(\frac {B \,x^{2}}{2 c^{2}}+\frac {b A}{2 c^{2} \left (c \,x^{2}+b \right )}-\frac {b^{2} B}{2 c^{3} \left (c \,x^{2}+b \right )}+\frac {\ln \left (c \,x^{2}+b \right ) A}{2 c^{2}}-\frac {\ln \left (c \,x^{2}+b \right ) B b}{c^{3}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 60, normalized size = 0.98 \begin {gather*} \frac {B x^{2}}{2 \, c^{2}} - \frac {B b^{2} - A b c}{2 \, {\left (c^{4} x^{2} + b c^{3}\right )}} - \frac {{\left (2 \, B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.41, size = 81, normalized size = 1.33 \begin {gather*} \frac {B c^{2} x^{4} + B b c x^{2} - B b^{2} + A b c - {\left (2 \, B b^{2} - A b c + {\left (2 \, B b c - A c^{2}\right )} x^{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 = 0.31, size = 56, normalized size = 0.92 \begin {gather*} \frac {B x^{2}}{2 c^{2}} + \frac {A b c - B b^{2}}{2 b c^{3} + 2 c^{4} x^{2}} - \frac {\left (- A c + 2 B b\right ) \log {\left (b + c x^{2} \right )}}{2 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 70, normalized size = 1.15 \begin {gather*} \frac {B x^{2}}{2 \, c^{2}} - \frac {{\left (2 \, B b - A c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} + \frac {2 \, B b c x^{2} - A c^{2} x^{2} + B 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 = 0.12, size = 62, normalized size = 1.02 \begin {gather*} \frac {B\,x^2}{2\,c^2}+\frac {\ln \left (c\,x^2+b\right )\,\left (A\,c-2\,B\,b\right )}{2\,c^3}-\frac {B\,b^2-A\,b\,c}{2\,c\,\left (c^3\,x^2+b\,c^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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