Optimal. Leaf size=105 \[ \frac {b (3 b B-2 A c) x^2}{2 c^4}-\frac {(2 b B-A c) x^4}{4 c^3}+\frac {B x^6}{6 c^2}-\frac {b^3 (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac {b^2 (4 b B-3 A c) \log \left (b+c x^2\right )}{2 c^5} \]
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Rubi [A]
time = 0.10, antiderivative size = 105, 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^3 (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac {b^2 (4 b B-3 A c) \log \left (b+c x^2\right )}{2 c^5}+\frac {b x^2 (3 b B-2 A c)}{2 c^4}-\frac {x^4 (2 b B-A c)}{4 c^3}+\frac {B x^6}{6 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^{11} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^7 \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 (A+B x)}{(b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {b (3 b B-2 A c)}{c^4}+\frac {(-2 b B+A c) x}{c^3}+\frac {B x^2}{c^2}+\frac {b^3 (b B-A c)}{c^4 (b+c x)^2}-\frac {b^2 (4 b B-3 A c)}{c^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b (3 b B-2 A c) x^2}{2 c^4}-\frac {(2 b B-A c) x^4}{4 c^3}+\frac {B x^6}{6 c^2}-\frac {b^3 (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac {b^2 (4 b B-3 A c) \log \left (b+c x^2\right )}{2 c^5}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 93, normalized size = 0.89 \begin {gather*} \frac {6 b c (3 b B-2 A c) x^2+3 c^2 (-2 b B+A c) x^4+2 B c^3 x^6+\frac {6 b^3 (-b B+A c)}{b+c x^2}+6 b^2 (-4 b B+3 A c) \log \left (b+c x^2\right )}{12 c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 103, normalized size = 0.98
method | result | size |
default | \(-\frac {-\frac {B \,c^{2} x^{6}}{6}+\frac {\left (-A \,c^{2}+2 b B c \right ) x^{4}}{4}+\frac {\left (2 A b c -3 b^{2} B \right ) x^{2}}{2}}{c^{4}}+\frac {b^{2} \left (\frac {\left (3 A c -4 B b \right ) \ln \left (c \,x^{2}+b \right )}{c}+\frac {b \left (A c -B b \right )}{c \left (c \,x^{2}+b \right )}\right )}{2 c^{4}}\) | \(103\) |
norman | \(\frac {\frac {B \,x^{11}}{6 c}+\frac {\left (3 A c -4 B b \right ) x^{9}}{12 c^{2}}-\frac {b \left (3 A c -4 B b \right ) x^{7}}{4 c^{3}}+\frac {b \left (3 A \,b^{2} c -4 B \,b^{3}\right ) x^{3}}{2 c^{5}}}{x^{3} \left (c \,x^{2}+b \right )}+\frac {b^{2} \left (3 A c -4 B b \right ) \ln \left (c \,x^{2}+b \right )}{2 c^{5}}\) | \(108\) |
risch | \(\frac {B \,x^{6}}{6 c^{2}}+\frac {A \,x^{4}}{4 c^{2}}-\frac {x^{4} b B}{2 c^{3}}-\frac {A b \,x^{2}}{c^{3}}+\frac {3 b^{2} B \,x^{2}}{2 c^{4}}+\frac {b^{3} A}{2 c^{4} \left (c \,x^{2}+b \right )}-\frac {b^{4} B}{2 c^{5} \left (c \,x^{2}+b \right )}+\frac {3 b^{2} \ln \left (c \,x^{2}+b \right ) A}{2 c^{4}}-\frac {2 b^{3} \ln \left (c \,x^{2}+b \right ) B}{c^{5}}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 107, normalized size = 1.02 \begin {gather*} -\frac {B b^{4} - A b^{3} c}{2 \, {\left (c^{6} x^{2} + b c^{5}\right )}} + \frac {2 \, B c^{2} x^{6} - 3 \, {\left (2 \, B b c - A c^{2}\right )} x^{4} + 6 \, {\left (3 \, B b^{2} - 2 \, A b c\right )} x^{2}}{12 \, c^{4}} - \frac {{\left (4 \, B b^{3} - 3 \, A b^{2} c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 148, normalized size = 1.41 \begin {gather*} \frac {2 \, B c^{4} x^{8} - {\left (4 \, B b c^{3} - 3 \, A c^{4}\right )} x^{6} - 6 \, B b^{4} + 6 \, A b^{3} c + 3 \, {\left (4 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{4} + 6 \, {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} - 6 \, {\left (4 \, B b^{4} - 3 \, A b^{3} c + {\left (4 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x^{2}\right )} \log \left (c x^{2} + b\right )}{12 \, {\left (c^{6} x^{2} + b c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 104, normalized size = 0.99 \begin {gather*} \frac {B x^{6}}{6 c^{2}} - \frac {b^{2} \left (- 3 A c + 4 B b\right ) \log {\left (b + c x^{2} \right )}}{2 c^{5}} + x^{4} \left (\frac {A}{4 c^{2}} - \frac {B b}{2 c^{3}}\right ) + x^{2} \left (- \frac {A b}{c^{3}} + \frac {3 B b^{2}}{2 c^{4}}\right ) + \frac {A b^{3} c - B b^{4}}{2 b c^{5} + 2 c^{6} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.74, size = 135, normalized size = 1.29 \begin {gather*} -\frac {{\left (4 \, B b^{3} - 3 \, A b^{2} c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{5}} + \frac {2 \, B c^{4} x^{6} - 6 \, B b c^{3} x^{4} + 3 \, A c^{4} x^{4} + 18 \, B b^{2} c^{2} x^{2} - 12 \, A b c^{3} x^{2}}{12 \, c^{6}} + \frac {4 \, B b^{3} c x^{2} - 3 \, A b^{2} c^{2} x^{2} + 3 \, B b^{4} - 2 \, A b^{3} c}{2 \, {\left (c x^{2} + b\right )} c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 121, normalized size = 1.15 \begin {gather*} x^4\,\left (\frac {A}{4\,c^2}-\frac {B\,b}{2\,c^3}\right )-x^2\,\left (\frac {b\,\left (\frac {A}{c^2}-\frac {2\,B\,b}{c^3}\right )}{c}+\frac {B\,b^2}{2\,c^4}\right )+\frac {B\,x^6}{6\,c^2}-\frac {\ln \left (c\,x^2+b\right )\,\left (4\,B\,b^3-3\,A\,b^2\,c\right )}{2\,c^5}-\frac {B\,b^4-A\,b^3\,c}{2\,c\,\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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