Optimal. Leaf size=111 \[ -\frac {b^3 (b B-A c)}{4 c^5 \left (b+c x^2\right )^2}+\frac {b^2 (4 b B-3 A c)}{2 c^5 \left (b+c x^2\right )}+\frac {3 b (2 b B-A c) \log \left (b+c x^2\right )}{2 c^5}-\frac {x^2 (3 b B-A c)}{2 c^4}+\frac {B x^4}{4 c^3} \]
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Rubi [A] time = 0.14, antiderivative size = 111, 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 = {1584, 446, 77} \begin {gather*} \frac {b^2 (4 b B-3 A c)}{2 c^5 \left (b+c x^2\right )}-\frac {b^3 (b B-A c)}{4 c^5 \left (b+c x^2\right )^2}-\frac {x^2 (3 b B-A c)}{2 c^4}+\frac {3 b (2 b B-A c) \log \left (b+c x^2\right )}{2 c^5}+\frac {B x^4}{4 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{13} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^7 \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 (A+B x)}{(b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {-3 b B+A c}{c^4}+\frac {B x}{c^3}+\frac {b^3 (b B-A c)}{c^4 (b+c x)^3}-\frac {b^2 (4 b B-3 A c)}{c^4 (b+c x)^2}+\frac {3 b (2 b B-A c)}{c^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {(3 b B-A c) x^2}{2 c^4}+\frac {B x^4}{4 c^3}-\frac {b^3 (b B-A c)}{4 c^5 \left (b+c x^2\right )^2}+\frac {b^2 (4 b B-3 A c)}{2 c^5 \left (b+c x^2\right )}+\frac {3 b (2 b B-A c) \log \left (b+c x^2\right )}{2 c^5}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 94, normalized size = 0.85 \begin {gather*} \frac {\frac {b^3 (A c-b B)}{\left (b+c x^2\right )^2}+\frac {2 b^2 (4 b B-3 A c)}{b+c x^2}+2 c x^2 (A c-3 b B)+6 b (2 b B-A c) \log \left (b+c x^2\right )+B c^2 x^4}{4 c^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{13} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.37, size = 179, normalized size = 1.61 \begin {gather*} \frac {B c^{4} x^{8} - 2 \, {\left (2 \, B b c^{3} - A c^{4}\right )} x^{6} + 7 \, B b^{4} - 5 \, A b^{3} c - {\left (11 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} x^{4} + 2 \, {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} + 6 \, {\left (2 \, B b^{4} - A b^{3} c + {\left (2 \, B b^{2} c^{2} - A b c^{3}\right )} x^{4} + 2 \, {\left (2 \, B b^{3} c - A b^{2} c^{2}\right )} x^{2}\right )} \log \left (c x^{2} + b\right )}{4 \, {\left (c^{7} x^{4} + 2 \, b c^{6} x^{2} + b^{2} c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 132, normalized size = 1.19 \begin {gather*} \frac {3 \, {\left (2 \, B b^{2} - A b c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{5}} + \frac {B c^{3} x^{4} - 6 \, B b c^{2} x^{2} + 2 \, A c^{3} x^{2}}{4 \, c^{6}} - \frac {18 \, B b^{2} c^{2} x^{4} - 9 \, A b c^{3} x^{4} + 28 \, B b^{3} c x^{2} - 12 \, A b^{2} c^{2} x^{2} + 11 \, B b^{4} - 4 \, A b^{3} c}{4 \, {\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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maple [A] time = 0.06, size = 134, normalized size = 1.21 \begin {gather*} \frac {B \,x^{4}}{4 c^{3}}+\frac {A \,b^{3}}{4 \left (c \,x^{2}+b \right )^{2} c^{4}}+\frac {A \,x^{2}}{2 c^{3}}-\frac {B \,b^{4}}{4 \left (c \,x^{2}+b \right )^{2} c^{5}}-\frac {3 B b \,x^{2}}{2 c^{4}}-\frac {3 A \,b^{2}}{2 \left (c \,x^{2}+b \right ) c^{4}}-\frac {3 A b \ln \left (c \,x^{2}+b \right )}{2 c^{4}}+\frac {2 B \,b^{3}}{\left (c \,x^{2}+b \right ) c^{5}}+\frac {3 B \,b^{2} \ln \left (c \,x^{2}+b \right )}{c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 116, normalized size = 1.05 \begin {gather*} \frac {7 \, B b^{4} - 5 \, A b^{3} c + 2 \, {\left (4 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x^{2}}{4 \, {\left (c^{7} x^{4} + 2 \, b c^{6} x^{2} + b^{2} c^{5}\right )}} + \frac {B c x^{4} - 2 \, {\left (3 \, B b - A c\right )} x^{2}}{4 \, c^{4}} + \frac {3 \, {\left (2 \, B b^{2} - A b 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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mupad [B] time = 0.12, size = 118, normalized size = 1.06 \begin {gather*} \frac {\frac {7\,B\,b^4-5\,A\,b^3\,c}{4\,c}+x^2\,\left (2\,B\,b^3-\frac {3\,A\,b^2\,c}{2}\right )}{b^2\,c^4+2\,b\,c^5\,x^2+c^6\,x^4}+x^2\,\left (\frac {A}{2\,c^3}-\frac {3\,B\,b}{2\,c^4}\right )+\frac {\ln \left (c\,x^2+b\right )\,\left (6\,B\,b^2-3\,A\,b\,c\right )}{2\,c^5}+\frac {B\,x^4}{4\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.52, size = 119, normalized size = 1.07 \begin {gather*} \frac {B x^{4}}{4 c^{3}} + \frac {3 b \left (- A c + 2 B b\right ) \log {\left (b + c x^{2} \right )}}{2 c^{5}} + x^{2} \left (\frac {A}{2 c^{3}} - \frac {3 B b}{2 c^{4}}\right ) + \frac {- 5 A b^{3} c + 7 B b^{4} + x^{2} \left (- 6 A b^{2} c^{2} + 8 B b^{3} c\right )}{4 b^{2} c^{5} + 8 b c^{6} x^{2} + 4 c^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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