Optimal. Leaf size=71 \[ -\frac {A b^3}{6 x^6}-\frac {b^2 (3 A c+b B)}{4 x^4}+c^2 \log (x) (A c+3 b B)-\frac {3 b c (A c+b B)}{2 x^2}+\frac {1}{2} B c^3 x^2 \]
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Rubi [A] time = 0.06, antiderivative size = 71, 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, 76} \[ -\frac {b^2 (3 A c+b B)}{4 x^4}-\frac {A b^3}{6 x^6}+c^2 \log (x) (A c+3 b B)-\frac {3 b c (A c+b B)}{2 x^2}+\frac {1}{2} B c^3 x^2 \]
Antiderivative was successfully verified.
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Rule 76
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{13}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^7} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) (b+c x)^3}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (B c^3+\frac {A b^3}{x^4}+\frac {b^2 (b B+3 A c)}{x^3}+\frac {3 b c (b B+A c)}{x^2}+\frac {c^2 (3 b B+A c)}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {A b^3}{6 x^6}-\frac {b^2 (b B+3 A c)}{4 x^4}-\frac {3 b c (b B+A c)}{2 x^2}+\frac {1}{2} B c^3 x^2+c^2 (3 b B+A c) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 71, normalized size = 1.00 \[ -\frac {A b^3}{6 x^6}-\frac {b^2 (3 A c+b B)}{4 x^4}+c^2 \log (x) (A c+3 b B)-\frac {3 b c (A c+b B)}{2 x^2}+\frac {1}{2} B c^3 x^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 77, normalized size = 1.08 \[ \frac {6 \, B c^{3} x^{8} + 12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} \log \relax (x) - 18 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} - 2 \, A b^{3} - 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 99, normalized size = 1.39 \[ \frac {1}{2} \, B c^{3} x^{2} + \frac {1}{2} \, {\left (3 \, B b c^{2} + A c^{3}\right )} \log \left (x^{2}\right ) - \frac {33 \, B b c^{2} x^{6} + 11 \, A c^{3} x^{6} + 18 \, B b^{2} c x^{4} + 18 \, A b c^{2} x^{4} + 3 \, B b^{3} x^{2} + 9 \, A b^{2} c x^{2} + 2 \, A b^{3}}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 75, normalized size = 1.06 \[ \frac {B \,c^{3} x^{2}}{2}+A \,c^{3} \ln \relax (x )+3 B b \,c^{2} \ln \relax (x )-\frac {3 A b \,c^{2}}{2 x^{2}}-\frac {3 B \,b^{2} c}{2 x^{2}}-\frac {3 A \,b^{2} c}{4 x^{4}}-\frac {B \,b^{3}}{4 x^{4}}-\frac {A \,b^{3}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 77, normalized size = 1.08 \[ \frac {1}{2} \, B c^{3} x^{2} + \frac {1}{2} \, {\left (3 \, B b c^{2} + A c^{3}\right )} \log \left (x^{2}\right ) - \frac {18 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 2 \, A b^{3} + 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 75, normalized size = 1.06 \[ \ln \relax (x)\,\left (A\,c^3+3\,B\,b\,c^2\right )-\frac {x^4\,\left (\frac {3\,B\,b^2\,c}{2}+\frac {3\,A\,b\,c^2}{2}\right )+\frac {A\,b^3}{6}+x^2\,\left (\frac {B\,b^3}{4}+\frac {3\,A\,c\,b^2}{4}\right )}{x^6}+\frac {B\,c^3\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.29, size = 78, normalized size = 1.10 \[ \frac {B c^{3} x^{2}}{2} + c^{2} \left (A c + 3 B b\right ) \log {\relax (x )} + \frac {- 2 A b^{3} + x^{4} \left (- 18 A b c^{2} - 18 B b^{2} c\right ) + x^{2} \left (- 9 A b^{2} c - 3 B b^{3}\right )}{12 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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