Optimal. Leaf size=72 \[ -\frac {A b^3}{4 x^4}-\frac {b^2 (3 A c+b B)}{2 x^2}+\frac {1}{2} c^2 x^2 (A c+3 b B)+3 b c \log (x) (A c+b B)+\frac {1}{4} B c^3 x^4 \]
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Rubi [A] time = 0.07, antiderivative size = 72, 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} \[ -\frac {b^2 (3 A c+b B)}{2 x^2}-\frac {A b^3}{4 x^4}+\frac {1}{2} c^2 x^2 (A c+3 b B)+3 b c \log (x) (A c+b B)+\frac {1}{4} B c^3 x^4 \]
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^{11}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^5} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) (b+c x)^3}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (c^2 (3 b B+A c)+\frac {A b^3}{x^3}+\frac {b^2 (b B+3 A c)}{x^2}+\frac {3 b c (b B+A c)}{x}+B c^3 x\right ) \, dx,x,x^2\right )\\ &=-\frac {A b^3}{4 x^4}-\frac {b^2 (b B+3 A c)}{2 x^2}+\frac {1}{2} c^2 (3 b B+A c) x^2+\frac {1}{4} B c^3 x^4+3 b c (b B+A c) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 1.01 \[ \frac {B x^2 \left (-2 b^3+6 b c^2 x^4+c^3 x^6\right )-A \left (b^3+6 b^2 c x^2-2 c^3 x^6\right )}{4 x^4}+3 b c \log (x) (A c+b B) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 76, normalized size = 1.06 \[ \frac {B c^{3} x^{8} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 12 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} \log \relax (x) - A b^{3} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 98, normalized size = 1.36 \[ \frac {1}{4} \, B c^{3} x^{4} + \frac {3}{2} \, B b c^{2} x^{2} + \frac {1}{2} \, A c^{3} x^{2} + \frac {3}{2} \, {\left (B b^{2} c + A b c^{2}\right )} \log \left (x^{2}\right ) - \frac {9 \, B b^{2} c x^{4} + 9 \, A b c^{2} x^{4} + 2 \, B b^{3} x^{2} + 6 \, A b^{2} c x^{2} + A b^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 76, normalized size = 1.06 \[ \frac {B \,c^{3} x^{4}}{4}+\frac {A \,c^{3} x^{2}}{2}+\frac {3 B b \,c^{2} x^{2}}{2}+3 A b \,c^{2} \ln \relax (x )+3 B \,b^{2} c \ln \relax (x )-\frac {3 A \,b^{2} c}{2 x^{2}}-\frac {B \,b^{3}}{2 x^{2}}-\frac {A \,b^{3}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 76, normalized size = 1.06 \[ \frac {1}{4} \, B c^{3} x^{4} + \frac {1}{2} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{2} + \frac {3}{2} \, {\left (B b^{2} c + A b c^{2}\right )} \log \left (x^{2}\right ) - \frac {A b^{3} + 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 76, normalized size = 1.06 \[ \ln \relax (x)\,\left (3\,B\,b^2\,c+3\,A\,b\,c^2\right )-\frac {\frac {A\,b^3}{4}+x^2\,\left (\frac {B\,b^3}{2}+\frac {3\,A\,c\,b^2}{2}\right )}{x^4}+x^2\,\left (\frac {A\,c^3}{2}+\frac {3\,B\,b\,c^2}{2}\right )+\frac {B\,c^3\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 75, normalized size = 1.04 \[ \frac {B c^{3} x^{4}}{4} + 3 b c \left (A c + B b\right ) \log {\relax (x )} + x^{2} \left (\frac {A c^{3}}{2} + \frac {3 B b c^{2}}{2}\right ) + \frac {- A b^{3} + x^{2} \left (- 6 A b^{2} c - 2 B b^{3}\right )}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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