Optimal. Leaf size=48 \[ \frac {\left (a+b x^2\right )^3 (A b-4 a B)}{24 a^2 x^6}-\frac {A \left (a+b x^2\right )^3}{8 a x^8} \]
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Rubi [A] time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {446, 78, 37} \[ \frac {\left (a+b x^2\right )^3 (A b-4 a B)}{24 a^2 x^6}-\frac {A \left (a+b x^2\right )^3}{8 a x^8} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^3}{8 a x^8}+\frac {(-A b+4 a B) \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^4} \, dx,x,x^2\right )}{8 a}\\ &=-\frac {A \left (a+b x^2\right )^3}{8 a x^8}+\frac {(A b-4 a B) \left (a+b x^2\right )^3}{24 a^2 x^6}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 1.15 \[ -\frac {a^2 \left (3 A+4 B x^2\right )+4 a b x^2 \left (2 A+3 B x^2\right )+6 b^2 x^4 \left (A+2 B x^2\right )}{24 x^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 53, normalized size = 1.10 \[ -\frac {12 \, B b^{2} x^{6} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 55, normalized size = 1.15 \[ -\frac {12 \, B b^{2} x^{6} + 12 \, B a b x^{4} + 6 \, A b^{2} x^{4} + 4 \, B a^{2} x^{2} + 8 \, A a b x^{2} + 3 \, A a^{2}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 1.00 \[ -\frac {B \,b^{2}}{2 x^{2}}-\frac {\left (A b +2 B a \right ) b}{4 x^{4}}-\frac {A \,a^{2}}{8 x^{8}}-\frac {\left (2 A b +B a \right ) a}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 53, normalized size = 1.10 \[ -\frac {12 \, B b^{2} x^{6} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 53, normalized size = 1.10 \[ -\frac {x^2\,\left (\frac {B\,a^2}{6}+\frac {A\,b\,a}{3}\right )+x^4\,\left (\frac {A\,b^2}{4}+\frac {B\,a\,b}{2}\right )+\frac {A\,a^2}{8}+\frac {B\,b^2\,x^6}{2}}{x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 58, normalized size = 1.21 \[ \frac {- 3 A a^{2} - 12 B b^{2} x^{6} + x^{4} \left (- 6 A b^{2} - 12 B a b\right ) + x^{2} \left (- 8 A a b - 4 B a^{2}\right )}{24 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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