Optimal. Leaf size=99 \[ -\frac{a^3 (a B+4 A b)}{7 x^7}-\frac{a^2 b (2 a B+3 A b)}{3 x^6}-\frac{a^4 A}{8 x^8}-\frac{2 a b^2 (3 a B+2 A b)}{5 x^5}-\frac{b^3 (4 a B+A b)}{4 x^4}-\frac{b^4 B}{3 x^3} \]
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Rubi [A] time = 0.0462386, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ -\frac{a^3 (a B+4 A b)}{7 x^7}-\frac{a^2 b (2 a B+3 A b)}{3 x^6}-\frac{a^4 A}{8 x^8}-\frac{2 a b^2 (3 a B+2 A b)}{5 x^5}-\frac{b^3 (4 a B+A b)}{4 x^4}-\frac{b^4 B}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^9} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{x^9} \, dx\\ &=\int \left (\frac{a^4 A}{x^9}+\frac{a^3 (4 A b+a B)}{x^8}+\frac{2 a^2 b (3 A b+2 a B)}{x^7}+\frac{2 a b^2 (2 A b+3 a B)}{x^6}+\frac{b^3 (A b+4 a B)}{x^5}+\frac{b^4 B}{x^4}\right ) \, dx\\ &=-\frac{a^4 A}{8 x^8}-\frac{a^3 (4 A b+a B)}{7 x^7}-\frac{a^2 b (3 A b+2 a B)}{3 x^6}-\frac{2 a b^2 (2 A b+3 a B)}{5 x^5}-\frac{b^3 (A b+4 a B)}{4 x^4}-\frac{b^4 B}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0257552, size = 88, normalized size = 0.89 \[ -\frac{168 a^2 b^2 x^2 (5 A+6 B x)+80 a^3 b x (6 A+7 B x)+15 a^4 (7 A+8 B x)+168 a b^3 x^3 (4 A+5 B x)+70 b^4 x^4 (3 A+4 B x)}{840 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 88, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{4}}{8\,{x}^{8}}}-{\frac{{a}^{3} \left ( 4\,Ab+aB \right ) }{7\,{x}^{7}}}-{\frac{{a}^{2}b \left ( 3\,Ab+2\,aB \right ) }{3\,{x}^{6}}}-{\frac{2\,a{b}^{2} \left ( 2\,Ab+3\,aB \right ) }{5\,{x}^{5}}}-{\frac{{b}^{3} \left ( Ab+4\,aB \right ) }{4\,{x}^{4}}}-{\frac{{b}^{4}B}{3\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994885, size = 134, normalized size = 1.35 \begin{align*} -\frac{280 \, B b^{4} x^{5} + 105 \, A a^{4} + 210 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 336 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 280 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 120 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{840 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28031, size = 230, normalized size = 2.32 \begin{align*} -\frac{280 \, B b^{4} x^{5} + 105 \, A a^{4} + 210 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 336 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 280 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 120 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{840 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.65636, size = 102, normalized size = 1.03 \begin{align*} - \frac{105 A a^{4} + 280 B b^{4} x^{5} + x^{4} \left (210 A b^{4} + 840 B a b^{3}\right ) + x^{3} \left (672 A a b^{3} + 1008 B a^{2} b^{2}\right ) + x^{2} \left (840 A a^{2} b^{2} + 560 B a^{3} b\right ) + x \left (480 A a^{3} b + 120 B a^{4}\right )}{840 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16342, size = 134, normalized size = 1.35 \begin{align*} -\frac{280 \, B b^{4} x^{5} + 840 \, B a b^{3} x^{4} + 210 \, A b^{4} x^{4} + 1008 \, B a^{2} b^{2} x^{3} + 672 \, A a b^{3} x^{3} + 560 \, B a^{3} b x^{2} + 840 \, A a^{2} b^{2} x^{2} + 120 \, B a^{4} x + 480 \, A a^{3} b x + 105 \, A a^{4}}{840 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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