Optimal. Leaf size=99 \[ -\frac {a^4 A}{8 x^8}-\frac {a^3 (a B+4 A b)}{7 x^7}-\frac {a^2 b (2 a B+3 A b)}{3 x^6}-\frac {b^3 (4 a B+A b)}{4 x^4}-\frac {2 a b^2 (3 a B+2 A b)}{5 x^5}-\frac {b^4 B}{3 x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}
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*}
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Mathematica [A] time = 0.03, size = 88, normalized size = 0.89 \begin {gather*} -\frac {15 a^4 (7 A+8 B x)+80 a^3 b x (6 A+7 B x)+168 a^2 b^2 x^2 (5 A+6 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} \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 {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^9} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 99, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 99, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 88, normalized size = 0.89 \begin {gather*} -\frac {B \,b^{4}}{3 x^{3}}-\frac {\left (A b +4 B a \right ) b^{3}}{4 x^{4}}-\frac {2 \left (2 A b +3 B a \right ) a \,b^{2}}{5 x^{5}}-\frac {A \,a^{4}}{8 x^{8}}-\frac {\left (3 A b +2 B a \right ) a^{2} b}{3 x^{6}}-\frac {\left (4 A b +B a \right ) a^{3}}{7 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 99, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 95, normalized size = 0.96 \begin {gather*} -\frac {x\,\left (\frac {B\,a^4}{7}+\frac {4\,A\,b\,a^3}{7}\right )+\frac {A\,a^4}{8}+x^2\,\left (\frac {2\,B\,a^3\,b}{3}+A\,a^2\,b^2\right )+x^3\,\left (\frac {6\,B\,a^2\,b^2}{5}+\frac {4\,A\,a\,b^3}{5}\right )+x^4\,\left (\frac {A\,b^4}{4}+B\,a\,b^3\right )+\frac {B\,b^4\,x^5}{3}}{x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.71, size = 107, normalized size = 1.08 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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