Optimal. Leaf size=55 \[ -\frac {a^2 A}{7 x^7}-\frac {a (a B+2 A b)}{6 x^6}-\frac {b (2 a B+A b)}{5 x^5}-\frac {b^2 B}{4 x^4} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \begin {gather*} -\frac {a^2 A}{7 x^7}-\frac {a (a B+2 A b)}{6 x^6}-\frac {b (2 a B+A b)}{5 x^5}-\frac {b^2 B}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx &=\int \left (\frac {a^2 A}{x^8}+\frac {a (2 A b+a B)}{x^7}+\frac {b (A b+2 a B)}{x^6}+\frac {b^2 B}{x^5}\right ) \, dx\\ &=-\frac {a^2 A}{7 x^7}-\frac {a (2 A b+a B)}{6 x^6}-\frac {b (A b+2 a B)}{5 x^5}-\frac {b^2 B}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 0.91 \begin {gather*} -\frac {10 a^2 (6 A+7 B x)+28 a b x (5 A+6 B x)+21 b^2 x^2 (4 A+5 B x)}{420 x^7} \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)^2 (A+B x)}{x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.40, size = 51, normalized size = 0.93 \begin {gather*} -\frac {105 \, B b^{2} x^{3} + 60 \, A a^{2} + 84 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 70 \, {\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 51, normalized size = 0.93 \begin {gather*} -\frac {105 \, B b^{2} x^{3} + 168 \, B a b x^{2} + 84 \, A b^{2} x^{2} + 70 \, B a^{2} x + 140 \, A a b x + 60 \, A a^{2}}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.87 \begin {gather*} -\frac {B \,b^{2}}{4 x^{4}}-\frac {A \,a^{2}}{7 x^{7}}-\frac {\left (A b +2 B a \right ) b}{5 x^{5}}-\frac {\left (2 A b +B a \right ) a}{6 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 51, normalized size = 0.93 \begin {gather*} -\frac {105 \, B b^{2} x^{3} + 60 \, A a^{2} + 84 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 70 \, {\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 51, normalized size = 0.93 \begin {gather*} -\frac {x^2\,\left (\frac {A\,b^2}{5}+\frac {2\,B\,a\,b}{5}\right )+\frac {A\,a^2}{7}+x\,\left (\frac {B\,a^2}{6}+\frac {A\,b\,a}{3}\right )+\frac {B\,b^2\,x^3}{4}}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.44, size = 56, normalized size = 1.02 \begin {gather*} \frac {- 60 A a^{2} - 105 B b^{2} x^{3} + x^{2} \left (- 84 A b^{2} - 168 B a b\right ) + x \left (- 140 A a b - 70 B a^{2}\right )}{420 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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