Optimal. Leaf size=55 \[ -\frac {A b^2}{5 x^5}-\frac {b (2 A c+b B)}{4 x^4}-\frac {c (A c+2 b B)}{3 x^3}-\frac {B c^2}{2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} -\frac {A b^2}{5 x^5}-\frac {b (2 A c+b B)}{4 x^4}-\frac {c (A c+2 b B)}{3 x^3}-\frac {B c^2}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{x^8} \, dx &=\int \left (\frac {A b^2}{x^6}+\frac {b (b B+2 A c)}{x^5}+\frac {c (2 b B+A c)}{x^4}+\frac {B c^2}{x^3}\right ) \, dx\\ &=-\frac {A b^2}{5 x^5}-\frac {b (b B+2 A c)}{4 x^4}-\frac {c (2 b B+A c)}{3 x^3}-\frac {B c^2}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.96 \begin {gather*} -\frac {2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )+5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )}{60 x^5} \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 (b x+c x^2\right )^2}{x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 51, normalized size = 0.93 \begin {gather*} -\frac {30 \, B c^{2} x^{3} + 12 \, A b^{2} + 20 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \, {\left (B b^{2} + 2 \, A b c\right )} x}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 51, normalized size = 0.93 \begin {gather*} -\frac {30 \, B c^{2} x^{3} + 40 \, B b c x^{2} + 20 \, A c^{2} x^{2} + 15 \, B b^{2} x + 30 \, A b c x + 12 \, A b^{2}}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 0.87 \begin {gather*} -\frac {B \,c^{2}}{2 x^{2}}-\frac {A \,b^{2}}{5 x^{5}}-\frac {\left (A c +2 b B \right ) c}{3 x^{3}}-\frac {\left (2 A c +b B \right ) b}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 51, normalized size = 0.93 \begin {gather*} -\frac {30 \, B c^{2} x^{3} + 12 \, A b^{2} + 20 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \, {\left (B b^{2} + 2 \, A b c\right )} x}{60 \, x^{5}} \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\,c^2}{3}+\frac {2\,B\,b\,c}{3}\right )+\frac {A\,b^2}{5}+x\,\left (\frac {B\,b^2}{4}+\frac {A\,c\,b}{2}\right )+\frac {B\,c^2\,x^3}{2}}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.86, size = 56, normalized size = 1.02 \begin {gather*} \frac {- 12 A b^{2} - 30 B c^{2} x^{3} + x^{2} \left (- 20 A c^{2} - 40 B b c\right ) + x \left (- 30 A b c - 15 B b^{2}\right )}{60 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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