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 = 0.027832, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ -\frac{A b^2}{5 x^5}-\frac{b (2 A c+b B)}{4 x^4}-\frac{c (A c+2 b B)}{3 x^3}-\frac{B c^2}{2 x^2} \]
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*}
Mathematica [A] time = 0.0145733, size = 53, normalized size = 0.96 \[ -\frac{2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )+5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )}{60 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 48, normalized size = 0.9 \begin{align*} -{\frac{A{b}^{2}}{5\,{x}^{5}}}-{\frac{b \left ( 2\,Ac+bB \right ) }{4\,{x}^{4}}}-{\frac{c \left ( Ac+2\,bB \right ) }{3\,{x}^{3}}}-{\frac{B{c}^{2}}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05425, size = 69, normalized size = 1.25 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64742, size = 120, normalized size = 2.18 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.10666, size = 54, normalized size = 0.98 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1173, size = 69, normalized size = 1.25 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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