Optimal. Leaf size=87 \[ -\frac{a^3 b^2 c^4}{3 x^6}-\frac{2 a^2 b^3 c^4}{5 x^5}+\frac{3 a^4 b c^4}{7 x^7}-\frac{a^5 c^4}{8 x^8}+\frac{3 a b^4 c^4}{4 x^4}-\frac{b^5 c^4}{3 x^3} \]
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Rubi [A] time = 0.0353926, antiderivative size = 87, 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 = {75} \[ -\frac{a^3 b^2 c^4}{3 x^6}-\frac{2 a^2 b^3 c^4}{5 x^5}+\frac{3 a^4 b c^4}{7 x^7}-\frac{a^5 c^4}{8 x^8}+\frac{3 a b^4 c^4}{4 x^4}-\frac{b^5 c^4}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin{align*} \int \frac{(a+b x) (a c-b c x)^4}{x^9} \, dx &=\int \left (\frac{a^5 c^4}{x^9}-\frac{3 a^4 b c^4}{x^8}+\frac{2 a^3 b^2 c^4}{x^7}+\frac{2 a^2 b^3 c^4}{x^6}-\frac{3 a b^4 c^4}{x^5}+\frac{b^5 c^4}{x^4}\right ) \, dx\\ &=-\frac{a^5 c^4}{8 x^8}+\frac{3 a^4 b c^4}{7 x^7}-\frac{a^3 b^2 c^4}{3 x^6}-\frac{2 a^2 b^3 c^4}{5 x^5}+\frac{3 a b^4 c^4}{4 x^4}-\frac{b^5 c^4}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0081894, size = 87, normalized size = 1. \[ -\frac{a^3 b^2 c^4}{3 x^6}-\frac{2 a^2 b^3 c^4}{5 x^5}+\frac{3 a^4 b c^4}{7 x^7}-\frac{a^5 c^4}{8 x^8}+\frac{3 a b^4 c^4}{4 x^4}-\frac{b^5 c^4}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 62, normalized size = 0.7 \begin{align*}{c}^{4} \left ( -{\frac{{b}^{5}}{3\,{x}^{3}}}-{\frac{2\,{a}^{2}{b}^{3}}{5\,{x}^{5}}}+{\frac{3\,a{b}^{4}}{4\,{x}^{4}}}-{\frac{{a}^{5}}{8\,{x}^{8}}}-{\frac{{a}^{3}{b}^{2}}{3\,{x}^{6}}}+{\frac{3\,{a}^{4}b}{7\,{x}^{7}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02743, size = 101, normalized size = 1.16 \begin{align*} -\frac{280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94555, size = 170, normalized size = 1.95 \begin{align*} -\frac{280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.668314, size = 82, normalized size = 0.94 \begin{align*} - \frac{105 a^{5} c^{4} - 360 a^{4} b c^{4} x + 280 a^{3} b^{2} c^{4} x^{2} + 336 a^{2} b^{3} c^{4} x^{3} - 630 a b^{4} c^{4} x^{4} + 280 b^{5} c^{4} x^{5}}{840 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30532, size = 101, normalized size = 1.16 \begin{align*} -\frac{280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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