Optimal. Leaf size=84 \[ -\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 84, 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 = {75} \[ -\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a^4 b c^4}{2 x^6}-\frac {a^5 c^4}{7 x^7}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \]
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^8} \, dx &=\int \left (\frac {a^5 c^4}{x^8}-\frac {3 a^4 b c^4}{x^7}+\frac {2 a^3 b^2 c^4}{x^6}+\frac {2 a^2 b^3 c^4}{x^5}-\frac {3 a b^4 c^4}{x^4}+\frac {b^5 c^4}{x^3}\right ) \, dx\\ &=-\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 84, normalized size = 1.00 \[ -\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 75, normalized size = 0.89 \[ -\frac {35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 75, normalized size = 0.89 \[ -\frac {35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 61, normalized size = 0.73 \[ \left (-\frac {b^{5}}{2 x^{2}}+\frac {a \,b^{4}}{x^{3}}-\frac {a^{2} b^{3}}{2 x^{4}}-\frac {2 a^{3} b^{2}}{5 x^{5}}+\frac {a^{4} b}{2 x^{6}}-\frac {a^{5}}{7 x^{7}}\right ) c^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 75, normalized size = 0.89 \[ -\frac {35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 75, normalized size = 0.89 \[ -\frac {\frac {a^5\,c^4}{7}-\frac {a^4\,b\,c^4\,x}{2}+\frac {2\,a^3\,b^2\,c^4\,x^2}{5}+\frac {a^2\,b^3\,c^4\,x^3}{2}-a\,b^4\,c^4\,x^4+\frac {b^5\,c^4\,x^5}{2}}{x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 80, normalized size = 0.95 \[ \frac {- 10 a^{5} c^{4} + 35 a^{4} b c^{4} x - 28 a^{3} b^{2} c^{4} x^{2} - 35 a^{2} b^{3} c^{4} x^{3} + 70 a b^{4} c^{4} x^{4} - 35 b^{5} c^{4} x^{5}}{70 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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