Optimal. Leaf size=50 \[ -\frac {a^4 c^3}{5 x^5}+\frac {a^3 b c^3}{2 x^4}-\frac {a b^3 c^3}{x^2}+\frac {b^4 c^3}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 50, 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} \[ \frac {a^3 b c^3}{2 x^4}-\frac {a^4 c^3}{5 x^5}-\frac {a b^3 c^3}{x^2}+\frac {b^4 c^3}{x} \]
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx &=\int \left (\frac {a^4 c^3}{x^6}-\frac {2 a^3 b c^3}{x^5}+\frac {2 a b^3 c^3}{x^3}-\frac {b^4 c^3}{x^2}\right ) \, dx\\ &=-\frac {a^4 c^3}{5 x^5}+\frac {a^3 b c^3}{2 x^4}-\frac {a b^3 c^3}{x^2}+\frac {b^4 c^3}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.84 \[ c^3 \left (-\frac {a^4}{5 x^5}+\frac {a^3 b}{2 x^4}-\frac {a b^3}{x^2}+\frac {b^4}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 47, normalized size = 0.94 \[ \frac {10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 47, normalized size = 0.94 \[ \frac {10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 0.78 \[ \left (\frac {b^{4}}{x}-\frac {a \,b^{3}}{x^{2}}+\frac {a^{3} b}{2 x^{4}}-\frac {a^{4}}{5 x^{5}}\right ) c^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 47, normalized size = 0.94 \[ \frac {10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 46, normalized size = 0.92 \[ -\frac {\frac {a^4\,c^3}{5}-\frac {a^3\,b\,c^3\,x}{2}+a\,b^3\,c^3\,x^3-b^4\,c^3\,x^4}{x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 51, normalized size = 1.02 \[ - \frac {2 a^{4} c^{3} - 5 a^{3} b c^{3} x + 10 a b^{3} c^{3} x^{3} - 10 b^{4} c^{3} x^{4}}{10 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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