Optimal. Leaf size=38 \[ \frac {2 a}{3 b c^4 (a-b x)^3}-\frac {1}{2 b c^4 (a-b x)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \[ \frac {2 a}{3 b c^4 (a-b x)^3}-\frac {1}{2 b c^4 (a-b x)^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {a+b x}{(a c-b c x)^4} \, dx &=\int \left (\frac {2 a}{c^4 (a-b x)^4}-\frac {1}{c^4 (a-b x)^3}\right ) \, dx\\ &=\frac {2 a}{3 b c^4 (a-b x)^3}-\frac {1}{2 b c^4 (a-b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.66 \[ -\frac {a+3 b x}{6 b c^4 (b x-a)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 54, normalized size = 1.42 \[ -\frac {3 \, b x + a}{6 \, {\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 23, normalized size = 0.61 \[ -\frac {3 \, b x + a}{6 \, {\left (b x - a\right )}^{3} b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.92 \[ \frac {-\frac {2 a}{3 \left (b x -a \right )^{3} b}-\frac {1}{2 \left (b x -a \right )^{2} b}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 54, normalized size = 1.42 \[ -\frac {3 \, b x + a}{6 \, {\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 54, normalized size = 1.42 \[ \frac {\frac {x}{2}+\frac {a}{6\,b}}{a^3\,c^4-3\,a^2\,b\,c^4\,x+3\,a\,b^2\,c^4\,x^2-b^3\,c^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 56, normalized size = 1.47 \[ \frac {- a - 3 b x}{- 6 a^{3} b c^{4} + 18 a^{2} b^{2} c^{4} x - 18 a b^{3} c^{4} x^{2} + 6 b^{4} c^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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