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 = 0.0187421, antiderivative size = 38, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0113416, 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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Maple [A] time = 0.005, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{\frac{2\,a}{3\,b \left ( bx-a \right ) ^{3}}}-{\frac{1}{2\,b \left ( bx-a \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00579, size = 73, normalized size = 1.92 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48177, size = 108, normalized size = 2.84 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.486193, size = 56, normalized size = 1.47 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05926, size = 31, normalized size = 0.82 \begin{align*} -\frac{3 \, b x + a}{6 \,{\left (b x - a\right )}^{3} b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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