Optimal. Leaf size=38 \[ \frac{a}{2 b c^5 (a-b x)^4}-\frac{1}{3 b c^5 (a-b x)^3} \]
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Rubi [A] time = 0.018712, 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} \[ \frac{a}{2 b c^5 (a-b x)^4}-\frac{1}{3 b c^5 (a-b x)^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{(a c-b c x)^5} \, dx &=\int \left (\frac{2 a}{c^5 (a-b x)^5}-\frac{1}{c^5 (a-b x)^4}\right ) \, dx\\ &=\frac{a}{2 b c^5 (a-b x)^4}-\frac{1}{3 b c^5 (a-b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0105647, size = 24, normalized size = 0.63 \[ \frac{a+2 b x}{6 b c^5 (a-b x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{a}{2\,b \left ( bx-a \right ) ^{4}}}+{\frac{1}{3\,b \left ( bx-a \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01628, size = 90, normalized size = 2.37 \begin{align*} \frac{2 \, b x + a}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45573, size = 134, normalized size = 3.53 \begin{align*} \frac{2 \, b x + a}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.52499, size = 70, normalized size = 1.84 \begin{align*} \frac{a + 2 b x}{6 a^{4} b c^{5} - 24 a^{3} b^{2} c^{5} x + 36 a^{2} b^{3} c^{5} x^{2} - 24 a b^{4} c^{5} x^{3} + 6 b^{5} c^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09206, size = 54, normalized size = 1.42 \begin{align*} \frac{a}{2 \,{\left (b c x - a c\right )}^{4} b c} + \frac{1}{3 \,{\left (b c x - a c\right )}^{3} b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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