Optimal. Leaf size=32 \[ \frac{2 a c (a+b x)^3}{3 b}-\frac{c (a+b x)^4}{4 b} \]
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Rubi [A] time = 0.0149207, antiderivative size = 32, 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} \[ \frac{2 a c (a+b x)^3}{3 b}-\frac{c (a+b x)^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^2 (a c-b c x) \, dx &=\int \left (2 a c (a+b x)^2-c (a+b x)^3\right ) \, dx\\ &=\frac{2 a c (a+b x)^3}{3 b}-\frac{c (a+b x)^4}{4 b}\\ \end{align*}
Mathematica [A] time = 0.001673, size = 40, normalized size = 1.25 \[ c \left (\frac{1}{2} a^2 b x^2+a^3 x-\frac{1}{3} a b^2 x^3-\frac{1}{4} b^3 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 37, normalized size = 1.2 \begin{align*} -{\frac{{b}^{3}c{x}^{4}}{4}}-{\frac{a{b}^{2}c{x}^{3}}{3}}+{\frac{{a}^{2}bc{x}^{2}}{2}}+{a}^{3}cx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01329, size = 49, normalized size = 1.53 \begin{align*} -\frac{1}{4} \, b^{3} c x^{4} - \frac{1}{3} \, a b^{2} c x^{3} + \frac{1}{2} \, a^{2} b c x^{2} + a^{3} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35025, size = 84, normalized size = 2.62 \begin{align*} -\frac{1}{4} x^{4} c b^{3} - \frac{1}{3} x^{3} c b^{2} a + \frac{1}{2} x^{2} c b a^{2} + x c a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.066783, size = 39, normalized size = 1.22 \begin{align*} a^{3} c x + \frac{a^{2} b c x^{2}}{2} - \frac{a b^{2} c x^{3}}{3} - \frac{b^{3} c x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05419, size = 49, normalized size = 1.53 \begin{align*} -\frac{1}{4} \, b^{3} c x^{4} - \frac{1}{3} \, a b^{2} c x^{3} + \frac{1}{2} \, a^{2} b c x^{2} + a^{3} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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