Optimal. Leaf size=38 \[ \frac{c^2 (a-b x)^4}{4 b}-\frac{2 a c^2 (a-b x)^3}{3 b} \]
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Rubi [A] time = 0.0170622, 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} \[ \frac{c^2 (a-b x)^4}{4 b}-\frac{2 a c^2 (a-b x)^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x) (a c-b c x)^2 \, dx &=\int \left (2 a (a c-b c x)^2-\frac{(a c-b c x)^3}{c}\right ) \, dx\\ &=-\frac{2 a c^2 (a-b x)^3}{3 b}+\frac{c^2 (a-b x)^4}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0023617, size = 42, normalized size = 1.11 \[ c^2 \left (-\frac{1}{2} a^2 b x^2+a^3 x-\frac{1}{3} a b^2 x^3+\frac{b^3 x^4}{4}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 45, normalized size = 1.2 \begin{align*}{\frac{{b}^{3}{c}^{2}{x}^{4}}{4}}-{\frac{a{b}^{2}{c}^{2}{x}^{3}}{3}}-{\frac{{a}^{2}{c}^{2}b{x}^{2}}{2}}+{a}^{3}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00264, size = 59, normalized size = 1.55 \begin{align*} \frac{1}{4} \, b^{3} c^{2} x^{4} - \frac{1}{3} \, a b^{2} c^{2} x^{3} - \frac{1}{2} \, a^{2} b c^{2} x^{2} + a^{3} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31821, size = 93, normalized size = 2.45 \begin{align*} \frac{1}{4} x^{4} c^{2} b^{3} - \frac{1}{3} x^{3} c^{2} b^{2} a - \frac{1}{2} x^{2} c^{2} b a^{2} + x c^{2} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.065695, size = 46, normalized size = 1.21 \begin{align*} a^{3} c^{2} x - \frac{a^{2} b c^{2} x^{2}}{2} - \frac{a b^{2} c^{2} x^{3}}{3} + \frac{b^{3} c^{2} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05925, size = 59, normalized size = 1.55 \begin{align*} \frac{1}{4} \, b^{3} c^{2} x^{4} - \frac{1}{3} \, a b^{2} c^{2} x^{3} - \frac{1}{2} \, a^{2} b c^{2} x^{2} + a^{3} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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