Optimal. Leaf size=38 \[ -\frac{2}{3} a^2 b^2 c^2 x^3+a^4 c^2 x+\frac{1}{5} b^4 c^2 x^5 \]
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Rubi [A] time = 0.0171346, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {41, 194} \[ -\frac{2}{3} a^2 b^2 c^2 x^3+a^4 c^2 x+\frac{1}{5} b^4 c^2 x^5 \]
Antiderivative was successfully verified.
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Rule 41
Rule 194
Rubi steps
\begin{align*} \int (a+b x)^2 (a c-b c x)^2 \, dx &=\int \left (a^2 c-b^2 c x^2\right )^2 \, dx\\ &=\int \left (a^4 c^2-2 a^2 b^2 c^2 x^2+b^4 c^2 x^4\right ) \, dx\\ &=a^4 c^2 x-\frac{2}{3} a^2 b^2 c^2 x^3+\frac{1}{5} b^4 c^2 x^5\\ \end{align*}
Mathematica [A] time = 0.0016806, size = 38, normalized size = 1. \[ -\frac{2}{3} a^2 b^2 c^2 x^3+a^4 c^2 x+\frac{1}{5} b^4 c^2 x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 35, normalized size = 0.9 \begin{align*}{a}^{4}{c}^{2}x-{\frac{2\,{a}^{2}{b}^{2}{c}^{2}{x}^{3}}{3}}+{\frac{{b}^{4}{c}^{2}{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02374, size = 46, normalized size = 1.21 \begin{align*} \frac{1}{5} \, b^{4} c^{2} x^{5} - \frac{2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28935, size = 69, normalized size = 1.82 \begin{align*} \frac{1}{5} x^{5} c^{2} b^{4} - \frac{2}{3} x^{3} c^{2} b^{2} a^{2} + x c^{2} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.067727, size = 36, normalized size = 0.95 \begin{align*} a^{4} c^{2} x - \frac{2 a^{2} b^{2} c^{2} x^{3}}{3} + \frac{b^{4} c^{2} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0603, size = 46, normalized size = 1.21 \begin{align*} \frac{1}{5} \, b^{4} c^{2} x^{5} - \frac{2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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