Optimal. Leaf size=38 \[ \frac{c^4 (a-b x)^6}{6 b}-\frac{2 a c^4 (a-b x)^5}{5 b} \]
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Rubi [A] time = 0.0114237, 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} \[ \frac{c^4 (a-b x)^6}{6 b}-\frac{2 a c^4 (a-b x)^5}{5 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x) (a c-b c x)^4 \, dx &=\int \left (2 a (a c-b c x)^4-\frac{(a c-b c x)^5}{c}\right ) \, dx\\ &=-\frac{2 a c^4 (a-b x)^5}{5 b}+\frac{c^4 (a-b x)^6}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0027008, size = 68, normalized size = 1.79 \[ c^4 \left (\frac{1}{2} a^2 b^3 x^4+\frac{2}{3} a^3 b^2 x^3-\frac{3}{2} a^4 b x^2+a^5 x-\frac{3}{5} a b^4 x^5+\frac{b^5 x^6}{6}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 73, normalized size = 1.9 \begin{align*}{\frac{{b}^{5}{c}^{4}{x}^{6}}{6}}-{\frac{3\,a{b}^{4}{c}^{4}{x}^{5}}{5}}+{\frac{{a}^{2}{b}^{3}{c}^{4}{x}^{4}}{2}}+{\frac{2\,{a}^{3}{b}^{2}{c}^{4}{x}^{3}}{3}}-{\frac{3\,{a}^{4}b{c}^{4}{x}^{2}}{2}}+{a}^{5}{c}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05802, size = 97, normalized size = 2.55 \begin{align*} \frac{1}{6} \, b^{5} c^{4} x^{6} - \frac{3}{5} \, a b^{4} c^{4} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{4} x^{4} + \frac{2}{3} \, a^{3} b^{2} c^{4} x^{3} - \frac{3}{2} \, a^{4} b c^{4} x^{2} + a^{5} c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73831, size = 153, normalized size = 4.03 \begin{align*} \frac{1}{6} x^{6} c^{4} b^{5} - \frac{3}{5} x^{5} c^{4} b^{4} a + \frac{1}{2} x^{4} c^{4} b^{3} a^{2} + \frac{2}{3} x^{3} c^{4} b^{2} a^{3} - \frac{3}{2} x^{2} c^{4} b a^{4} + x c^{4} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.079036, size = 82, normalized size = 2.16 \begin{align*} a^{5} c^{4} x - \frac{3 a^{4} b c^{4} x^{2}}{2} + \frac{2 a^{3} b^{2} c^{4} x^{3}}{3} + \frac{a^{2} b^{3} c^{4} x^{4}}{2} - \frac{3 a b^{4} c^{4} x^{5}}{5} + \frac{b^{5} c^{4} x^{6}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21504, size = 97, normalized size = 2.55 \begin{align*} \frac{1}{6} \, b^{5} c^{4} x^{6} - \frac{3}{5} \, a b^{4} c^{4} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{4} x^{4} + \frac{2}{3} \, a^{3} b^{2} c^{4} x^{3} - \frac{3}{2} \, a^{4} b c^{4} x^{2} + a^{5} c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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