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 = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 a c (a+b x)^3}{3 b}-\frac {c (a+b x)^4}{4 b} \end {gather*}
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*}
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Mathematica [A] time = 0.00, size = 40, normalized size = 1.25 \begin {gather*} c \left (a^3 x+\frac {1}{2} a^2 b x^2-\frac {1}{3} a b^2 x^3-\frac {1}{4} b^3 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^2 (a c-b c x) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.13, size = 36, normalized size = 1.12 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 36, normalized size = 1.12 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 1.16 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 36, normalized size = 1.12 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 36, normalized size = 1.12 \begin {gather*} c\,a^3\,x+\frac {c\,a^2\,b\,x^2}{2}-\frac {c\,a\,b^2\,x^3}{3}-\frac {c\,b^3\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 39, normalized size = 1.22 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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