Optimal. Leaf size=38 \[ \frac {c^3 (a-b x)^5}{5 b}-\frac {a c^3 (a-b x)^4}{2 b} \]
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Rubi [A] time = 0.01, antiderivative size = 38, 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} \[ \frac {c^3 (a-b x)^5}{5 b}-\frac {a c^3 (a-b x)^4}{2 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x) (a c-b c x)^3 \, dx &=\int \left (2 a (a c-b c x)^3-\frac {(a c-b c x)^4}{c}\right ) \, dx\\ &=-\frac {a c^3 (a-b x)^4}{2 b}+\frac {c^3 (a-b x)^5}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 40, normalized size = 1.05 \[ c^3 \left (a^4 x-a^3 b x^2+\frac {1}{2} a b^3 x^4-\frac {1}{5} b^4 x^5\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 44, normalized size = 1.16 \[ -\frac {1}{5} x^{5} c^{3} b^{4} + \frac {1}{2} x^{4} c^{3} b^{3} a - x^{2} c^{3} b a^{3} + x c^{3} a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 44, normalized size = 1.16 \[ -\frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 1.18 \[ -\frac {1}{5} b^{4} c^{3} x^{5}+\frac {1}{2} a \,b^{3} c^{3} x^{4}-a^{3} b \,c^{3} x^{2}+a^{4} c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 44, normalized size = 1.16 \[ -\frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 44, normalized size = 1.16 \[ a^4\,c^3\,x-a^3\,b\,c^3\,x^2+\frac {a\,b^3\,c^3\,x^4}{2}-\frac {b^4\,c^3\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 44, normalized size = 1.16 \[ a^{4} c^{3} x - a^{3} b c^{3} x^{2} + \frac {a b^{3} c^{3} x^{4}}{2} - \frac {b^{4} c^{3} x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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