Optimal. Leaf size=38 \[ -\frac {2 a c^2 (a-b x)^3}{3 b}+\frac {c^2 (a-b x)^4}{4 b} \]
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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 = {45}
\begin {gather*} \frac {c^2 (a-b x)^4}{4 b}-\frac {2 a c^2 (a-b x)^3}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
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*}
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Mathematica [A]
time = 0.00, size = 42, normalized size = 1.11 \begin {gather*} c^2 \left (a^3 x-\frac {1}{2} a^2 b x^2-\frac {1}{3} a b^2 x^3+\frac {b^3 x^4}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 45, normalized size = 1.18
method | result | size |
gosper | \(\frac {x \left (3 b^{3} x^{3}-4 a \,b^{2} x^{2}-6 a^{2} b x +12 a^{3}\right ) c^{2}}{12}\) | \(37\) |
default | \(\frac {1}{4} b^{3} c^{2} x^{4}-\frac {1}{3} a \,b^{2} c^{2} x^{3}-\frac {1}{2} a^{2} b \,c^{2} x^{2}+a^{3} c^{2} x\) | \(45\) |
norman | \(\frac {1}{4} b^{3} c^{2} x^{4}-\frac {1}{3} a \,b^{2} c^{2} x^{3}-\frac {1}{2} a^{2} b \,c^{2} x^{2}+a^{3} c^{2} x\) | \(45\) |
risch | \(\frac {1}{4} b^{3} c^{2} x^{4}-\frac {1}{3} a \,b^{2} c^{2} x^{3}-\frac {1}{2} a^{2} b \,c^{2} x^{2}+a^{3} c^{2} x\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 44, normalized size = 1.16 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 44, normalized size = 1.16 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 46, normalized size = 1.21 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.53, size = 44, normalized size = 1.16 \begin {gather*} \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 {gather*}
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 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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