Optimal. Leaf size=53 \[ -\frac {2 a (a c-b c x)^{1+n}}{b c (1+n)}+\frac {(a c-b c x)^{2+n}}{b c^2 (2+n)} \]
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Rubi [A]
time = 0.01, antiderivative size = 53, 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 {(a c-b c x)^{n+2}}{b c^2 (n+2)}-\frac {2 a (a c-b c x)^{n+1}}{b c (n+1)} \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)^n \, dx &=\int \left (2 a (a c-b c x)^n-\frac {(a c-b c x)^{1+n}}{c}\right ) \, dx\\ &=-\frac {2 a (a c-b c x)^{1+n}}{b c (1+n)}+\frac {(a c-b c x)^{2+n}}{b c^2 (2+n)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 43, normalized size = 0.81 \begin {gather*} \frac {(c (a-b x))^n (-a+b x) (a (3+n)+b (1+n) x)}{b (1+n) (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.80, size = 256, normalized size = 4.83 \begin {gather*} \text {Piecewise}\left [\left \{\left \{a x \left (a c\right )^n,b\text {==}0\right \},\left \{\frac {a \left (2+\text {Log}\left [\frac {-a+b x}{b}\right ]\right )-b x \text {Log}\left [\frac {-a+b x}{b}\right ]}{b c^2 \left (a-b x\right )},n\text {==}-2\right \},\left \{\frac {-2 a \text {Log}\left [-\frac {a}{b}+x\right ]-b x}{b c},n\text {==}-1\right \}\right \},\frac {-3 a^2 \left (a c-b c x\right )^n}{2 b+3 b n+b n^2}-\frac {a^2 n \left (a c-b c x\right )^n}{2 b+3 b n+b n^2}+\frac {2 a b x \left (a c-b c x\right )^n}{2 b+3 b n+b n^2}+\frac {b^2 x^2 \left (a c-b c x\right )^n}{2 b+3 b n+b n^2}+\frac {b^2 n x^2 \left (a c-b c x\right )^n}{2 b+3 b n+b n^2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 47, normalized size = 0.89
method | result | size |
gosper | \(-\frac {\left (-b c x +a c \right )^{n} \left (b n x +a n +b x +3 a \right ) \left (-b x +a \right )}{b \left (n^{2}+3 n +2\right )}\) | \(47\) |
risch | \(-\frac {\left (-b^{2} n \,x^{2}-x^{2} b^{2}+a^{2} n -2 a b x +3 a^{2}\right ) \left (c \left (-b x +a \right )\right )^{n}}{\left (2+n \right ) \left (1+n \right ) b}\) | \(59\) |
norman | \(\frac {b \,x^{2} {\mathrm e}^{n \ln \left (-b c x +a c \right )}}{2+n}+\frac {2 a x \,{\mathrm e}^{n \ln \left (-b c x +a c \right )}}{n^{2}+3 n +2}-\frac {a^{2} \left (3+n \right ) {\mathrm e}^{n \ln \left (-b c x +a c \right )}}{b \left (n^{2}+3 n +2\right )}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 81, normalized size = 1.53 \begin {gather*} \frac {{\left (b^{2} c^{n} {\left (n + 1\right )} x^{2} - a b c^{n} n x - a^{2} c^{n}\right )} {\left (-b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b} - \frac {{\left (-b c x + a c\right )}^{n + 1} a}{b c {\left (n + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 58, normalized size = 1.09 \begin {gather*} -\frac {{\left (a^{2} n - 2 \, a b x - {\left (b^{2} n + b^{2}\right )} x^{2} + 3 \, a^{2}\right )} {\left (-b c x + a c\right )}^{n}}{b n^{2} + 3 \, b n + 2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 245, normalized size = 4.62 \begin {gather*} \begin {cases} a x \left (a c\right )^{n} & \text {for}\: b = 0 \\- \frac {a \log {\left (- \frac {a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} - \frac {2 a}{- a b c^{2} + b^{2} c^{2} x} + \frac {b x \log {\left (- \frac {a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} & \text {for}\: n = -2 \\- \frac {2 a \log {\left (- \frac {a}{b} + x \right )}}{b c} - \frac {x}{c} & \text {for}\: n = -1 \\- \frac {a^{2} n \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} - \frac {3 a^{2} \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} + \frac {2 a b x \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} + \frac {b^{2} n x^{2} \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} + \frac {b^{2} x^{2} \left (a c - b c x\right )^{n}}{b n^{2} + 3 b n + 2 b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 113, normalized size = 2.13 \begin {gather*} \frac {-a^{2} n \mathrm {e}^{n \ln \left (a c-b c x\right )}-3 a^{2} \mathrm {e}^{n \ln \left (a c-b c x\right )}+2 a b x \mathrm {e}^{n \ln \left (a c-b c x\right )}+b^{2} n x^{2} \mathrm {e}^{n \ln \left (a c-b c x\right )}+b^{2} x^{2} \mathrm {e}^{n \ln \left (a c-b c x\right )}}{b n^{2}+3 b n+2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 66, normalized size = 1.25 \begin {gather*} {\left (a\,c-b\,c\,x\right )}^n\,\left (\frac {2\,a\,x}{n^2+3\,n+2}-\frac {a^2\,\left (n+3\right )}{b\,\left (n^2+3\,n+2\right )}+\frac {b\,x^2\,\left (n+1\right )}{n^2+3\,n+2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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