Optimal. Leaf size=83 \[ -\frac {4 a^2 (a c-b c x)^{n+1}}{b c (n+1)}-\frac {(a c-b c x)^{n+3}}{b c^3 (n+3)}+\frac {4 a (a c-b c x)^{n+2}}{b c^2 (n+2)} \]
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Rubi [A] time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {43} \[ -\frac {4 a^2 (a c-b c x)^{n+1}}{b c (n+1)}+\frac {4 a (a c-b c x)^{n+2}}{b c^2 (n+2)}-\frac {(a c-b c x)^{n+3}}{b c^3 (n+3)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x)^2 (a c-b c x)^n \, dx &=\int \left (4 a^2 (a c-b c x)^n-\frac {4 a (a c-b c x)^{1+n}}{c}+\frac {(a c-b c x)^{2+n}}{c^2}\right ) \, dx\\ &=-\frac {4 a^2 (a c-b c x)^{1+n}}{b c (1+n)}+\frac {4 a (a c-b c x)^{2+n}}{b c^2 (2+n)}-\frac {(a c-b c x)^{3+n}}{b c^3 (3+n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 77, normalized size = 0.93 \[ \frac {(b x-a) \left (a^2 \left (n^2+7 n+14\right )+2 a b \left (n^2+5 n+4\right ) x+b^2 \left (n^2+3 n+2\right ) x^2\right ) (c (a-b x))^n}{b (n+1) (n+2) (n+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 128, normalized size = 1.54 \[ -\frac {{\left (a^{3} n^{2} + 7 \, a^{3} n - {\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3} + 14 \, a^{3} - {\left (a b^{2} n^{2} + 7 \, a b^{2} n + 6 \, a b^{2}\right )} x^{2} + {\left (a^{2} b n^{2} + 3 \, a^{2} b n - 6 \, a^{2} b\right )} x\right )} {\left (-b c x + a c\right )}^{n}}{b n^{3} + 6 \, b n^{2} + 11 \, b n + 6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.15, size = 256, normalized size = 3.08 \[ \frac {{\left (-b c x + a c\right )}^{n} b^{3} n^{2} x^{3} + {\left (-b c x + a c\right )}^{n} a b^{2} n^{2} x^{2} + 3 \, {\left (-b c x + a c\right )}^{n} b^{3} n x^{3} - {\left (-b c x + a c\right )}^{n} a^{2} b n^{2} x + 7 \, {\left (-b c x + a c\right )}^{n} a b^{2} n x^{2} + 2 \, {\left (-b c x + a c\right )}^{n} b^{3} x^{3} - {\left (-b c x + a c\right )}^{n} a^{3} n^{2} - 3 \, {\left (-b c x + a c\right )}^{n} a^{2} b n x + 6 \, {\left (-b c x + a c\right )}^{n} a b^{2} x^{2} - 7 \, {\left (-b c x + a c\right )}^{n} a^{3} n + 6 \, {\left (-b c x + a c\right )}^{n} a^{2} b x - 14 \, {\left (-b c x + a c\right )}^{n} a^{3}}{b n^{3} + 6 \, b n^{2} + 11 \, b n + 6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 103, normalized size = 1.24 \[ -\frac {\left (-b x +a \right ) \left (b^{2} n^{2} x^{2}+2 a b \,n^{2} x +3 b^{2} n \,x^{2}+a^{2} n^{2}+10 a b n x +2 b^{2} x^{2}+7 a^{2} n +8 a b x +14 a^{2}\right ) \left (-b c x +a c \right )^{n}}{\left (n^{3}+6 n^{2}+11 n +6\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.53, size = 167, normalized size = 2.01 \[ \frac {2 \, {\left (b^{2} c^{n} {\left (n + 1\right )} x^{2} - a b c^{n} n x - a^{2} c^{n}\right )} {\left (-b x + a\right )}^{n} a}{{\left (n^{2} + 3 \, n + 2\right )} b} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} c^{n} x^{3} - {\left (n^{2} + n\right )} a b^{2} c^{n} x^{2} - 2 \, a^{2} b c^{n} n x - 2 \, a^{3} c^{n}\right )} {\left (-b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b} - \frac {{\left (-b c x + a c\right )}^{n + 1} a^{2}}{b c {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 133, normalized size = 1.60 \[ -{\left (a\,c-b\,c\,x\right )}^n\,\left (\frac {a^2\,x\,\left (n^2+3\,n-6\right )}{n^3+6\,n^2+11\,n+6}+\frac {a^3\,\left (n^2+7\,n+14\right )}{b\,\left (n^3+6\,n^2+11\,n+6\right )}-\frac {b^2\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}-\frac {a\,b\,x^2\,\left (n^2+7\,n+6\right )}{n^3+6\,n^2+11\,n+6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.30, size = 819, normalized size = 9.87 \[ \begin {cases} a^{2} x \left (a c\right )^{n} & \text {for}\: b = 0 \\- \frac {a^{2} \log {\left (- \frac {a}{b} + x \right )}}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} - \frac {2 a^{2}}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} + \frac {2 a b x \log {\left (- \frac {a}{b} + x \right )}}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} + \frac {4 a b x}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} - \frac {b^{2} x^{2} \log {\left (- \frac {a}{b} + x \right )}}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} & \text {for}\: n = -3 \\- \frac {4 a^{2} \log {\left (- \frac {a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} - \frac {5 a^{2}}{- a b c^{2} + b^{2} c^{2} x} + \frac {4 a b x \log {\left (- \frac {a}{b} + x \right )}}{- a b c^{2} + b^{2} c^{2} x} + \frac {b^{2} x^{2}}{- a b c^{2} + b^{2} c^{2} x} & \text {for}\: n = -2 \\- \frac {4 a^{2} \log {\left (- \frac {a}{b} + x \right )}}{b c} - \frac {3 a x}{c} - \frac {b x^{2}}{2 c} & \text {for}\: n = -1 \\- \frac {a^{3} n^{2} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} - \frac {7 a^{3} n \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} - \frac {14 a^{3} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} - \frac {a^{2} b n^{2} x \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} - \frac {3 a^{2} b n x \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {6 a^{2} b x \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {a b^{2} n^{2} x^{2} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {7 a b^{2} n x^{2} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {6 a b^{2} x^{2} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {b^{3} n^{2} x^{3} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {3 b^{3} n x^{3} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} + \frac {2 b^{3} x^{3} \left (a c - b c x\right )^{n}}{b n^{3} + 6 b n^{2} + 11 b n + 6 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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