Optimal. Leaf size=83 \[ -\frac {4 a^2 (a c-b c x)^{1+n}}{b c (1+n)}+\frac {4 a (a c-b c x)^{2+n}}{b c^2 (2+n)}-\frac {(a c-b c x)^{3+n}}{b c^3 (3+n)} \]
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Rubi [A]
time = 0.02, 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 = {45}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
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 = 0.06, size = 77, normalized size = 0.93 \begin {gather*} \frac {(c (a-b x))^n (-a+b x) \left (a^2 \left (14+7 n+n^2\right )+2 a b \left (4+5 n+n^2\right ) x+b^2 \left (2+3 n+n^2\right ) x^2\right )}{b (1+n) (2+n) (3+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 = 4.95, size = 693, normalized size = 8.35 \begin {gather*} \text {Piecewise}\left [\left \{\left \{a^2 x \left (a c\right )^n,b\text {==}0\right \},\left \{\frac {-a^2 \left (2+\text {Log}\left [\frac {-a+b x}{b}\right ]\right )+b x \left (2 a \text {Log}\left [\frac {-a+b x}{b}\right ]+4 a-b x \text {Log}\left [\frac {-a+b x}{b}\right ]\right )}{b c^3 \left (a^2-2 a b x+b^2 x^2\right )},n\text {==}-3\right \},\left \{\frac {a^2 \left (5+4 \text {Log}\left [\frac {-a+b x}{b}\right ]\right )-b x \left (4 a \text {Log}\left [\frac {-a+b x}{b}\right ]+b x\right )}{b c^2 \left (a-b x\right )},n\text {==}-2\right \},\left \{\frac {-8 a^2 \text {Log}\left [\frac {-a+b x}{b}\right ]+b x \left (-6 a-b x\right )}{2 b c},n\text {==}-1\right \}\right \},\frac {-14 a^3 \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}-\frac {7 a^3 n \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}-\frac {a^3 n^2 \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}+\frac {6 a^2 b x \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}-\frac {3 a^2 b n x \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}-\frac {a^2 b n^2 x \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}+\frac {6 a b^2 x^2 \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}+\frac {7 a b^2 n x^2 \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}+\frac {a b^2 n^2 x^2 \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}+\frac {2 b^3 x^3 \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}+\frac {3 b^3 n x^3 \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}+\frac {b^3 n^2 x^3 \left (a c-b c x\right )^n}{6 b+11 b n+6 b n^2+b n^3}\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 103, normalized size = 1.24
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (b^{2} n^{2} x^{2}+2 a b \,n^{2} x +3 b^{2} n \,x^{2}+a^{2} n^{2}+10 a b n x +2 x^{2} b^{2}+7 a^{2} n +8 a b x +14 a^{2}\right ) \left (-b c x +a c \right )^{n}}{b \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(103\) |
risch | \(-\frac {\left (-b^{3} n^{2} x^{3}-a \,b^{2} n^{2} x^{2}-3 n \,b^{3} x^{3}+a^{2} b \,n^{2} x -7 a \,b^{2} n \,x^{2}-2 b^{3} x^{3}+a^{3} n^{2}+3 a^{2} b n x -6 a \,b^{2} x^{2}+7 a^{3} n -6 a^{2} b x +14 a^{3}\right ) \left (c \left (-b x +a \right )\right )^{n}}{\left (2+n \right ) \left (3+n \right ) b \left (1+n \right )}\) | \(133\) |
norman | \(\frac {b^{2} x^{3} {\mathrm e}^{n \ln \left (-b c x +a c \right )}}{3+n}+\frac {a b \left (6+n \right ) x^{2} {\mathrm e}^{n \ln \left (-b c x +a c \right )}}{n^{2}+5 n +6}-\frac {a^{2} \left (n^{2}+3 n -6\right ) x \,{\mathrm e}^{n \ln \left (-b c x +a c \right )}}{n^{3}+6 n^{2}+11 n +6}-\frac {a^{3} \left (n^{2}+7 n +14\right ) {\mathrm e}^{n \ln \left (-b c x +a c \right )}}{b \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs.
\(2 (83) = 166\).
time = 0.27, size = 167, normalized size = 2.01 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 128, normalized size = 1.54 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 819, normalized size = 9.87 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 256 vs.
\(2 (83) = 166\).
time = 0.00, size = 282, normalized size = 3.40 \begin {gather*} \frac {-a^{3} n^{2} \mathrm {e}^{n \ln \left (a c-b c x\right )}-7 a^{3} n \mathrm {e}^{n \ln \left (a c-b c x\right )}-14 a^{3} \mathrm {e}^{n \ln \left (a c-b c x\right )}-a^{2} b n^{2} x \mathrm {e}^{n \ln \left (a c-b c x\right )}-3 a^{2} b n x \mathrm {e}^{n \ln \left (a c-b c x\right )}+6 a^{2} b x \mathrm {e}^{n \ln \left (a c-b c x\right )}+a b^{2} n^{2} x^{2} \mathrm {e}^{n \ln \left (a c-b c x\right )}+7 a b^{2} n x^{2} \mathrm {e}^{n \ln \left (a c-b c x\right )}+6 a b^{2} x^{2} \mathrm {e}^{n \ln \left (a c-b c x\right )}+b^{3} n^{2} x^{3} \mathrm {e}^{n \ln \left (a c-b c x\right )}+3 b^{3} n x^{3} \mathrm {e}^{n \ln \left (a c-b c x\right )}+2 b^{3} x^{3} \mathrm {e}^{n \ln \left (a c-b c x\right )}}{b n^{3}+6 b n^{2}+11 b n+6 b} \end {gather*}
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 \begin {gather*} -{\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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