Optimal. Leaf size=49 \[ \frac {1}{12} \log (3-x) (27 a+9 b+3 c+d)-\frac {1}{4} \log (x+1) (a-b+c-d)+a x-\frac {1}{3} d \log (x) \]
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Rubi [A] time = 0.08, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {1612} \[ \frac {1}{12} \log (3-x) (27 a+9 b+3 c+d)-\frac {1}{4} \log (x+1) (a-b+c-d)+a x-\frac {1}{3} d \log (x) \]
Antiderivative was successfully verified.
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Rule 1612
Rubi steps
\begin {align*} \int \frac {d+c x+b x^2+a x^3}{(-3+x) x (1+x)} \, dx &=\int \left (a+\frac {27 a+9 b+3 c+d}{12 (-3+x)}-\frac {d}{3 x}+\frac {-a+b-c+d}{4 (1+x)}\right ) \, dx\\ &=a x+\frac {1}{12} (27 a+9 b+3 c+d) \log (3-x)-\frac {1}{3} d \log (x)-\frac {1}{4} (a-b+c-d) \log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \[ \frac {1}{12} \log (3-x) (27 a+9 b+3 c+d)+\frac {1}{4} \log (x+1) (-a+b-c+d)+a x-\frac {1}{3} d \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 41, normalized size = 0.84 \[ a x - \frac {1}{4} \, {\left (a - b + c - d\right )} \log \left (x + 1\right ) + \frac {1}{12} \, {\left (27 \, a + 9 \, b + 3 \, c + d\right )} \log \left (x - 3\right ) - \frac {1}{3} \, d \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 44, normalized size = 0.90 \[ a x - \frac {1}{4} \, {\left (a - b + c - d\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{12} \, {\left (27 \, a + 9 \, b + 3 \, c + d\right )} \log \left ({\left | x - 3 \right |}\right ) - \frac {1}{3} \, d \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 1.35 \[ a x +\frac {9 a \ln \left (x -3\right )}{4}-\frac {a \ln \left (x +1\right )}{4}+\frac {3 b \ln \left (x -3\right )}{4}+\frac {b \ln \left (x +1\right )}{4}+\frac {c \ln \left (x -3\right )}{4}-\frac {c \ln \left (x +1\right )}{4}-\frac {d \ln \relax (x )}{3}+\frac {d \ln \left (x -3\right )}{12}+\frac {d \ln \left (x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 41, normalized size = 0.84 \[ a x - \frac {1}{4} \, {\left (a - b + c - d\right )} \log \left (x + 1\right ) + \frac {1}{12} \, {\left (27 \, a + 9 \, b + 3 \, c + d\right )} \log \left (x - 3\right ) - \frac {1}{3} \, d \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 46, normalized size = 0.94 \[ \ln \left (x-3\right )\,\left (\frac {9\,a}{4}+\frac {3\,b}{4}+\frac {c}{4}+\frac {d}{12}\right )-\ln \left (x+1\right )\,\left (\frac {a}{4}-\frac {b}{4}+\frac {c}{4}-\frac {d}{4}\right )+a\,x-\frac {d\,\ln \relax (x)}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 86.23, size = 762, normalized size = 15.55 \[ a x - \frac {d \log {\relax (x )}}{3} - \frac {\left (a - b + c - d\right ) \log {\left (x + \frac {- 1512 a^{2} d + 1134 a^{2} \left (a - b + c - d\right ) - 864 a b d + 648 a b \left (a - b + c - d\right ) - 432 a c d + 324 a c \left (a - b + c - d\right ) - 144 a d^{2} + 81 a \left (a - b + c - d\right )^{2} - 216 b^{2} d + 162 b^{2} \left (a - b + c - d\right ) - 288 b d^{2} + 108 b d \left (a - b + c - d\right ) + 81 b \left (a - b + c - d\right )^{2} - 72 c^{2} d + 54 c^{2} \left (a - b + c - d\right ) + 144 c d^{2} - 72 c d \left (a - b + c - d\right ) - 27 c \left (a - b + c - d\right )^{2} - 136 d^{3} - 54 d^{2} \left (a - b + c - d\right ) + 117 d \left (a - b + c - d\right )^{2}}{1215 a^{3} - 567 a^{2} b + 1593 a^{2} c - 2691 a^{2} d - 567 a b^{2} + 378 a b c - 1638 a b d + 405 a c^{2} - 702 a c d - 351 a d^{2} - 81 b^{3} - 27 b^{2} c - 207 b^{2} d + 81 b c^{2} - 270 b c d - 27 b d^{2} + 27 c^{3} - 27 c^{2} d - 99 c d^{2} + 35 d^{3}} \right )}}{4} + \frac {\left (27 a + 9 b + 3 c + d\right ) \log {\left (x + \frac {- 1512 a^{2} d - 378 a^{2} \left (27 a + 9 b + 3 c + d\right ) - 864 a b d - 216 a b \left (27 a + 9 b + 3 c + d\right ) - 432 a c d - 108 a c \left (27 a + 9 b + 3 c + d\right ) - 144 a d^{2} + 9 a \left (27 a + 9 b + 3 c + d\right )^{2} - 216 b^{2} d - 54 b^{2} \left (27 a + 9 b + 3 c + d\right ) - 288 b d^{2} - 36 b d \left (27 a + 9 b + 3 c + d\right ) + 9 b \left (27 a + 9 b + 3 c + d\right )^{2} - 72 c^{2} d - 18 c^{2} \left (27 a + 9 b + 3 c + d\right ) + 144 c d^{2} + 24 c d \left (27 a + 9 b + 3 c + d\right ) - 3 c \left (27 a + 9 b + 3 c + d\right )^{2} - 136 d^{3} + 18 d^{2} \left (27 a + 9 b + 3 c + d\right ) + 13 d \left (27 a + 9 b + 3 c + d\right )^{2}}{1215 a^{3} - 567 a^{2} b + 1593 a^{2} c - 2691 a^{2} d - 567 a b^{2} + 378 a b c - 1638 a b d + 405 a c^{2} - 702 a c d - 351 a d^{2} - 81 b^{3} - 27 b^{2} c - 207 b^{2} d + 81 b c^{2} - 270 b c d - 27 b d^{2} + 27 c^{3} - 27 c^{2} d - 99 c d^{2} + 35 d^{3}} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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