3.272 \(\int \frac{d+c x+b x^2+a x^3}{(-3+x) x (1+x)} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{12} \log (3-x) (27 a+9 b+3 c+d)-\frac{1}{4} \log (x+1) (a-b+c-d)+a x-\frac{1}{3} d \log (x) \]

[Out]

a*x + ((27*a + 9*b + 3*c + d)*Log[3 - x])/12 - (d*Log[x])/3 - ((a - b + c - d)*Log[1 + x])/4

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Rubi [A]  time = 0.0778306, antiderivative size = 49, normalized size of antiderivative = 1., 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.

[In]

Int[(d + c*x + b*x^2 + a*x^3)/((-3 + x)*x*(1 + x)),x]

[Out]

a*x + ((27*a + 9*b + 3*c + d)*Log[3 - x])/12 - (d*Log[x])/3 - ((a - b + c - d)*Log[1 + x])/4

Rule 1612

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

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*}

Mathematica [A]  time = 0.0260068, size = 49, normalized size = 1. \[ \frac{1}{12} \log (3-x) (27 a+9 b+3 c+d)+\frac{1}{4} \log (x+1) (-a+b-c+d)+a x-\frac{1}{3} d \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + c*x + b*x^2 + a*x^3)/((-3 + x)*x*(1 + x)),x]

[Out]

a*x + ((27*a + 9*b + 3*c + d)*Log[3 - x])/12 - (d*Log[x])/3 + ((-a + b - c + d)*Log[1 + x])/4

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Maple [A]  time = 0.006, size = 66, normalized size = 1.4 \begin{align*} ax-{\frac{d\ln \left ( x \right ) }{3}}-{\frac{\ln \left ( 1+x \right ) a}{4}}+{\frac{\ln \left ( 1+x \right ) b}{4}}-{\frac{\ln \left ( 1+x \right ) c}{4}}+{\frac{\ln \left ( 1+x \right ) d}{4}}+{\frac{9\,\ln \left ( -3+x \right ) a}{4}}+{\frac{3\,\ln \left ( -3+x \right ) b}{4}}+{\frac{\ln \left ( -3+x \right ) c}{4}}+{\frac{\ln \left ( -3+x \right ) d}{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3+b*x^2+c*x+d)/(-3+x)/x/(1+x),x)

[Out]

a*x-1/3*d*ln(x)-1/4*ln(1+x)*a+1/4*ln(1+x)*b-1/4*ln(1+x)*c+1/4*ln(1+x)*d+9/4*ln(-3+x)*a+3/4*ln(-3+x)*b+1/4*ln(-
3+x)*c+1/12*ln(-3+x)*d

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Maxima [A]  time = 0.936996, size = 55, normalized size = 1.12 \begin{align*} 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 \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+b*x^2+c*x+d)/(-3+x)/x/(1+x),x, algorithm="maxima")

[Out]

a*x - 1/4*(a - b + c - d)*log(x + 1) + 1/12*(27*a + 9*b + 3*c + d)*log(x - 3) - 1/3*d*log(x)

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Fricas [A]  time = 1.97565, size = 127, normalized size = 2.59 \begin{align*} 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 \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+b*x^2+c*x+d)/(-3+x)/x/(1+x),x, algorithm="fricas")

[Out]

a*x - 1/4*(a - b + c - d)*log(x + 1) + 1/12*(27*a + 9*b + 3*c + d)*log(x - 3) - 1/3*d*log(x)

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Sympy [B]  time = 25.1378, size = 762, normalized size = 15.55 \begin{align*} a x - \frac{d \log{\left (x \right )}}{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**3+b*x**2+c*x+d)/(-3+x)/x/(1+x),x)

[Out]

a*x - d*log(x)/3 - (a - b + c - d)*log(x + (-1512*a**2*d + 1134*a**2*(a - b + c - d) - 864*a*b*d + 648*a*b*(a
- b + c - d) - 432*a*c*d + 324*a*c*(a - b + c - d) - 144*a*d**2 + 81*a*(a - b + c - d)**2 - 216*b**2*d + 162*b
**2*(a - b + c - d) - 288*b*d**2 + 108*b*d*(a - b + c - d) + 81*b*(a - b + c - d)**2 - 72*c**2*d + 54*c**2*(a
- b + c - d) + 144*c*d**2 - 72*c*d*(a - b + c - d) - 27*c*(a - b + c - d)**2 - 136*d**3 - 54*d**2*(a - b + c -
d) + 117*d*(a - b + c - d)**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))/4 + (27*a + 9*b + 3*c + d)*log(x + (-1512*a**2*d - 37
8*a**2*(27*a + 9*b + 3*c + d) - 864*a*b*d - 216*a*b*(27*a + 9*b + 3*c + d) - 432*a*c*d - 108*a*c*(27*a + 9*b +
3*c + d) - 144*a*d**2 + 9*a*(27*a + 9*b + 3*c + d)**2 - 216*b**2*d - 54*b**2*(27*a + 9*b + 3*c + d) - 288*b*d
**2 - 36*b*d*(27*a + 9*b + 3*c + d) + 9*b*(27*a + 9*b + 3*c + d)**2 - 72*c**2*d - 18*c**2*(27*a + 9*b + 3*c +
d) + 144*c*d**2 + 24*c*d*(27*a + 9*b + 3*c + d) - 3*c*(27*a + 9*b + 3*c + d)**2 - 136*d**3 + 18*d**2*(27*a + 9
*b + 3*c + d) + 13*d*(27*a + 9*b + 3*c + d)**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))/12

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Giac [A]  time = 1.0817, size = 59, normalized size = 1.2 \begin{align*} 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+b*x^2+c*x+d)/(-3+x)/x/(1+x),x, algorithm="giac")

[Out]

a*x - 1/4*(a - b + c - d)*log(abs(x + 1)) + 1/12*(27*a + 9*b + 3*c + d)*log(abs(x - 3)) - 1/3*d*log(abs(x))