3.1654 \(\int \frac{(3+5 x)^2}{(1-2 x)^3 (2+3 x)^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{22}{343 (1-2 x)}-\frac{1}{343 (3 x+2)}+\frac{121}{196 (1-2 x)^2}+\frac{64 \log (1-2 x)}{2401}-\frac{64 \log (3 x+2)}{2401} \]

[Out]

121/(196*(1 - 2*x)^2) - 22/(343*(1 - 2*x)) - 1/(343*(2 + 3*x)) + (64*Log[1 - 2*x])/2401 - (64*Log[2 + 3*x])/24
01

________________________________________________________________________________________

Rubi [A]  time = 0.0237, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ -\frac{22}{343 (1-2 x)}-\frac{1}{343 (3 x+2)}+\frac{121}{196 (1-2 x)^2}+\frac{64 \log (1-2 x)}{2401}-\frac{64 \log (3 x+2)}{2401} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/((1 - 2*x)^3*(2 + 3*x)^2),x]

[Out]

121/(196*(1 - 2*x)^2) - 22/(343*(1 - 2*x)) - 1/(343*(2 + 3*x)) + (64*Log[1 - 2*x])/2401 - (64*Log[2 + 3*x])/24
01

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(3+5 x)^2}{(1-2 x)^3 (2+3 x)^2} \, dx &=\int \left (-\frac{121}{49 (-1+2 x)^3}-\frac{44}{343 (-1+2 x)^2}+\frac{128}{2401 (-1+2 x)}+\frac{3}{343 (2+3 x)^2}-\frac{192}{2401 (2+3 x)}\right ) \, dx\\ &=\frac{121}{196 (1-2 x)^2}-\frac{22}{343 (1-2 x)}-\frac{1}{343 (2+3 x)}+\frac{64 \log (1-2 x)}{2401}-\frac{64 \log (2+3 x)}{2401}\\ \end{align*}

Mathematica [A]  time = 0.0355578, size = 47, normalized size = 0.87 \[ \frac{\frac{7 \left (512 x^2+2645 x+1514\right )}{(1-2 x)^2 (3 x+2)}+256 \log (1-2 x)-256 \log (6 x+4)}{9604} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^2/((1 - 2*x)^3*(2 + 3*x)^2),x]

[Out]

((7*(1514 + 2645*x + 512*x^2))/((1 - 2*x)^2*(2 + 3*x)) + 256*Log[1 - 2*x] - 256*Log[4 + 6*x])/9604

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 45, normalized size = 0.8 \begin{align*}{\frac{121}{196\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{22}{686\,x-343}}+{\frac{64\,\ln \left ( 2\,x-1 \right ) }{2401}}-{\frac{1}{686+1029\,x}}-{\frac{64\,\ln \left ( 2+3\,x \right ) }{2401}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^2/(1-2*x)^3/(2+3*x)^2,x)

[Out]

121/196/(2*x-1)^2+22/343/(2*x-1)+64/2401*ln(2*x-1)-1/343/(2+3*x)-64/2401*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.73538, size = 62, normalized size = 1.15 \begin{align*} \frac{512 \, x^{2} + 2645 \, x + 1514}{1372 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} - \frac{64}{2401} \, \log \left (3 \, x + 2\right ) + \frac{64}{2401} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^3/(2+3*x)^2,x, algorithm="maxima")

[Out]

1/1372*(512*x^2 + 2645*x + 1514)/(12*x^3 - 4*x^2 - 5*x + 2) - 64/2401*log(3*x + 2) + 64/2401*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.57619, size = 211, normalized size = 3.91 \begin{align*} \frac{3584 \, x^{2} - 256 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 256 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (2 \, x - 1\right ) + 18515 \, x + 10598}{9604 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^3/(2+3*x)^2,x, algorithm="fricas")

[Out]

1/9604*(3584*x^2 - 256*(12*x^3 - 4*x^2 - 5*x + 2)*log(3*x + 2) + 256*(12*x^3 - 4*x^2 - 5*x + 2)*log(2*x - 1) +
18515*x + 10598)/(12*x^3 - 4*x^2 - 5*x + 2)

________________________________________________________________________________________

Sympy [A]  time = 0.15803, size = 44, normalized size = 0.81 \begin{align*} \frac{512 x^{2} + 2645 x + 1514}{16464 x^{3} - 5488 x^{2} - 6860 x + 2744} + \frac{64 \log{\left (x - \frac{1}{2} \right )}}{2401} - \frac{64 \log{\left (x + \frac{2}{3} \right )}}{2401} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**2/(1-2*x)**3/(2+3*x)**2,x)

[Out]

(512*x**2 + 2645*x + 1514)/(16464*x**3 - 5488*x**2 - 6860*x + 2744) + 64*log(x - 1/2)/2401 - 64*log(x + 2/3)/2
401

________________________________________________________________________________________

Giac [A]  time = 1.81876, size = 69, normalized size = 1.28 \begin{align*} -\frac{1}{343 \,{\left (3 \, x + 2\right )}} + \frac{33 \,{\left (\frac{203}{3 \, x + 2} - 25\right )}}{2401 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}^{2}} + \frac{64}{2401} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^3/(2+3*x)^2,x, algorithm="giac")

[Out]

-1/343/(3*x + 2) + 33/2401*(203/(3*x + 2) - 25)/(7/(3*x + 2) - 2)^2 + 64/2401*log(abs(-7/(3*x + 2) + 2))