3.1463 \(\int \frac{(3+5 x)^2}{(1-2 x) (2+3 x)^4} \, dx\)

Optimal. Leaf size=54 \[ -\frac{121}{343 (3 x+2)}+\frac{34}{441 (3 x+2)^2}-\frac{1}{189 (3 x+2)^3}-\frac{242 \log (1-2 x)}{2401}+\frac{242 \log (3 x+2)}{2401} \]

[Out]

-1/(189*(2 + 3*x)^3) + 34/(441*(2 + 3*x)^2) - 121/(343*(2 + 3*x)) - (242*Log[1 - 2*x])/2401 + (242*Log[2 + 3*x
])/2401

________________________________________________________________________________________

Rubi [A]  time = 0.021555, 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} \[ -\frac{121}{343 (3 x+2)}+\frac{34}{441 (3 x+2)^2}-\frac{1}{189 (3 x+2)^3}-\frac{242 \log (1-2 x)}{2401}+\frac{242 \log (3 x+2)}{2401} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

-1/(189*(2 + 3*x)^3) + 34/(441*(2 + 3*x)^2) - 121/(343*(2 + 3*x)) - (242*Log[1 - 2*x])/2401 + (242*Log[2 + 3*x
])/2401

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) (2+3 x)^4} \, dx &=\int \left (-\frac{484}{2401 (-1+2 x)}+\frac{1}{21 (2+3 x)^4}-\frac{68}{147 (2+3 x)^3}+\frac{363}{343 (2+3 x)^2}+\frac{726}{2401 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{189 (2+3 x)^3}+\frac{34}{441 (2+3 x)^2}-\frac{121}{343 (2+3 x)}-\frac{242 \log (1-2 x)}{2401}+\frac{242 \log (2+3 x)}{2401}\\ \end{align*}

Mathematica [A]  time = 0.022006, size = 40, normalized size = 0.74 \[ \frac{-\frac{7 \left (29403 x^2+37062 x+11689\right )}{(3 x+2)^3}-6534 \log (1-2 x)+6534 \log (6 x+4)}{64827} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

((-7*(11689 + 37062*x + 29403*x^2))/(2 + 3*x)^3 - 6534*Log[1 - 2*x] + 6534*Log[4 + 6*x])/64827

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 45, normalized size = 0.8 \begin{align*} -{\frac{242\,\ln \left ( 2\,x-1 \right ) }{2401}}-{\frac{1}{189\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{34}{441\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{121}{686+1029\,x}}+{\frac{242\,\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)/(2+3*x)^4,x)

[Out]

-242/2401*ln(2*x-1)-1/189/(2+3*x)^3+34/441/(2+3*x)^2-121/343/(2+3*x)+242/2401*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.06188, size = 62, normalized size = 1.15 \begin{align*} -\frac{29403 \, x^{2} + 37062 \, x + 11689}{9261 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{242}{2401} \, \log \left (3 \, x + 2\right ) - \frac{242}{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)/(2+3*x)^4,x, algorithm="maxima")

[Out]

-1/9261*(29403*x^2 + 37062*x + 11689)/(27*x^3 + 54*x^2 + 36*x + 8) + 242/2401*log(3*x + 2) - 242/2401*log(2*x
- 1)

________________________________________________________________________________________

Fricas [A]  time = 1.32705, size = 228, normalized size = 4.22 \begin{align*} -\frac{205821 \, x^{2} - 6534 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 6534 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x - 1\right ) + 259434 \, x + 81823}{64827 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/64827*(205821*x^2 - 6534*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) + 6534*(27*x^3 + 54*x^2 + 36*x + 8)*log(
2*x - 1) + 259434*x + 81823)/(27*x^3 + 54*x^2 + 36*x + 8)

________________________________________________________________________________________

Sympy [A]  time = 0.16002, size = 44, normalized size = 0.81 \begin{align*} - \frac{29403 x^{2} + 37062 x + 11689}{250047 x^{3} + 500094 x^{2} + 333396 x + 74088} - \frac{242 \log{\left (x - \frac{1}{2} \right )}}{2401} + \frac{242 \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)/(2+3*x)**4,x)

[Out]

-(29403*x**2 + 37062*x + 11689)/(250047*x**3 + 500094*x**2 + 333396*x + 74088) - 242*log(x - 1/2)/2401 + 242*l
og(x + 2/3)/2401

________________________________________________________________________________________

Giac [A]  time = 1.22196, size = 51, normalized size = 0.94 \begin{align*} -\frac{29403 \, x^{2} + 37062 \, x + 11689}{9261 \,{\left (3 \, x + 2\right )}^{3}} + \frac{242}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{242}{2401} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^4,x, algorithm="giac")

[Out]

-1/9261*(29403*x^2 + 37062*x + 11689)/(3*x + 2)^3 + 242/2401*log(abs(3*x + 2)) - 242/2401*log(abs(2*x - 1))