3.1565 \(\int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac {121}{343 (1-2 x)}+\frac {22}{343 (3 x+2)}-\frac {1}{294 (3 x+2)^2}-\frac {319 \log (1-2 x)}{2401}+\frac {319 \log (3 x+2)}{2401} \]

[Out]

121/343/(1-2*x)-1/294/(2+3*x)^2+22/343/(2+3*x)-319/2401*ln(1-2*x)+319/2401*ln(2+3*x)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, 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} \[ \frac {121}{343 (1-2 x)}+\frac {22}{343 (3 x+2)}-\frac {1}{294 (3 x+2)^2}-\frac {319 \log (1-2 x)}{2401}+\frac {319 \log (3 x+2)}{2401} \]

Antiderivative was successfully verified.

[In]

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

[Out]

121/(343*(1 - 2*x)) - 1/(294*(2 + 3*x)^2) + 22/(343*(2 + 3*x)) - (319*Log[1 - 2*x])/2401 + (319*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 (2+3 x)^3} \, dx &=\int \left (\frac {242}{343 (-1+2 x)^2}-\frac {638}{2401 (-1+2 x)}+\frac {1}{49 (2+3 x)^3}-\frac {66}{343 (2+3 x)^2}+\frac {957}{2401 (2+3 x)}\right ) \, dx\\ &=\frac {121}{343 (1-2 x)}-\frac {1}{294 (2+3 x)^2}+\frac {22}{343 (2+3 x)}-\frac {319 \log (1-2 x)}{2401}+\frac {319 \log (2+3 x)}{2401}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 47, normalized size = 0.87 \[ \frac {-\frac {7 \left (5742 x^2+8594 x+3161\right )}{(2 x-1) (3 x+2)^2}-1914 \log (1-2 x)+1914 \log (6 x+4)}{14406} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(3161 + 8594*x + 5742*x^2))/((-1 + 2*x)*(2 + 3*x)^2) - 1914*Log[1 - 2*x] + 1914*Log[4 + 6*x])/14406

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 75, normalized size = 1.39 \[ -\frac {40194 \, x^{2} - 1914 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (3 \, x + 2\right ) + 1914 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (2 \, x - 1\right ) + 60158 \, x + 22127}{14406 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14406*(40194*x^2 - 1914*(18*x^3 + 15*x^2 - 4*x - 4)*log(3*x + 2) + 1914*(18*x^3 + 15*x^2 - 4*x - 4)*log(2*x
- 1) + 60158*x + 22127)/(18*x^3 + 15*x^2 - 4*x - 4)

________________________________________________________________________________________

giac [A]  time = 0.83, size = 51, normalized size = 0.94 \[ -\frac {121}{343 \, {\left (2 \, x - 1\right )}} - \frac {2 \, {\left (\frac {448}{2 \, x - 1} + 195\right )}}{2401 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{2}} + \frac {319}{2401} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/343/(2*x - 1) - 2/2401*(448/(2*x - 1) + 195)/(7/(2*x - 1) + 3)^2 + 319/2401*log(abs(-7/(2*x - 1) - 3))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 45, normalized size = 0.83 \[ -\frac {319 \ln \left (2 x -1\right )}{2401}+\frac {319 \ln \left (3 x +2\right )}{2401}-\frac {1}{294 \left (3 x +2\right )^{2}}+\frac {22}{343 \left (3 x +2\right )}-\frac {121}{343 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/294/(3*x+2)^2+22/343/(3*x+2)+319/2401*ln(3*x+2)-121/343/(2*x-1)-319/2401*ln(2*x-1)

________________________________________________________________________________________

maxima [A]  time = 0.60, size = 46, normalized size = 0.85 \[ -\frac {5742 \, x^{2} + 8594 \, x + 3161}{2058 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} + \frac {319}{2401} \, \log \left (3 \, x + 2\right ) - \frac {319}{2401} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2058*(5742*x^2 + 8594*x + 3161)/(18*x^3 + 15*x^2 - 4*x - 4) + 319/2401*log(3*x + 2) - 319/2401*log(2*x - 1)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 37, normalized size = 0.69 \[ \frac {638\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{2401}+\frac {\frac {319\,x^2}{2058}+\frac {4297\,x}{18522}+\frac {3161}{37044}}{-x^3-\frac {5\,x^2}{6}+\frac {2\,x}{9}+\frac {2}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(638*atanh((12*x)/7 + 1/7))/2401 + ((4297*x)/18522 + (319*x^2)/2058 + 3161/37044)/((2*x)/9 - (5*x^2)/6 - x^3 +
2/9)

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 46, normalized size = 0.85 \[ \frac {- 5742 x^{2} - 8594 x - 3161}{37044 x^{3} + 30870 x^{2} - 8232 x - 8232} - \frac {319 \log {\left (x - \frac {1}{2} \right )}}{2401} + \frac {319 \log {\left (x + \frac {2}{3} \right )}}{2401} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-5742*x**2 - 8594*x - 3161)/(37044*x**3 + 30870*x**2 - 8232*x - 8232) - 319*log(x - 1/2)/2401 + 319*log(x + 2
/3)/2401

________________________________________________________________________________________