3.1446 \(\int \frac{3+5 x}{(1-2 x) (2+3 x)^4} \, dx\)

Optimal. Leaf size=54 \[ -\frac{22}{343 (3 x+2)}-\frac{11}{98 (3 x+2)^2}+\frac{1}{63 (3 x+2)^3}-\frac{44 \log (1-2 x)}{2401}+\frac{44 \log (3 x+2)}{2401} \]

[Out]

1/(63*(2 + 3*x)^3) - 11/(98*(2 + 3*x)^2) - 22/(343*(2 + 3*x)) - (44*Log[1 - 2*x])/2401 + (44*Log[2 + 3*x])/240
1

________________________________________________________________________________________

Rubi [A]  time = 0.0210959, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{22}{343 (3 x+2)}-\frac{11}{98 (3 x+2)^2}+\frac{1}{63 (3 x+2)^3}-\frac{44 \log (1-2 x)}{2401}+\frac{44 \log (3 x+2)}{2401} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(63*(2 + 3*x)^3) - 11/(98*(2 + 3*x)^2) - 22/(343*(2 + 3*x)) - (44*Log[1 - 2*x])/2401 + (44*Log[2 + 3*x])/240
1

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{3+5 x}{(1-2 x) (2+3 x)^4} \, dx &=\int \left (-\frac{88}{2401 (-1+2 x)}-\frac{1}{7 (2+3 x)^4}+\frac{33}{49 (2+3 x)^3}+\frac{66}{343 (2+3 x)^2}+\frac{132}{2401 (2+3 x)}\right ) \, dx\\ &=\frac{1}{63 (2+3 x)^3}-\frac{11}{98 (2+3 x)^2}-\frac{22}{343 (2+3 x)}-\frac{44 \log (1-2 x)}{2401}+\frac{44 \log (2+3 x)}{2401}\\ \end{align*}

Mathematica [A]  time = 0.02141, size = 40, normalized size = 0.74 \[ \frac{-\frac{7 \left (3564 x^2+6831 x+2872\right )}{(3 x+2)^3}-792 \log (3-6 x)+792 \log (3 x+2)}{43218} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(2872 + 6831*x + 3564*x^2))/(2 + 3*x)^3 - 792*Log[3 - 6*x] + 792*Log[2 + 3*x])/43218

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 45, normalized size = 0.8 \begin{align*} -{\frac{44\,\ln \left ( 2\,x-1 \right ) }{2401}}+{\frac{1}{63\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{11}{98\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{22}{686+1029\,x}}+{\frac{44\,\ln \left ( 2+3\,x \right ) }{2401}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)/(1-2*x)/(2+3*x)^4,x)

[Out]

-44/2401*ln(2*x-1)+1/63/(2+3*x)^3-11/98/(2+3*x)^2-22/343/(2+3*x)+44/2401*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.06565, size = 62, normalized size = 1.15 \begin{align*} -\frac{3564 \, x^{2} + 6831 \, x + 2872}{6174 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{44}{2401} \, \log \left (3 \, x + 2\right ) - \frac{44}{2401} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6174*(3564*x^2 + 6831*x + 2872)/(27*x^3 + 54*x^2 + 36*x + 8) + 44/2401*log(3*x + 2) - 44/2401*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.32163, size = 223, normalized size = 4.13 \begin{align*} -\frac{24948 \, x^{2} - 792 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 792 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x - 1\right ) + 47817 \, x + 20104}{43218 \,{\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)/(1-2*x)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/43218*(24948*x^2 - 792*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) + 792*(27*x^3 + 54*x^2 + 36*x + 8)*log(2*x
- 1) + 47817*x + 20104)/(27*x^3 + 54*x^2 + 36*x + 8)

________________________________________________________________________________________

Sympy [A]  time = 0.151297, size = 44, normalized size = 0.81 \begin{align*} - \frac{3564 x^{2} + 6831 x + 2872}{166698 x^{3} + 333396 x^{2} + 222264 x + 49392} - \frac{44 \log{\left (x - \frac{1}{2} \right )}}{2401} + \frac{44 \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)/(1-2*x)/(2+3*x)**4,x)

[Out]

-(3564*x**2 + 6831*x + 2872)/(166698*x**3 + 333396*x**2 + 222264*x + 49392) - 44*log(x - 1/2)/2401 + 44*log(x
+ 2/3)/2401

________________________________________________________________________________________

Giac [A]  time = 1.94863, size = 51, normalized size = 0.94 \begin{align*} -\frac{3564 \, x^{2} + 6831 \, x + 2872}{6174 \,{\left (3 \, x + 2\right )}^{3}} + \frac{44}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{44}{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)/(1-2*x)/(2+3*x)^4,x, algorithm="giac")

[Out]

-1/6174*(3564*x^2 + 6831*x + 2872)/(3*x + 2)^3 + 44/2401*log(abs(3*x + 2)) - 44/2401*log(abs(2*x - 1))