3.1639 \(\int \frac{3+5 x}{(1-2 x)^3 (2+3 x)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{1}{49 (1-2 x)}+\frac{11}{28 (1-2 x)^2}+\frac{3}{343} \log (1-2 x)-\frac{3}{343} \log (3 x+2) \]

[Out]

11/(28*(1 - 2*x)^2) - 1/(49*(1 - 2*x)) + (3*Log[1 - 2*x])/343 - (3*Log[2 + 3*x])/343

________________________________________________________________________________________

Rubi [A]  time = 0.021647, antiderivative size = 43, 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} \[ -\frac{1}{49 (1-2 x)}+\frac{11}{28 (1-2 x)^2}+\frac{3}{343} \log (1-2 x)-\frac{3}{343} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

11/(28*(1 - 2*x)^2) - 1/(49*(1 - 2*x)) + (3*Log[1 - 2*x])/343 - (3*Log[2 + 3*x])/343

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)^3 (2+3 x)} \, dx &=\int \left (-\frac{11}{7 (-1+2 x)^3}-\frac{2}{49 (-1+2 x)^2}+\frac{6}{343 (-1+2 x)}-\frac{9}{343 (2+3 x)}\right ) \, dx\\ &=\frac{11}{28 (1-2 x)^2}-\frac{1}{49 (1-2 x)}+\frac{3}{343} \log (1-2 x)-\frac{3}{343} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0181143, size = 35, normalized size = 0.81 \[ \frac{\frac{7 (8 x+73)}{(1-2 x)^2}+12 \log (1-2 x)-12 \log (6 x+4)}{1372} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*(73 + 8*x))/(1 - 2*x)^2 + 12*Log[1 - 2*x] - 12*Log[4 + 6*x])/1372

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \begin{align*}{\frac{11}{28\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{1}{98\,x-49}}+{\frac{3\,\ln \left ( 2\,x-1 \right ) }{343}}-{\frac{3\,\ln \left ( 2+3\,x \right ) }{343}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/28/(2*x-1)^2+1/49/(2*x-1)+3/343*ln(2*x-1)-3/343*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 2.29175, size = 49, normalized size = 1.14 \begin{align*} \frac{8 \, x + 73}{196 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{3}{343} \, \log \left (3 \, x + 2\right ) + \frac{3}{343} \, \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)^3/(2+3*x),x, algorithm="maxima")

[Out]

1/196*(8*x + 73)/(4*x^2 - 4*x + 1) - 3/343*log(3*x + 2) + 3/343*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.27989, size = 151, normalized size = 3.51 \begin{align*} -\frac{12 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (3 \, x + 2\right ) - 12 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 56 \, x - 511}{1372 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1372*(12*(4*x^2 - 4*x + 1)*log(3*x + 2) - 12*(4*x^2 - 4*x + 1)*log(2*x - 1) - 56*x - 511)/(4*x^2 - 4*x + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.132194, size = 34, normalized size = 0.79 \begin{align*} \frac{8 x + 73}{784 x^{2} - 784 x + 196} + \frac{3 \log{\left (x - \frac{1}{2} \right )}}{343} - \frac{3 \log{\left (x + \frac{2}{3} \right )}}{343} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*x + 73)/(784*x**2 - 784*x + 196) + 3*log(x - 1/2)/343 - 3*log(x + 2/3)/343

________________________________________________________________________________________

Giac [A]  time = 3.86074, size = 45, normalized size = 1.05 \begin{align*} \frac{8 \, x + 73}{196 \,{\left (2 \, x - 1\right )}^{2}} - \frac{3}{343} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{3}{343} \, \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)^3/(2+3*x),x, algorithm="giac")

[Out]

1/196*(8*x + 73)/(2*x - 1)^2 - 3/343*log(abs(3*x + 2)) + 3/343*log(abs(2*x - 1))