3.1550 \(\int \frac{3+5 x}{(1-2 x)^2 (2+3 x)^4} \, dx\)

Optimal. Leaf size=65 \[ \frac{44}{2401 (1-2 x)}-\frac{128}{2401 (3 x+2)}-\frac{31}{686 (3 x+2)^2}+\frac{1}{147 (3 x+2)^3}-\frac{388 \log (1-2 x)}{16807}+\frac{388 \log (3 x+2)}{16807} \]

[Out]

44/(2401*(1 - 2*x)) + 1/(147*(2 + 3*x)^3) - 31/(686*(2 + 3*x)^2) - 128/(2401*(2 + 3*x)) - (388*Log[1 - 2*x])/1
6807 + (388*Log[2 + 3*x])/16807

________________________________________________________________________________________

Rubi [A]  time = 0.0306685, antiderivative size = 65, 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} \[ \frac{44}{2401 (1-2 x)}-\frac{128}{2401 (3 x+2)}-\frac{31}{686 (3 x+2)^2}+\frac{1}{147 (3 x+2)^3}-\frac{388 \log (1-2 x)}{16807}+\frac{388 \log (3 x+2)}{16807} \]

Antiderivative was successfully verified.

[In]

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

[Out]

44/(2401*(1 - 2*x)) + 1/(147*(2 + 3*x)^3) - 31/(686*(2 + 3*x)^2) - 128/(2401*(2 + 3*x)) - (388*Log[1 - 2*x])/1
6807 + (388*Log[2 + 3*x])/16807

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 (2+3 x)^4} \, dx &=\int \left (\frac{88}{2401 (-1+2 x)^2}-\frac{776}{16807 (-1+2 x)}-\frac{3}{49 (2+3 x)^4}+\frac{93}{343 (2+3 x)^3}+\frac{384}{2401 (2+3 x)^2}+\frac{1164}{16807 (2+3 x)}\right ) \, dx\\ &=\frac{44}{2401 (1-2 x)}+\frac{1}{147 (2+3 x)^3}-\frac{31}{686 (2+3 x)^2}-\frac{128}{2401 (2+3 x)}-\frac{388 \log (1-2 x)}{16807}+\frac{388 \log (2+3 x)}{16807}\\ \end{align*}

Mathematica [A]  time = 0.0304381, size = 52, normalized size = 0.8 \[ \frac{-\frac{7 \left (20952 x^3+29682 x^2+6887 x-2164\right )}{(2 x-1) (3 x+2)^3}-2328 \log (3-6 x)+2328 \log (3 x+2)}{100842} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(-2164 + 6887*x + 29682*x^2 + 20952*x^3))/((-1 + 2*x)*(2 + 3*x)^3) - 2328*Log[3 - 6*x] + 2328*Log[2 + 3*x
])/100842

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 54, normalized size = 0.8 \begin{align*} -{\frac{44}{4802\,x-2401}}-{\frac{388\,\ln \left ( 2\,x-1 \right ) }{16807}}+{\frac{1}{147\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{31}{686\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{128}{4802+7203\,x}}+{\frac{388\,\ln \left ( 2+3\,x \right ) }{16807}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-44/2401/(2*x-1)-388/16807*ln(2*x-1)+1/147/(2+3*x)^3-31/686/(2+3*x)^2-128/2401/(2+3*x)+388/16807*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.10815, size = 76, normalized size = 1.17 \begin{align*} -\frac{20952 \, x^{3} + 29682 \, x^{2} + 6887 \, x - 2164}{14406 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} + \frac{388}{16807} \, \log \left (3 \, x + 2\right ) - \frac{388}{16807} \, \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/(2+3*x)^4,x, algorithm="maxima")

[Out]

-1/14406*(20952*x^3 + 29682*x^2 + 6887*x - 2164)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8) + 388/16807*log(3*x + 2
) - 388/16807*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.13507, size = 282, normalized size = 4.34 \begin{align*} -\frac{146664 \, x^{3} + 207774 \, x^{2} - 2328 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (3 \, x + 2\right ) + 2328 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (2 \, x - 1\right ) + 48209 \, x - 15148}{100842 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/100842*(146664*x^3 + 207774*x^2 - 2328*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(3*x + 2) + 2328*(54*x^4 +
81*x^3 + 18*x^2 - 20*x - 8)*log(2*x - 1) + 48209*x - 15148)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)

________________________________________________________________________________________

Sympy [A]  time = 0.154955, size = 54, normalized size = 0.83 \begin{align*} - \frac{20952 x^{3} + 29682 x^{2} + 6887 x - 2164}{777924 x^{4} + 1166886 x^{3} + 259308 x^{2} - 288120 x - 115248} - \frac{388 \log{\left (x - \frac{1}{2} \right )}}{16807} + \frac{388 \log{\left (x + \frac{2}{3} \right )}}{16807} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20952*x**3 + 29682*x**2 + 6887*x - 2164)/(777924*x**4 + 1166886*x**3 + 259308*x**2 - 288120*x - 115248) - 38
8*log(x - 1/2)/16807 + 388*log(x + 2/3)/16807

________________________________________________________________________________________

Giac [A]  time = 2.87681, size = 81, normalized size = 1.25 \begin{align*} -\frac{44}{2401 \,{\left (2 \, x - 1\right )}} + \frac{18 \,{\left (\frac{2415}{2 \, x - 1} + \frac{3038}{{\left (2 \, x - 1\right )}^{2}} + 473\right )}}{16807 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{3}} + \frac{388}{16807} \, \log \left ({\left | -\frac{7}{2 \, x - 1} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-44/2401/(2*x - 1) + 18/16807*(2415/(2*x - 1) + 3038/(2*x - 1)^2 + 473)/(7/(2*x - 1) + 3)^3 + 388/16807*log(ab
s(-7/(2*x - 1) - 3))