3.1551 \(\int \frac{3+5 x}{(1-2 x)^2 (2+3 x)^5} \, dx\)

Optimal. Leaf size=76 \[ \frac{88}{16807 (1-2 x)}-\frac{388}{16807 (3 x+2)}-\frac{64}{2401 (3 x+2)^2}-\frac{31}{1029 (3 x+2)^3}+\frac{1}{196 (3 x+2)^4}-\frac{1040 \log (1-2 x)}{117649}+\frac{1040 \log (3 x+2)}{117649} \]

[Out]

88/(16807*(1 - 2*x)) + 1/(196*(2 + 3*x)^4) - 31/(1029*(2 + 3*x)^3) - 64/(2401*(2 + 3*x)^2) - 388/(16807*(2 + 3
*x)) - (1040*Log[1 - 2*x])/117649 + (1040*Log[2 + 3*x])/117649

________________________________________________________________________________________

Rubi [A]  time = 0.0368517, antiderivative size = 76, 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} \[ \frac{88}{16807 (1-2 x)}-\frac{388}{16807 (3 x+2)}-\frac{64}{2401 (3 x+2)^2}-\frac{31}{1029 (3 x+2)^3}+\frac{1}{196 (3 x+2)^4}-\frac{1040 \log (1-2 x)}{117649}+\frac{1040 \log (3 x+2)}{117649} \]

Antiderivative was successfully verified.

[In]

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

[Out]

88/(16807*(1 - 2*x)) + 1/(196*(2 + 3*x)^4) - 31/(1029*(2 + 3*x)^3) - 64/(2401*(2 + 3*x)^2) - 388/(16807*(2 + 3
*x)) - (1040*Log[1 - 2*x])/117649 + (1040*Log[2 + 3*x])/117649

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)^5} \, dx &=\int \left (\frac{176}{16807 (-1+2 x)^2}-\frac{2080}{117649 (-1+2 x)}-\frac{3}{49 (2+3 x)^5}+\frac{93}{343 (2+3 x)^4}+\frac{384}{2401 (2+3 x)^3}+\frac{1164}{16807 (2+3 x)^2}+\frac{3120}{117649 (2+3 x)}\right ) \, dx\\ &=\frac{88}{16807 (1-2 x)}+\frac{1}{196 (2+3 x)^4}-\frac{31}{1029 (2+3 x)^3}-\frac{64}{2401 (2+3 x)^2}-\frac{388}{16807 (2+3 x)}-\frac{1040 \log (1-2 x)}{117649}+\frac{1040 \log (2+3 x)}{117649}\\ \end{align*}

Mathematica [A]  time = 0.0420935, size = 59, normalized size = 0.78 \[ \frac{4 \left (-\frac{7 \left (336960 x^4+702000 x^3+429000 x^2-9230 x-52979\right )}{16 (2 x-1) (3 x+2)^4}-780 \log (1-2 x)+780 \log (6 x+4)\right )}{352947} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-7*(-52979 - 9230*x + 429000*x^2 + 702000*x^3 + 336960*x^4))/(16*(-1 + 2*x)*(2 + 3*x)^4) - 780*Log[1 - 2*
x] + 780*Log[4 + 6*x]))/352947

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 63, normalized size = 0.8 \begin{align*} -{\frac{88}{33614\,x-16807}}-{\frac{1040\,\ln \left ( 2\,x-1 \right ) }{117649}}+{\frac{1}{196\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{31}{1029\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{64}{2401\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{388}{33614+50421\,x}}+{\frac{1040\,\ln \left ( 2+3\,x \right ) }{117649}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-88/16807/(2*x-1)-1040/117649*ln(2*x-1)+1/196/(2+3*x)^4-31/1029/(2+3*x)^3-64/2401/(2+3*x)^2-388/16807/(2+3*x)+
1040/117649*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.07021, size = 89, normalized size = 1.17 \begin{align*} -\frac{336960 \, x^{4} + 702000 \, x^{3} + 429000 \, x^{2} - 9230 \, x - 52979}{201684 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac{1040}{117649} \, \log \left (3 \, x + 2\right ) - \frac{1040}{117649} \, \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)^5,x, algorithm="maxima")

[Out]

-1/201684*(336960*x^4 + 702000*x^3 + 429000*x^2 - 9230*x - 52979)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x
- 16) + 1040/117649*log(3*x + 2) - 1040/117649*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.2635, size = 362, normalized size = 4.76 \begin{align*} -\frac{2358720 \, x^{4} + 4914000 \, x^{3} + 3003000 \, x^{2} - 12480 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) + 12480 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) - 64610 \, x - 370853}{1411788 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\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)^5,x, algorithm="fricas")

[Out]

-1/1411788*(2358720*x^4 + 4914000*x^3 + 3003000*x^2 - 12480*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)
*log(3*x + 2) + 12480*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(2*x - 1) - 64610*x - 370853)/(162
*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

________________________________________________________________________________________

Sympy [A]  time = 0.172513, size = 65, normalized size = 0.86 \begin{align*} - \frac{336960 x^{4} + 702000 x^{3} + 429000 x^{2} - 9230 x - 52979}{32672808 x^{5} + 70791084 x^{4} + 43563744 x^{3} - 4840416 x^{2} - 12907776 x - 3226944} - \frac{1040 \log{\left (x - \frac{1}{2} \right )}}{117649} + \frac{1040 \log{\left (x + \frac{2}{3} \right )}}{117649} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(336960*x**4 + 702000*x**3 + 429000*x**2 - 9230*x - 52979)/(32672808*x**5 + 70791084*x**4 + 43563744*x**3 - 4
840416*x**2 - 12907776*x - 3226944) - 1040*log(x - 1/2)/117649 + 1040*log(x + 2/3)/117649

________________________________________________________________________________________

Giac [A]  time = 1.99797, size = 90, normalized size = 1.18 \begin{align*} -\frac{388}{16807 \,{\left (3 \, x + 2\right )}} + \frac{528}{117649 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}} - \frac{64}{2401 \,{\left (3 \, x + 2\right )}^{2}} - \frac{31}{1029 \,{\left (3 \, x + 2\right )}^{3}} + \frac{1}{196 \,{\left (3 \, x + 2\right )}^{4}} - \frac{1040}{117649} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 2 \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)^5,x, algorithm="giac")

[Out]

-388/16807/(3*x + 2) + 528/117649/(7/(3*x + 2) - 2) - 64/2401/(3*x + 2)^2 - 31/1029/(3*x + 2)^3 + 1/196/(3*x +
2)^4 - 1040/117649*log(abs(-7/(3*x + 2) + 2))