3.1642 \(\int \frac{3+5 x}{(1-2 x)^3 (2+3 x)^4} \, dx\)

Optimal. Leaf size=76 \[ \frac{388}{16807 (1-2 x)}-\frac{558}{16807 (3 x+2)}+\frac{22}{2401 (1-2 x)^2}-\frac{87}{4802 (3 x+2)^2}+\frac{1}{343 (3 x+2)^3}-\frac{2280 \log (1-2 x)}{117649}+\frac{2280 \log (3 x+2)}{117649} \]

[Out]

22/(2401*(1 - 2*x)^2) + 388/(16807*(1 - 2*x)) + 1/(343*(2 + 3*x)^3) - 87/(4802*(2 + 3*x)^2) - 558/(16807*(2 +
3*x)) - (2280*Log[1 - 2*x])/117649 + (2280*Log[2 + 3*x])/117649

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Rubi [A]  time = 0.0417642, 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} \[ \frac{388}{16807 (1-2 x)}-\frac{558}{16807 (3 x+2)}+\frac{22}{2401 (1-2 x)^2}-\frac{87}{4802 (3 x+2)^2}+\frac{1}{343 (3 x+2)^3}-\frac{2280 \log (1-2 x)}{117649}+\frac{2280 \log (3 x+2)}{117649} \]

Antiderivative was successfully verified.

[In]

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

[Out]

22/(2401*(1 - 2*x)^2) + 388/(16807*(1 - 2*x)) + 1/(343*(2 + 3*x)^3) - 87/(4802*(2 + 3*x)^2) - 558/(16807*(2 +
3*x)) - (2280*Log[1 - 2*x])/117649 + (2280*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)^3 (2+3 x)^4} \, dx &=\int \left (-\frac{88}{2401 (-1+2 x)^3}+\frac{776}{16807 (-1+2 x)^2}-\frac{4560}{117649 (-1+2 x)}-\frac{9}{343 (2+3 x)^4}+\frac{261}{2401 (2+3 x)^3}+\frac{1674}{16807 (2+3 x)^2}+\frac{6840}{117649 (2+3 x)}\right ) \, dx\\ &=\frac{22}{2401 (1-2 x)^2}+\frac{388}{16807 (1-2 x)}+\frac{1}{343 (2+3 x)^3}-\frac{87}{4802 (2+3 x)^2}-\frac{558}{16807 (2+3 x)}-\frac{2280 \log (1-2 x)}{117649}+\frac{2280 \log (2+3 x)}{117649}\\ \end{align*}

Mathematica [A]  time = 0.0332864, size = 57, normalized size = 0.75 \[ \frac{-\frac{7 \left (82080 x^4+75240 x^3-31160 x^2-33725 x-3088\right )}{(1-2 x)^2 (3 x+2)^3}-4560 \log (3-6 x)+4560 \log (3 x+2)}{235298} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(-3088 - 33725*x - 31160*x^2 + 75240*x^3 + 82080*x^4))/((1 - 2*x)^2*(2 + 3*x)^3) - 4560*Log[3 - 6*x] + 45
60*Log[2 + 3*x])/235298

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Maple [A]  time = 0.009, size = 63, normalized size = 0.8 \begin{align*}{\frac{22}{2401\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{388}{33614\,x-16807}}-{\frac{2280\,\ln \left ( 2\,x-1 \right ) }{117649}}+{\frac{1}{343\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{87}{4802\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{558}{33614+50421\,x}}+{\frac{2280\,\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)^3/(2+3*x)^4,x)

[Out]

22/2401/(2*x-1)^2-388/16807/(2*x-1)-2280/117649*ln(2*x-1)+1/343/(2+3*x)^3-87/4802/(2+3*x)^2-558/16807/(2+3*x)+
2280/117649*ln(2+3*x)

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Maxima [A]  time = 1.41624, size = 89, normalized size = 1.17 \begin{align*} -\frac{82080 \, x^{4} + 75240 \, x^{3} - 31160 \, x^{2} - 33725 \, x - 3088}{33614 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} + \frac{2280}{117649} \, \log \left (3 \, x + 2\right ) - \frac{2280}{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)^3/(2+3*x)^4,x, algorithm="maxima")

[Out]

-1/33614*(82080*x^4 + 75240*x^3 - 31160*x^2 - 33725*x - 3088)/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)
+ 2280/117649*log(3*x + 2) - 2280/117649*log(2*x - 1)

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Fricas [A]  time = 1.19273, size = 342, normalized size = 4.5 \begin{align*} -\frac{574560 \, x^{4} + 526680 \, x^{3} - 218120 \, x^{2} - 4560 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 4560 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (2 \, x - 1\right ) - 236075 \, x - 21616}{235298 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\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)^4,x, algorithm="fricas")

[Out]

-1/235298*(574560*x^4 + 526680*x^3 - 218120*x^2 - 4560*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(3*x
+ 2) + 4560*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(2*x - 1) - 236075*x - 21616)/(108*x^5 + 108*x
^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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Sympy [A]  time = 0.17391, size = 65, normalized size = 0.86 \begin{align*} - \frac{82080 x^{4} + 75240 x^{3} - 31160 x^{2} - 33725 x - 3088}{3630312 x^{5} + 3630312 x^{4} - 1512630 x^{3} - 1949612 x^{2} + 134456 x + 268912} - \frac{2280 \log{\left (x - \frac{1}{2} \right )}}{117649} + \frac{2280 \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)**3/(2+3*x)**4,x)

[Out]

-(82080*x**4 + 75240*x**3 - 31160*x**2 - 33725*x - 3088)/(3630312*x**5 + 3630312*x**4 - 1512630*x**3 - 1949612
*x**2 + 134456*x + 268912) - 2280*log(x - 1/2)/117649 + 2280*log(x + 2/3)/117649

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Giac [A]  time = 1.80844, size = 74, normalized size = 0.97 \begin{align*} -\frac{82080 \, x^{4} + 75240 \, x^{3} - 31160 \, x^{2} - 33725 \, x - 3088}{33614 \,{\left (3 \, x + 2\right )}^{3}{\left (2 \, x - 1\right )}^{2}} + \frac{2280}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{2280}{117649} \, \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)^4,x, algorithm="giac")

[Out]

-1/33614*(82080*x^4 + 75240*x^3 - 31160*x^2 - 33725*x - 3088)/((3*x + 2)^3*(2*x - 1)^2) + 2280/117649*log(abs(
3*x + 2)) - 2280/117649*log(abs(2*x - 1))