3.1674 \(\int \frac{(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=55 \[ -\frac{243 x^2}{80}-\frac{10287 x}{400}-\frac{156065}{1936 (1-2 x)}+\frac{16807}{704 (1-2 x)^2}-\frac{543655 \log (1-2 x)}{10648}+\frac{\log (5 x+3)}{166375} \]

[Out]

16807/(704*(1 - 2*x)^2) - 156065/(1936*(1 - 2*x)) - (10287*x)/400 - (243*x^2)/80 - (543655*Log[1 - 2*x])/10648
+ Log[3 + 5*x]/166375

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Rubi [A]  time = 0.0278863, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ -\frac{243 x^2}{80}-\frac{10287 x}{400}-\frac{156065}{1936 (1-2 x)}+\frac{16807}{704 (1-2 x)^2}-\frac{543655 \log (1-2 x)}{10648}+\frac{\log (5 x+3)}{166375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16807/(704*(1 - 2*x)^2) - 156065/(1936*(1 - 2*x)) - (10287*x)/400 - (243*x^2)/80 - (543655*Log[1 - 2*x])/10648
+ Log[3 + 5*x]/166375

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^5}{(1-2 x)^3 (3+5 x)} \, dx &=\int \left (-\frac{10287}{400}-\frac{243 x}{40}-\frac{16807}{176 (-1+2 x)^3}-\frac{156065}{968 (-1+2 x)^2}-\frac{543655}{5324 (-1+2 x)}+\frac{1}{33275 (3+5 x)}\right ) \, dx\\ &=\frac{16807}{704 (1-2 x)^2}-\frac{156065}{1936 (1-2 x)}-\frac{10287 x}{400}-\frac{243 x^2}{80}-\frac{543655 \log (1-2 x)}{10648}+\frac{\log (3+5 x)}{166375}\\ \end{align*}

Mathematica [A]  time = 0.0318272, size = 55, normalized size = 1. \[ \frac{-1293732 (5 x+3)^2-47005596 (5 x+3)+\frac{858357500}{2 x-1}+\frac{254205875}{(1-2 x)^2}-543655000 \log (5-10 x)+64 \log (5 x+3)}{10648000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(254205875/(1 - 2*x)^2 + 858357500/(-1 + 2*x) - 47005596*(3 + 5*x) - 1293732*(3 + 5*x)^2 - 543655000*Log[5 - 1
0*x] + 64*Log[3 + 5*x])/10648000

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Maple [A]  time = 0.007, size = 44, normalized size = 0.8 \begin{align*} -{\frac{243\,{x}^{2}}{80}}-{\frac{10287\,x}{400}}+{\frac{16807}{704\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{156065}{3872\,x-1936}}-{\frac{543655\,\ln \left ( 2\,x-1 \right ) }{10648}}+{\frac{\ln \left ( 3+5\,x \right ) }{166375}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/80*x^2-10287/400*x+16807/704/(2*x-1)^2+156065/1936/(2*x-1)-543655/10648*ln(2*x-1)+1/166375*ln(3+5*x)

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Maxima [A]  time = 1.07639, size = 59, normalized size = 1.07 \begin{align*} -\frac{243}{80} \, x^{2} - \frac{10287}{400} \, x + \frac{2401 \,{\left (520 \, x - 183\right )}}{7744 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1}{166375} \, \log \left (5 \, x + 3\right ) - \frac{543655}{10648} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/80*x^2 - 10287/400*x + 2401/7744*(520*x - 183)/(4*x^2 - 4*x + 1) + 1/166375*log(5*x + 3) - 543655/10648*l
og(2*x - 1)

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Fricas [A]  time = 1.53616, size = 251, normalized size = 4.56 \begin{align*} -\frac{129373200 \, x^{4} + 965986560 \, x^{3} - 1063016460 \, x^{2} - 64 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 543655000 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 1442875060 \, x + 604151625}{10648000 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10648000*(129373200*x^4 + 965986560*x^3 - 1063016460*x^2 - 64*(4*x^2 - 4*x + 1)*log(5*x + 3) + 543655000*(4
*x^2 - 4*x + 1)*log(2*x - 1) - 1442875060*x + 604151625)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.164062, size = 44, normalized size = 0.8 \begin{align*} - \frac{243 x^{2}}{80} - \frac{10287 x}{400} + \frac{1248520 x - 439383}{30976 x^{2} - 30976 x + 7744} - \frac{543655 \log{\left (x - \frac{1}{2} \right )}}{10648} + \frac{\log{\left (x + \frac{3}{5} \right )}}{166375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x**2/80 - 10287*x/400 + (1248520*x - 439383)/(30976*x**2 - 30976*x + 7744) - 543655*log(x - 1/2)/10648 +
log(x + 3/5)/166375

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Giac [A]  time = 1.38987, size = 55, normalized size = 1. \begin{align*} -\frac{243}{80} \, x^{2} - \frac{10287}{400} \, x + \frac{2401 \,{\left (520 \, x - 183\right )}}{7744 \,{\left (2 \, x - 1\right )}^{2}} + \frac{1}{166375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{543655}{10648} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/80*x^2 - 10287/400*x + 2401/7744*(520*x - 183)/(2*x - 1)^2 + 1/166375*log(abs(5*x + 3)) - 543655/10648*lo
g(abs(2*x - 1))