3.1503 \(\int \frac{(2+3 x)^6}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=58 \[ -\frac{729 x^4}{200}-\frac{8019 x^3}{500}-\frac{335097 x^2}{10000}-\frac{2682909 x}{50000}-\frac{1}{171875 (5 x+3)}-\frac{117649 \log (1-2 x)}{3872}+\frac{8 \log (5 x+3)}{75625} \]

[Out]

(-2682909*x)/50000 - (335097*x^2)/10000 - (8019*x^3)/500 - (729*x^4)/200 - 1/(171875*(3 + 5*x)) - (117649*Log[
1 - 2*x])/3872 + (8*Log[3 + 5*x])/75625

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Rubi [A]  time = 0.0262073, antiderivative size = 58, 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} \[ -\frac{729 x^4}{200}-\frac{8019 x^3}{500}-\frac{335097 x^2}{10000}-\frac{2682909 x}{50000}-\frac{1}{171875 (5 x+3)}-\frac{117649 \log (1-2 x)}{3872}+\frac{8 \log (5 x+3)}{75625} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6/((1 - 2*x)*(3 + 5*x)^2),x]

[Out]

(-2682909*x)/50000 - (335097*x^2)/10000 - (8019*x^3)/500 - (729*x^4)/200 - 1/(171875*(3 + 5*x)) - (117649*Log[
1 - 2*x])/3872 + (8*Log[3 + 5*x])/75625

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)^6}{(1-2 x) (3+5 x)^2} \, dx &=\int \left (-\frac{2682909}{50000}-\frac{335097 x}{5000}-\frac{24057 x^2}{500}-\frac{729 x^3}{50}-\frac{117649}{1936 (-1+2 x)}+\frac{1}{34375 (3+5 x)^2}+\frac{8}{15125 (3+5 x)}\right ) \, dx\\ &=-\frac{2682909 x}{50000}-\frac{335097 x^2}{10000}-\frac{8019 x^3}{500}-\frac{729 x^4}{200}-\frac{1}{171875 (3+5 x)}-\frac{117649 \log (1-2 x)}{3872}+\frac{8 \log (3+5 x)}{75625}\\ \end{align*}

Mathematica [A]  time = 0.0332528, size = 54, normalized size = 0.93 \[ \frac{-80190000 x^4-352836000 x^3-737213400 x^2-1180479960 x-\frac{128}{5 x+3}[phone]}{22000000}-\frac{117649 \log (1-2 x)}{3872}+\frac{8 \log (10 x+6)}{75625} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^6/((1 - 2*x)*(3 + 5*x)^2),x]

[Out]

(823659705 - 1180479960*x - 737213400*x^2 - 352836000*x^3 - 80190000*x^4 - 128/(3 + 5*x))/22000000 - (117649*L
og[1 - 2*x])/3872 + (8*Log[6 + 10*x])/75625

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Maple [A]  time = 0.007, size = 45, normalized size = 0.8 \begin{align*} -{\frac{729\,{x}^{4}}{200}}-{\frac{8019\,{x}^{3}}{500}}-{\frac{335097\,{x}^{2}}{10000}}-{\frac{2682909\,x}{50000}}-{\frac{117649\,\ln \left ( 2\,x-1 \right ) }{3872}}-{\frac{1}{515625+859375\,x}}+{\frac{8\,\ln \left ( 3+5\,x \right ) }{75625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6/(1-2*x)/(3+5*x)^2,x)

[Out]

-729/200*x^4-8019/500*x^3-335097/10000*x^2-2682909/50000*x-117649/3872*ln(2*x-1)-1/171875/(3+5*x)+8/75625*ln(3
+5*x)

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Maxima [A]  time = 1.08362, size = 59, normalized size = 1.02 \begin{align*} -\frac{729}{200} \, x^{4} - \frac{8019}{500} \, x^{3} - \frac{335097}{10000} \, x^{2} - \frac{2682909}{50000} \, x - \frac{1}{171875 \,{\left (5 \, x + 3\right )}} + \frac{8}{75625} \, \log \left (5 \, x + 3\right ) - \frac{117649}{3872} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)/(3+5*x)^2,x, algorithm="maxima")

[Out]

-729/200*x^4 - 8019/500*x^3 - 335097/10000*x^2 - 2682909/50000*x - 1/171875/(5*x + 3) + 8/75625*log(5*x + 3) -
117649/3872*log(2*x - 1)

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Fricas [A]  time = 1.2807, size = 243, normalized size = 4.19 \begin{align*} -\frac{1102612500 \, x^{5} + 5513062500 \, x^{4} + 13047581250 \, x^{3} + 22313610000 \, x^{2} - 6400 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 1838265625 \,{\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 9738959670 \, x + 352}{60500000 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/60500000*(1102612500*x^5 + 5513062500*x^4 + 13047581250*x^3 + 22313610000*x^2 - 6400*(5*x + 3)*log(5*x + 3)
+ 1838265625*(5*x + 3)*log(2*x - 1) + 9738959670*x + 352)/(5*x + 3)

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Sympy [A]  time = 0.148803, size = 51, normalized size = 0.88 \begin{align*} - \frac{729 x^{4}}{200} - \frac{8019 x^{3}}{500} - \frac{335097 x^{2}}{10000} - \frac{2682909 x}{50000} - \frac{117649 \log{\left (x - \frac{1}{2} \right )}}{3872} + \frac{8 \log{\left (x + \frac{3}{5} \right )}}{75625} - \frac{1}{859375 x + 515625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6/(1-2*x)/(3+5*x)**2,x)

[Out]

-729*x**4/200 - 8019*x**3/500 - 335097*x**2/10000 - 2682909*x/50000 - 117649*log(x - 1/2)/3872 + 8*log(x + 3/5
)/75625 - 1/(859375*x + 515625)

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Giac [A]  time = 1.80109, size = 109, normalized size = 1.88 \begin{align*} -\frac{27}{250000} \,{\left (5 \, x + 3\right )}^{4}{\left (\frac{540}{5 \, x + 3} + \frac{4635}{{\left (5 \, x + 3\right )}^{2}} + \frac{51145}{{\left (5 \, x + 3\right )}^{3}} + 54\right )} - \frac{1}{171875 \,{\left (5 \, x + 3\right )}} + \frac{607689}{20000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{117649}{3872} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)/(3+5*x)^2,x, algorithm="giac")

[Out]

-27/250000*(5*x + 3)^4*(540/(5*x + 3) + 4635/(5*x + 3)^2 + 51145/(5*x + 3)^3 + 54) - 1/171875/(5*x + 3) + 6076
89/20000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 117649/3872*log(abs(-11/(5*x + 3) + 2))