3.1518 \(\int \frac{(2+3 x)^6}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=62 \[ -\frac{243 x^3}{250}-\frac{19683 x^2}{5000}-\frac{216999 x}{25000}-\frac{8}{75625 (5 x+3)}-\frac{1}{343750 (5 x+3)^2}-\frac{117649 \log (1-2 x)}{21296}+\frac{3347 \log (5 x+3)}{4159375} \]

[Out]

(-216999*x)/25000 - (19683*x^2)/5000 - (243*x^3)/250 - 1/(343750*(3 + 5*x)^2) - 8/(75625*(3 + 5*x)) - (117649*
Log[1 - 2*x])/21296 + (3347*Log[3 + 5*x])/4159375

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Rubi [A]  time = 0.0280522, antiderivative size = 62, 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} \[ -\frac{243 x^3}{250}-\frac{19683 x^2}{5000}-\frac{216999 x}{25000}-\frac{8}{75625 (5 x+3)}-\frac{1}{343750 (5 x+3)^2}-\frac{117649 \log (1-2 x)}{21296}+\frac{3347 \log (5 x+3)}{4159375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-216999*x)/25000 - (19683*x^2)/5000 - (243*x^3)/250 - 1/(343750*(3 + 5*x)^2) - 8/(75625*(3 + 5*x)) - (117649*
Log[1 - 2*x])/21296 + (3347*Log[3 + 5*x])/4159375

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)^3} \, dx &=\int \left (-\frac{216999}{25000}-\frac{19683 x}{2500}-\frac{729 x^2}{250}-\frac{117649}{10648 (-1+2 x)}+\frac{1}{34375 (3+5 x)^3}+\frac{8}{15125 (3+5 x)^2}+\frac{3347}{831875 (3+5 x)}\right ) \, dx\\ &=-\frac{216999 x}{25000}-\frac{19683 x^2}{5000}-\frac{243 x^3}{250}-\frac{1}{343750 (3+5 x)^2}-\frac{8}{75625 (3+5 x)}-\frac{117649 \log (1-2 x)}{21296}+\frac{3347 \log (3+5 x)}{4159375}\\ \end{align*}

Mathematica [A]  time = 0.0416947, size = 56, normalized size = 0.9 \[ \frac{11 \left (-58806000 x^3-238164300 x^2-525137580 x-\frac{6400}{5 x+3}-\frac{176}{(5 x+3)^2}[phone]\right )-3676531250 \log (1-2 x)+535520 \log (10 x+6)}{665500000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*(329460615 - 525137580*x - 238164300*x^2 - 58806000*x^3 - 176/(3 + 5*x)^2 - 6400/(3 + 5*x)) - 3676531250*L
og[1 - 2*x] + 535520*Log[6 + 10*x])/665500000

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Maple [A]  time = 0.009, size = 49, normalized size = 0.8 \begin{align*} -{\frac{243\,{x}^{3}}{250}}-{\frac{19683\,{x}^{2}}{5000}}-{\frac{216999\,x}{25000}}-{\frac{117649\,\ln \left ( 2\,x-1 \right ) }{21296}}-{\frac{1}{343750\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{8}{226875+378125\,x}}+{\frac{3347\,\ln \left ( 3+5\,x \right ) }{4159375}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/250*x^3-19683/5000*x^2-216999/25000*x-117649/21296*ln(2*x-1)-1/343750/(3+5*x)^2-8/75625/(3+5*x)+3347/4159
375*ln(3+5*x)

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Maxima [A]  time = 1.07037, size = 66, normalized size = 1.06 \begin{align*} -\frac{243}{250} \, x^{3} - \frac{19683}{5000} \, x^{2} - \frac{216999}{25000} \, x - \frac{2000 \, x + 1211}{3781250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{3347}{4159375} \, \log \left (5 \, x + 3\right ) - \frac{117649}{21296} \, \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)^3,x, algorithm="maxima")

[Out]

-243/250*x^3 - 19683/5000*x^2 - 216999/25000*x - 1/3781250*(2000*x + 1211)/(25*x^2 + 30*x + 9) + 3347/4159375*
log(5*x + 3) - 117649/21296*log(2*x - 1)

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Fricas [A]  time = 1.28982, size = 296, normalized size = 4.77 \begin{align*} -\frac{8085825000 \, x^{5} + 42450581250 \, x^{4} + 114414423750 \, x^{3} + 98436833550 \, x^{2} - 267760 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 1838265625 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 25994486210 \, x + 106568}{332750000 \,{\left (25 \, x^{2} + 30 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

-1/332750000*(8085825000*x^5 + 42450581250*x^4 + 114414423750*x^3 + 98436833550*x^2 - 267760*(25*x^2 + 30*x +
9)*log(5*x + 3) + 1838265625*(25*x^2 + 30*x + 9)*log(2*x - 1) + 25994486210*x + 106568)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.169284, size = 53, normalized size = 0.85 \begin{align*} - \frac{243 x^{3}}{250} - \frac{19683 x^{2}}{5000} - \frac{216999 x}{25000} - \frac{2000 x + 1211}{94531250 x^{2} + 113437500 x + 34031250} - \frac{117649 \log{\left (x - \frac{1}{2} \right )}}{21296} + \frac{3347 \log{\left (x + \frac{3}{5} \right )}}{4159375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x**3/250 - 19683*x**2/5000 - 216999*x/25000 - (2000*x + 1211)/(94531250*x**2 + 113437500*x + 34031250) -
117649*log(x - 1/2)/21296 + 3347*log(x + 3/5)/4159375

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Giac [A]  time = 1.23283, size = 62, normalized size = 1. \begin{align*} -\frac{243}{250} \, x^{3} - \frac{19683}{5000} \, x^{2} - \frac{216999}{25000} \, x - \frac{2000 \, x + 1211}{3781250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{3347}{4159375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{117649}{21296} \, \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)^6/(1-2*x)/(3+5*x)^3,x, algorithm="giac")

[Out]

-243/250*x^3 - 19683/5000*x^2 - 216999/25000*x - 1/3781250*(2000*x + 1211)/(5*x + 3)^2 + 3347/4159375*log(abs(
5*x + 3)) - 117649/21296*log(abs(2*x - 1))