3.1672 \(\int \frac{(2+3 x)^7}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=69 \[ -\frac{2187 x^4}{160}-\frac{40581 x^3}{400}-\frac{792423 x^2}{2000}-\frac{26161299 x}{20000}-\frac{5764801}{3872 (1-2 x)}+\frac{823543}{2816 (1-2 x)^2}-\frac{269063263 \log (1-2 x)}{170368}+\frac{\log (5 x+3)}{4159375} \]

[Out]

823543/(2816*(1 - 2*x)^2) - 5764801/(3872*(1 - 2*x)) - (26161299*x)/20000 - (792423*x^2)/2000 - (40581*x^3)/40
0 - (2187*x^4)/160 - (269063263*Log[1 - 2*x])/170368 + Log[3 + 5*x]/4159375

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Rubi [A]  time = 0.0342934, antiderivative size = 69, 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} \[ -\frac{2187 x^4}{160}-\frac{40581 x^3}{400}-\frac{792423 x^2}{2000}-\frac{26161299 x}{20000}-\frac{5764801}{3872 (1-2 x)}+\frac{823543}{2816 (1-2 x)^2}-\frac{269063263 \log (1-2 x)}{170368}+\frac{\log (5 x+3)}{4159375} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)),x]

[Out]

823543/(2816*(1 - 2*x)^2) - 5764801/(3872*(1 - 2*x)) - (26161299*x)/20000 - (792423*x^2)/2000 - (40581*x^3)/40
0 - (2187*x^4)/160 - (269063263*Log[1 - 2*x])/170368 + 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)^7}{(1-2 x)^3 (3+5 x)} \, dx &=\int \left (-\frac{26161299}{20000}-\frac{792423 x}{1000}-\frac{121743 x^2}{400}-\frac{2187 x^3}{40}-\frac{823543}{704 (-1+2 x)^3}-\frac{5764801}{1936 (-1+2 x)^2}-\frac{269063263}{85184 (-1+2 x)}+\frac{1}{831875 (3+5 x)}\right ) \, dx\\ &=\frac{823543}{2816 (1-2 x)^2}-\frac{5764801}{3872 (1-2 x)}-\frac{26161299 x}{20000}-\frac{792423 x^2}{2000}-\frac{40581 x^3}{400}-\frac{2187 x^4}{160}-\frac{269063263 \log (1-2 x)}{170368}+\frac{\log (3+5 x)}{4159375}\\ \end{align*}

Mathematica [A]  time = 0.0353426, size = 60, normalized size = 0.87 \[ \frac{-\frac{55 \left (1058508000 x^6+6797973600 x^5+23090763960 x^4+72578051568 x^3-42333890544 x^2-83615877112 x+35985148011\right )}{(1-2 x)^2}-1681645393750 \log (5-10 x)+256 \log (5 x+3)}{1064800000} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)),x]

[Out]

((-55*(35985148011 - 83615877112*x - 42333890544*x^2 + 72578051568*x^3 + 23090763960*x^4 + 6797973600*x^5 + 10
58508000*x^6))/(1 - 2*x)^2 - 1681645393750*Log[5 - 10*x] + 256*Log[3 + 5*x])/1064800000

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Maple [A]  time = 0.008, size = 54, normalized size = 0.8 \begin{align*} -{\frac{2187\,{x}^{4}}{160}}-{\frac{40581\,{x}^{3}}{400}}-{\frac{792423\,{x}^{2}}{2000}}-{\frac{26161299\,x}{20000}}+{\frac{823543}{2816\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{5764801}{7744\,x-3872}}-{\frac{269063263\,\ln \left ( 2\,x-1 \right ) }{170368}}+{\frac{\ln \left ( 3+5\,x \right ) }{4159375}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^7/(1-2*x)^3/(3+5*x),x)

[Out]

-2187/160*x^4-40581/400*x^3-792423/2000*x^2-26161299/20000*x+823543/2816/(2*x-1)^2+5764801/3872/(2*x-1)-269063
263/170368*ln(2*x-1)+1/4159375*ln(3+5*x)

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Maxima [A]  time = 1.08527, size = 73, normalized size = 1.06 \begin{align*} -\frac{2187}{160} \, x^{4} - \frac{40581}{400} \, x^{3} - \frac{792423}{2000} \, x^{2} - \frac{26161299}{20000} \, x + \frac{823543 \,{\left (112 \, x - 45\right )}}{30976 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1}{4159375} \, \log \left (5 \, x + 3\right ) - \frac{269063263}{170368} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)^3/(3+5*x),x, algorithm="maxima")

[Out]

-2187/160*x^4 - 40581/400*x^3 - 792423/2000*x^2 - 26161299/20000*x + 823543/30976*(112*x - 45)/(4*x^2 - 4*x +
1) + 1/4159375*log(5*x + 3) - 269063263/170368*log(2*x - 1)

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Fricas [A]  time = 1.7851, size = 335, normalized size = 4.86 \begin{align*} -\frac{58217940000 \, x^{6} + 373888548000 \, x^{5} + 1269992017800 \, x^{4} + 3991792836240 \, x^{3} - 5149424229840 \, x^{2} - 256 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 1681645393750 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 1777812991240 \, x + 1273918078125}{1064800000 \,{\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)^7/(1-2*x)^3/(3+5*x),x, algorithm="fricas")

[Out]

-1/1064800000*(58217940000*x^6 + 373888548000*x^5 + 1269992017800*x^4 + 3991792836240*x^3 - 5149424229840*x^2
- 256*(4*x^2 - 4*x + 1)*log(5*x + 3) + 1681645393750*(4*x^2 - 4*x + 1)*log(2*x - 1) - 1777812991240*x + 127391
8078125)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.168051, size = 58, normalized size = 0.84 \begin{align*} - \frac{2187 x^{4}}{160} - \frac{40581 x^{3}}{400} - \frac{792423 x^{2}}{2000} - \frac{26161299 x}{20000} + \frac{92236816 x - 37059435}{123904 x^{2} - 123904 x + 30976} - \frac{269063263 \log{\left (x - \frac{1}{2} \right )}}{170368} + \frac{\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)**7/(1-2*x)**3/(3+5*x),x)

[Out]

-2187*x**4/160 - 40581*x**3/400 - 792423*x**2/2000 - 26161299*x/20000 + (92236816*x - 37059435)/(123904*x**2 -
123904*x + 30976) - 269063263*log(x - 1/2)/170368 + log(x + 3/5)/4159375

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Giac [A]  time = 2.79757, size = 69, normalized size = 1. \begin{align*} -\frac{2187}{160} \, x^{4} - \frac{40581}{400} \, x^{3} - \frac{792423}{2000} \, x^{2} - \frac{26161299}{20000} \, x + \frac{823543 \,{\left (112 \, x - 45\right )}}{30976 \,{\left (2 \, x - 1\right )}^{2}} + \frac{1}{4159375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{269063263}{170368} \, \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)^7/(1-2*x)^3/(3+5*x),x, algorithm="giac")

[Out]

-2187/160*x^4 - 40581/400*x^3 - 792423/2000*x^2 - 26161299/20000*x + 823543/30976*(112*x - 45)/(2*x - 1)^2 + 1
/4159375*log(abs(5*x + 3)) - 269063263/170368*log(abs(2*x - 1))