3.1517 \(\int \frac{(2+3 x)^7}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=69 \[ -\frac{2187 x^4}{1000}-\frac{24543 x^3}{2500}-\frac{1044657 x^2}{50000}-\frac{339309 x}{10000}-\frac{233}{9453125 (5 x+3)}-\frac{1}{1718750 (5 x+3)^2}-\frac{823543 \log (1-2 x)}{42592}+\frac{4667 \log (5 x+3)}{20796875} \]

[Out]

(-339309*x)/10000 - (1044657*x^2)/50000 - (24543*x^3)/2500 - (2187*x^4)/1000 - 1/(1718750*(3 + 5*x)^2) - 233/(
9453125*(3 + 5*x)) - (823543*Log[1 - 2*x])/42592 + (4667*Log[3 + 5*x])/20796875

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Rubi [A]  time = 0.0320366, 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} \[ -\frac{2187 x^4}{1000}-\frac{24543 x^3}{2500}-\frac{1044657 x^2}{50000}-\frac{339309 x}{10000}-\frac{233}{9453125 (5 x+3)}-\frac{1}{1718750 (5 x+3)^2}-\frac{823543 \log (1-2 x)}{42592}+\frac{4667 \log (5 x+3)}{20796875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-339309*x)/10000 - (1044657*x^2)/50000 - (24543*x^3)/2500 - (2187*x^4)/1000 - 1/(1718750*(3 + 5*x)^2) - 233/(
9453125*(3 + 5*x)) - (823543*Log[1 - 2*x])/42592 + (4667*Log[3 + 5*x])/20796875

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+5 x)^3} \, dx &=\int \left (-\frac{339309}{10000}-\frac{1044657 x}{25000}-\frac{73629 x^2}{2500}-\frac{2187 x^3}{250}-\frac{823543}{21296 (-1+2 x)}+\frac{1}{171875 (3+5 x)^3}+\frac{233}{1890625 (3+5 x)^2}+\frac{4667}{4159375 (3+5 x)}\right ) \, dx\\ &=-\frac{339309 x}{10000}-\frac{1044657 x^2}{50000}-\frac{24543 x^3}{2500}-\frac{2187 x^4}{1000}-\frac{1}{1718750 (3+5 x)^2}-\frac{233}{9453125 (3+5 x)}-\frac{823543 \log (1-2 x)}{42592}+\frac{4667 \log (3+5 x)}{20796875}\\ \end{align*}

Mathematica [A]  time = 0.0340121, size = 60, normalized size = 0.87 \[ \frac{-\frac{11 \left (66156750000 x^6+376358400000 x^5+1012198275000 x^4+1891740015000 x^3+746752646475 x^2-485450731630 x-256487424349\right )}{(5 x+3)^2}-257357187500 \log (1-2 x)+2986880 \log (10 x+6)}{13310000000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-256487424349 - 485450731630*x + 746752646475*x^2 + 1891740015000*x^3 + 1012198275000*x^4 + 37635840000
0*x^5 + 66156750000*x^6))/(3 + 5*x)^2 - 257357187500*Log[1 - 2*x] + 2986880*Log[6 + 10*x])/13310000000

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Maple [A]  time = 0.008, size = 54, normalized size = 0.8 \begin{align*} -{\frac{2187\,{x}^{4}}{1000}}-{\frac{24543\,{x}^{3}}{2500}}-{\frac{1044657\,{x}^{2}}{50000}}-{\frac{339309\,x}{10000}}-{\frac{823543\,\ln \left ( 2\,x-1 \right ) }{42592}}-{\frac{1}{1718750\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{233}{28359375+47265625\,x}}+{\frac{4667\,\ln \left ( 3+5\,x \right ) }{20796875}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2187/1000*x^4-24543/2500*x^3-1044657/50000*x^2-339309/10000*x-823543/42592*ln(2*x-1)-1/1718750/(3+5*x)^2-233/
9453125/(3+5*x)+4667/20796875*ln(3+5*x)

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Maxima [A]  time = 1.10428, size = 73, normalized size = 1.06 \begin{align*} -\frac{2187}{1000} \, x^{4} - \frac{24543}{2500} \, x^{3} - \frac{1044657}{50000} \, x^{2} - \frac{339309}{10000} \, x - \frac{2330 \, x + 1409}{18906250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{4667}{20796875} \, \log \left (5 \, x + 3\right ) - \frac{823543}{42592} \, \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+5*x)^3,x, algorithm="maxima")

[Out]

-2187/1000*x^4 - 24543/2500*x^3 - 1044657/50000*x^2 - 339309/10000*x - 1/18906250*(2330*x + 1409)/(25*x^2 + 30
*x + 9) + 4667/20796875*log(5*x + 3) - 823543/42592*log(2*x - 1)

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Fricas [A]  time = 1.39214, size = 338, normalized size = 4.9 \begin{align*} -\frac{181931062500 \, x^{6} + 1034985600000 \, x^{5} + 2783545256250 \, x^{4} + 5202285041250 \, x^{3} + 4012849402650 \, x^{2} - 746720 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 64339296875 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 1016146037830 \, x + 247984}{3327500000 \,{\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)^7/(1-2*x)/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/3327500000*(181931062500*x^6 + 1034985600000*x^5 + 2783545256250*x^4 + 5202285041250*x^3 + 4012849402650*x^
2 - 746720*(25*x^2 + 30*x + 9)*log(5*x + 3) + 64339296875*(25*x^2 + 30*x + 9)*log(2*x - 1) + 1016146037830*x +
247984)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.172626, size = 60, normalized size = 0.87 \begin{align*} - \frac{2187 x^{4}}{1000} - \frac{24543 x^{3}}{2500} - \frac{1044657 x^{2}}{50000} - \frac{339309 x}{10000} - \frac{2330 x + 1409}{472656250 x^{2} + 567187500 x + 170156250} - \frac{823543 \log{\left (x - \frac{1}{2} \right )}}{42592} + \frac{4667 \log{\left (x + \frac{3}{5} \right )}}{20796875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2187*x**4/1000 - 24543*x**3/2500 - 1044657*x**2/50000 - 339309*x/10000 - (2330*x + 1409)/(472656250*x**2 + 56
7187500*x + 170156250) - 823543*log(x - 1/2)/42592 + 4667*log(x + 3/5)/20796875

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Giac [A]  time = 2.86701, size = 69, normalized size = 1. \begin{align*} -\frac{2187}{1000} \, x^{4} - \frac{24543}{2500} \, x^{3} - \frac{1044657}{50000} \, x^{2} - \frac{339309}{10000} \, x - \frac{2330 \, x + 1409}{18906250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{4667}{20796875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{823543}{42592} \, \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+5*x)^3,x, algorithm="giac")

[Out]

-2187/1000*x^4 - 24543/2500*x^3 - 1044657/50000*x^2 - 339309/10000*x - 1/18906250*(2330*x + 1409)/(5*x + 3)^2
+ 4667/20796875*log(abs(5*x + 3)) - 823543/42592*log(abs(2*x - 1))