3.1585 \(\int \frac{(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx\)

Optimal. Leaf size=87 \[ \frac{5324}{117649 (1-2 x)}-\frac{14520}{117649 (3 x+2)}-\frac{3267}{33614 (3 x+2)^2}+\frac{121}{2401 (3 x+2)^3}-\frac{101}{12348 (3 x+2)^4}+\frac{1}{2205 (3 x+2)^5}-\frac{45012 \log (1-2 x)}{823543}+\frac{45012 \log (3 x+2)}{823543} \]

[Out]

5324/(117649*(1 - 2*x)) + 1/(2205*(2 + 3*x)^5) - 101/(12348*(2 + 3*x)^4) + 121/(2401*(2 + 3*x)^3) - 3267/(3361
4*(2 + 3*x)^2) - 14520/(117649*(2 + 3*x)) - (45012*Log[1 - 2*x])/823543 + (45012*Log[2 + 3*x])/823543

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Rubi [A]  time = 0.0422226, antiderivative size = 87, 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} \[ \frac{5324}{117649 (1-2 x)}-\frac{14520}{117649 (3 x+2)}-\frac{3267}{33614 (3 x+2)^2}+\frac{121}{2401 (3 x+2)^3}-\frac{101}{12348 (3 x+2)^4}+\frac{1}{2205 (3 x+2)^5}-\frac{45012 \log (1-2 x)}{823543}+\frac{45012 \log (3 x+2)}{823543} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5324/(117649*(1 - 2*x)) + 1/(2205*(2 + 3*x)^5) - 101/(12348*(2 + 3*x)^4) + 121/(2401*(2 + 3*x)^3) - 3267/(3361
4*(2 + 3*x)^2) - 14520/(117649*(2 + 3*x)) - (45012*Log[1 - 2*x])/823543 + (45012*Log[2 + 3*x])/823543

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{(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx &=\int \left (\frac{10648}{117649 (-1+2 x)^2}-\frac{90024}{823543 (-1+2 x)}-\frac{1}{147 (2+3 x)^6}+\frac{101}{1029 (2+3 x)^5}-\frac{1089}{2401 (2+3 x)^4}+\frac{9801}{16807 (2+3 x)^3}+\frac{43560}{117649 (2+3 x)^2}+\frac{135036}{823543 (2+3 x)}\right ) \, dx\\ &=\frac{5324}{117649 (1-2 x)}+\frac{1}{2205 (2+3 x)^5}-\frac{101}{12348 (2+3 x)^4}+\frac{121}{2401 (2+3 x)^3}-\frac{3267}{33614 (2+3 x)^2}-\frac{14520}{117649 (2+3 x)}-\frac{45012 \log (1-2 x)}{823543}+\frac{45012 \log (2+3 x)}{823543}\\ \end{align*}

Mathematica [A]  time = 0.0595437, size = 64, normalized size = 0.74 \[ \frac{2 \left (-\frac{7 \left (656274960 x^5+1804756140 x^4+1747028250 x^3+649342770 x^2+25985087 x-23684986\right )}{8 (2 x-1) (3 x+2)^5}-1012770 \log (1-2 x)+1012770 \log (6 x+4)\right )}{37059435} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-7*(-23684986 + 25985087*x + 649342770*x^2 + 1747028250*x^3 + 1804756140*x^4 + 656274960*x^5))/(8*(-1 + 2
*x)*(2 + 3*x)^5) - 1012770*Log[1 - 2*x] + 1012770*Log[4 + 6*x]))/37059435

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Maple [A]  time = 0.01, size = 72, normalized size = 0.8 \begin{align*} -{\frac{5324}{235298\,x-117649}}-{\frac{45012\,\ln \left ( 2\,x-1 \right ) }{823543}}+{\frac{1}{2205\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{101}{12348\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{121}{2401\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{3267}{33614\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{14520}{235298+352947\,x}}+{\frac{45012\,\ln \left ( 2+3\,x \right ) }{823543}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5324/117649/(2*x-1)-45012/823543*ln(2*x-1)+1/2205/(2+3*x)^5-101/12348/(2+3*x)^4+121/2401/(2+3*x)^3-3267/33614
/(2+3*x)^2-14520/117649/(2+3*x)+45012/823543*ln(2+3*x)

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Maxima [A]  time = 1.2992, size = 103, normalized size = 1.18 \begin{align*} -\frac{656274960 \, x^{5} + 1804756140 \, x^{4} + 1747028250 \, x^{3} + 649342770 \, x^{2} + 25985087 \, x - 23684986}{21176820 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} + \frac{45012}{823543} \, \log \left (3 \, x + 2\right ) - \frac{45012}{823543} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21176820*(656274960*x^5 + 1804756140*x^4 + 1747028250*x^3 + 649342770*x^2 + 25985087*x - 23684986)/(486*x^6
+ 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32) + 45012/823543*log(3*x + 2) - 45012/823543*log(2*x -
1)

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Fricas [A]  time = 1.3636, size = 474, normalized size = 5.45 \begin{align*} -\frac{4593924720 \, x^{5} + 12633292980 \, x^{4} + 12229197750 \, x^{3} + 4545399390 \, x^{2} - 8102160 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (3 \, x + 2\right ) + 8102160 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (2 \, x - 1\right ) + 181895609 \, x - 165794902}{148237740 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/148237740*(4593924720*x^5 + 12633292980*x^4 + 12229197750*x^3 + 4545399390*x^2 - 8102160*(486*x^6 + 1377*x^
5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*log(3*x + 2) + 8102160*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3
- 240*x^2 - 176*x - 32)*log(2*x - 1) + 181895609*x - 165794902)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 24
0*x^2 - 176*x - 32)

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Sympy [A]  time = 0.19444, size = 75, normalized size = 0.86 \begin{align*} - \frac{656274960 x^{5} + 1804756140 x^{4} + 1747028250 x^{3} + 649342770 x^{2} + 25985087 x - 23684986}{10291934520 x^{6} + 29160481140 x^{5} + 28588707000 x^{4} + 7623655200 x^{3} - 5082436800 x^{2} - 3727120320 x - 677658240} - \frac{45012 \log{\left (x - \frac{1}{2} \right )}}{823543} + \frac{45012 \log{\left (x + \frac{2}{3} \right )}}{823543} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(656274960*x**5 + 1804756140*x**4 + 1747028250*x**3 + 649342770*x**2 + 25985087*x - 23684986)/(10291934520*x*
*6 + 29160481140*x**5 + 28588707000*x**4 + 7623655200*x**3 - 5082436800*x**2 - 3727120320*x - 677658240) - 450
12*log(x - 1/2)/823543 + 45012*log(x + 2/3)/823543

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Giac [A]  time = 1.85396, size = 105, normalized size = 1.21 \begin{align*} -\frac{5324}{117649 \,{\left (2 \, x - 1\right )}} + \frac{2 \,{\left (\frac{204418935}{2 \, x - 1} + \frac{740244225}{{\left (2 \, x - 1\right )}^{2}} + \frac{1185622375}{{\left (2 \, x - 1\right )}^{3}} + \frac{709135350}{{\left (2 \, x - 1\right )}^{4}} + 21049983\right )}}{4117715 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{5}} + \frac{45012}{823543} \, \log \left ({\left | -\frac{7}{2 \, x - 1} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5324/117649/(2*x - 1) + 2/4117715*(204418935/(2*x - 1) + 740244225/(2*x - 1)^2 + 1185622375/(2*x - 1)^3 + 709
135350/(2*x - 1)^4 + 21049983)/(7/(2*x - 1) + 3)^5 + 45012/823543*log(abs(-7/(2*x - 1) - 3))