3.1658 \(\int \frac{(3+5 x)^2}{(1-2 x)^3 (2+3 x)^6} \, dx\)

Optimal. Leaf size=98 \[ \frac{11264}{823543 (1-2 x)}-\frac{24040}{823543 (3 x+2)}+\frac{484}{117649 (1-2 x)^2}-\frac{2875}{117649 (3 x+2)^2}-\frac{829}{50421 (3 x+2)^3}+\frac{16}{2401 (3 x+2)^4}-\frac{1}{1715 (3 x+2)^5}-\frac{11696 \log (1-2 x)}{823543}+\frac{11696 \log (3 x+2)}{823543} \]

[Out]

484/(117649*(1 - 2*x)^2) + 11264/(823543*(1 - 2*x)) - 1/(1715*(2 + 3*x)^5) + 16/(2401*(2 + 3*x)^4) - 829/(5042
1*(2 + 3*x)^3) - 2875/(117649*(2 + 3*x)^2) - 24040/(823543*(2 + 3*x)) - (11696*Log[1 - 2*x])/823543 + (11696*L
og[2 + 3*x])/823543

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Rubi [A]  time = 0.0511099, antiderivative size = 98, 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} \[ \frac{11264}{823543 (1-2 x)}-\frac{24040}{823543 (3 x+2)}+\frac{484}{117649 (1-2 x)^2}-\frac{2875}{117649 (3 x+2)^2}-\frac{829}{50421 (3 x+2)^3}+\frac{16}{2401 (3 x+2)^4}-\frac{1}{1715 (3 x+2)^5}-\frac{11696 \log (1-2 x)}{823543}+\frac{11696 \log (3 x+2)}{823543} \]

Antiderivative was successfully verified.

[In]

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

[Out]

484/(117649*(1 - 2*x)^2) + 11264/(823543*(1 - 2*x)) - 1/(1715*(2 + 3*x)^5) + 16/(2401*(2 + 3*x)^4) - 829/(5042
1*(2 + 3*x)^3) - 2875/(117649*(2 + 3*x)^2) - 24040/(823543*(2 + 3*x)) - (11696*Log[1 - 2*x])/823543 + (11696*L
og[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)^2}{(1-2 x)^3 (2+3 x)^6} \, dx &=\int \left (-\frac{1936}{117649 (-1+2 x)^3}+\frac{22528}{823543 (-1+2 x)^2}-\frac{23392}{823543 (-1+2 x)}+\frac{3}{343 (2+3 x)^6}-\frac{192}{2401 (2+3 x)^5}+\frac{2487}{16807 (2+3 x)^4}+\frac{17250}{117649 (2+3 x)^3}+\frac{72120}{823543 (2+3 x)^2}+\frac{35088}{823543 (2+3 x)}\right ) \, dx\\ &=\frac{484}{117649 (1-2 x)^2}+\frac{11264}{823543 (1-2 x)}-\frac{1}{1715 (2+3 x)^5}+\frac{16}{2401 (2+3 x)^4}-\frac{829}{50421 (2+3 x)^3}-\frac{2875}{117649 (2+3 x)^2}-\frac{24040}{823543 (2+3 x)}-\frac{11696 \log (1-2 x)}{823543}+\frac{11696 \log (2+3 x)}{823543}\\ \end{align*}

Mathematica [A]  time = 0.0593101, size = 69, normalized size = 0.7 \[ \frac{4 \left (-\frac{7 \left (28421280 x^6+63947880 x^5+36579240 x^4-14484765 x^3-19495039 x^2-4230956 x+258089\right )}{4 (1-2 x)^2 (3 x+2)^5}-43860 \log (1-2 x)+43860 \log (6 x+4)\right )}{12353145} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-7*(258089 - 4230956*x - 19495039*x^2 - 14484765*x^3 + 36579240*x^4 + 63947880*x^5 + 28421280*x^6))/(4*(1
- 2*x)^2*(2 + 3*x)^5) - 43860*Log[1 - 2*x] + 43860*Log[4 + 6*x]))/12353145

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Maple [A]  time = 0.009, size = 81, normalized size = 0.8 \begin{align*}{\frac{484}{117649\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{11264}{1647086\,x-823543}}-{\frac{11696\,\ln \left ( 2\,x-1 \right ) }{823543}}-{\frac{1}{1715\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{16}{2401\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{829}{50421\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{2875}{117649\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{24040}{1647086+2470629\,x}}+{\frac{11696\,\ln \left ( 2+3\,x \right ) }{823543}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

484/117649/(2*x-1)^2-11264/823543/(2*x-1)-11696/823543*ln(2*x-1)-1/1715/(2+3*x)^5+16/2401/(2+3*x)^4-829/50421/
(2+3*x)^3-2875/117649/(2+3*x)^2-24040/823543/(2+3*x)+11696/823543*ln(2+3*x)

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Maxima [A]  time = 1.02704, size = 116, normalized size = 1.18 \begin{align*} -\frac{28421280 \, x^{6} + 63947880 \, x^{5} + 36579240 \, x^{4} - 14484765 \, x^{3} - 19495039 \, x^{2} - 4230956 \, x + 258089}{1764735 \,{\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} + \frac{11696}{823543} \, \log \left (3 \, x + 2\right ) - \frac{11696}{823543} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1764735*(28421280*x^6 + 63947880*x^5 + 36579240*x^4 - 14484765*x^3 - 19495039*x^2 - 4230956*x + 258089)/(97
2*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32) + 11696/823543*log(3*x + 2) - 11696/82
3543*log(2*x - 1)

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Fricas [A]  time = 1.53059, size = 520, normalized size = 5.31 \begin{align*} -\frac{198948960 \, x^{6} + 447635160 \, x^{5} + 256054680 \, x^{4} - 101393355 \, x^{3} - 136465273 \, x^{2} - 175440 \,{\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 175440 \,{\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (2 \, x - 1\right ) - 29616692 \, x + 1806623}{12353145 \,{\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12353145*(198948960*x^6 + 447635160*x^5 + 256054680*x^4 - 101393355*x^3 - 136465273*x^2 - 175440*(972*x^7 +
2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)*log(3*x + 2) + 175440*(972*x^7 + 2268*x^6 + 1
323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)*log(2*x - 1) - 29616692*x + 1806623)/(972*x^7 + 2268*x^6 +
1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)

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Sympy [A]  time = 0.223885, size = 85, normalized size = 0.87 \begin{align*} - \frac{28421280 x^{6} + 63947880 x^{5} + 36579240 x^{4} - 14484765 x^{3} - 19495039 x^{2} - 4230956 x + 258089}{1715322420 x^{7} + 4002418980 x^{6} + 2334744405 x^{5} - 1111783050 x^{4} - 1482377400 x^{3} - 197650320 x^{2} + 197650320 x + 56471520} - \frac{11696 \log{\left (x - \frac{1}{2} \right )}}{823543} + \frac{11696 \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)**2/(1-2*x)**3/(2+3*x)**6,x)

[Out]

-(28421280*x**6 + 63947880*x**5 + 36579240*x**4 - 14484765*x**3 - 19495039*x**2 - 4230956*x + 258089)/(1715322
420*x**7 + 4002418980*x**6 + 2334744405*x**5 - 1111783050*x**4 - 1482377400*x**3 - 197650320*x**2 + 197650320*
x + 56471520) - 11696*log(x - 1/2)/823543 + 11696*log(x + 2/3)/823543

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Giac [A]  time = 1.96801, size = 88, normalized size = 0.9 \begin{align*} -\frac{28421280 \, x^{6} + 63947880 \, x^{5} + 36579240 \, x^{4} - 14484765 \, x^{3} - 19495039 \, x^{2} - 4230956 \, x + 258089}{1764735 \,{\left (3 \, x + 2\right )}^{5}{\left (2 \, x - 1\right )}^{2}} + \frac{11696}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{11696}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1764735*(28421280*x^6 + 63947880*x^5 + 36579240*x^4 - 14484765*x^3 - 19495039*x^2 - 4230956*x + 258089)/((3
*x + 2)^5*(2*x - 1)^2) + 11696/823543*log(abs(3*x + 2)) - 11696/823543*log(abs(2*x - 1))