3.1569 \(\int \frac{(3+5 x)^2}{(1-2 x)^2 (2+3 x)^7} \, dx\)

Optimal. Leaf size=98 \[ \frac{1936}{823543 (1-2 x)}-\frac{11264}{823543 (3 x+2)}-\frac{2090}{117649 (3 x+2)^2}-\frac{1364}{50421 (3 x+2)^3}-\frac{319}{9604 (3 x+2)^4}+\frac{22}{1715 (3 x+2)^5}-\frac{1}{882 (3 x+2)^6}-\frac{4048 \log (1-2 x)}{823543}+\frac{4048 \log (3 x+2)}{823543} \]

[Out]

1936/(823543*(1 - 2*x)) - 1/(882*(2 + 3*x)^6) + 22/(1715*(2 + 3*x)^5) - 319/(9604*(2 + 3*x)^4) - 1364/(50421*(
2 + 3*x)^3) - 2090/(117649*(2 + 3*x)^2) - 11264/(823543*(2 + 3*x)) - (4048*Log[1 - 2*x])/823543 + (4048*Log[2
+ 3*x])/823543

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Rubi [A]  time = 0.0488627, 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} \[ \frac{1936}{823543 (1-2 x)}-\frac{11264}{823543 (3 x+2)}-\frac{2090}{117649 (3 x+2)^2}-\frac{1364}{50421 (3 x+2)^3}-\frac{319}{9604 (3 x+2)^4}+\frac{22}{1715 (3 x+2)^5}-\frac{1}{882 (3 x+2)^6}-\frac{4048 \log (1-2 x)}{823543}+\frac{4048 \log (3 x+2)}{823543} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1936/(823543*(1 - 2*x)) - 1/(882*(2 + 3*x)^6) + 22/(1715*(2 + 3*x)^5) - 319/(9604*(2 + 3*x)^4) - 1364/(50421*(
2 + 3*x)^3) - 2090/(117649*(2 + 3*x)^2) - 11264/(823543*(2 + 3*x)) - (4048*Log[1 - 2*x])/823543 + (4048*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)^2}{(1-2 x)^2 (2+3 x)^7} \, dx &=\int \left (\frac{3872}{823543 (-1+2 x)^2}-\frac{8096}{823543 (-1+2 x)}+\frac{1}{49 (2+3 x)^7}-\frac{66}{343 (2+3 x)^6}+\frac{957}{2401 (2+3 x)^5}+\frac{4092}{16807 (2+3 x)^4}+\frac{12540}{117649 (2+3 x)^3}+\frac{33792}{823543 (2+3 x)^2}+\frac{12144}{823543 (2+3 x)}\right ) \, dx\\ &=\frac{1936}{823543 (1-2 x)}-\frac{1}{882 (2+3 x)^6}+\frac{22}{1715 (2+3 x)^5}-\frac{319}{9604 (2+3 x)^4}-\frac{1364}{50421 (2+3 x)^3}-\frac{2090}{117649 (2+3 x)^2}-\frac{11264}{823543 (2+3 x)}-\frac{4048 \log (1-2 x)}{823543}+\frac{4048 \log (2+3 x)}{823543}\\ \end{align*}

Mathematica [A]  time = 0.0560313, size = 69, normalized size = 0.7 \[ \frac{4 \left (-\frac{7 \left (177059520 x^6+604953360 x^5+795948120 x^4+459657990 x^3+48220029 x^2-60874336 x-18979078\right )}{16 (2 x-1) (3 x+2)^6}-45540 \log (1-2 x)+45540 \log (6 x+4)\right )}{37059435} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-7*(-18979078 - 60874336*x + 48220029*x^2 + 459657990*x^3 + 795948120*x^4 + 604953360*x^5 + 177059520*x^6
))/(16*(-1 + 2*x)*(2 + 3*x)^6) - 45540*Log[1 - 2*x] + 45540*Log[4 + 6*x]))/37059435

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Maple [A]  time = 0.01, size = 81, normalized size = 0.8 \begin{align*} -{\frac{1936}{1647086\,x-823543}}-{\frac{4048\,\ln \left ( 2\,x-1 \right ) }{823543}}-{\frac{1}{882\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{22}{1715\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{319}{9604\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{1364}{50421\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{2090}{117649\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{11264}{1647086+2470629\,x}}+{\frac{4048\,\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)^2/(2+3*x)^7,x)

[Out]

-1936/823543/(2*x-1)-4048/823543*ln(2*x-1)-1/882/(2+3*x)^6+22/1715/(2+3*x)^5-319/9604/(2+3*x)^4-1364/50421/(2+
3*x)^3-2090/117649/(2+3*x)^2-11264/823543/(2+3*x)+4048/823543*ln(2+3*x)

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Maxima [A]  time = 2.5574, size = 109, normalized size = 1.11 \begin{align*} -\frac{177059520 \, x^{6} + 604953360 \, x^{5} + 795948120 \, x^{4} + 459657990 \, x^{3} + 48220029 \, x^{2} - 60874336 \, x - 18979078}{21176820 \,{\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} + \frac{4048}{823543} \, \log \left (3 \, x + 2\right ) - \frac{4048}{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)^2/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/21176820*(177059520*x^6 + 604953360*x^5 + 795948120*x^4 + 459657990*x^3 + 48220029*x^2 - 60874336*x - 18979
078)/(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64) + 4048/823543*log(3*x + 2) - 4048/823
543*log(2*x - 1)

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Fricas [A]  time = 1.32344, size = 502, normalized size = 5.12 \begin{align*} -\frac{1239416640 \, x^{6} + 4234673520 \, x^{5} + 5571636840 \, x^{4} + 3217605930 \, x^{3} + 337540203 \, x^{2} - 728640 \,{\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (3 \, x + 2\right ) + 728640 \,{\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (2 \, x - 1\right ) - 426120352 \, x - 132853546}{148237740 \,{\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/148237740*(1239416640*x^6 + 4234673520*x^5 + 5571636840*x^4 + 3217605930*x^3 + 337540203*x^2 - 728640*(1458
*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(3*x + 2) + 728640*(1458*x^7 + 5103*x^6 + 68
04*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(2*x - 1) - 426120352*x - 132853546)/(1458*x^7 + 5103*x^6 + 6804
*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)

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Sympy [A]  time = 0.206948, size = 80, normalized size = 0.82 \begin{align*} - \frac{177059520 x^{6} + 604953360 x^{5} + 795948120 x^{4} + 459657990 x^{3} + 48220029 x^{2} - 60874336 x - 18979078}{30875803560 x^{7} + 108065312460 x^{6} + 144087083280 x^{5} + 80048379600 x^{4} - 21346234560 x^{2} - 9487215360 x - 1355316480} - \frac{4048 \log{\left (x - \frac{1}{2} \right )}}{823543} + \frac{4048 \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)**2/(2+3*x)**7,x)

[Out]

-(177059520*x**6 + 604953360*x**5 + 795948120*x**4 + 459657990*x**3 + 48220029*x**2 - 60874336*x - 18979078)/(
30875803560*x**7 + 108065312460*x**6 + 144087083280*x**5 + 80048379600*x**4 - 21346234560*x**2 - 9487215360*x
- 1355316480) - 4048*log(x - 1/2)/823543 + 4048*log(x + 2/3)/823543

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Giac [A]  time = 2.84799, size = 117, normalized size = 1.19 \begin{align*} -\frac{1936}{823543 \,{\left (2 \, x - 1\right )}} + \frac{4 \,{\left (\frac{407084454}{2 \, x - 1} + \frac{2053765665}{{\left (2 \, x - 1\right )}^{2}} + \frac{5220014100}{{\left (2 \, x - 1\right )}^{3}} + \frac{6680782500}{{\left (2 \, x - 1\right )}^{4}} + \frac{3440056760}{{\left (2 \, x - 1\right )}^{5}} + 32498901\right )}}{28824005 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{6}} + \frac{4048}{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)^2/(1-2*x)^2/(2+3*x)^7,x, algorithm="giac")

[Out]

-1936/823543/(2*x - 1) + 4/28824005*(407084454/(2*x - 1) + 2053765665/(2*x - 1)^2 + 5220014100/(2*x - 1)^3 + 6
680782500/(2*x - 1)^4 + 3440056760/(2*x - 1)^5 + 32498901)/(7/(2*x - 1) + 3)^6 + 4048/823543*log(abs(-7/(2*x -
1) - 3))