∫ (3+5x)3 ________________________

3.1586 \(\int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^7} \, dx\)

Optimal. Leaf size=98 \[ \frac {10648}{823543 (1-2 x)}-\frac {45012}{823543 (3 x+2)}-\frac {7260}{117649 (3 x+2)^2}-\frac {1089}{16807 (3 x+2)^3}+\frac {363}{9604 (3 x+2)^4}-\frac {101}{15435 (3 x+2)^5}+\frac {1}{2646 (3 x+2)^6}-\frac {17424 \log (1-2 x)}{823543}+\frac {17424 \log (3 x+2)}{823543} \]

[Out]

10648/823543/(1-2*x)+1/2646/(2+3*x)^6-101/15435/(2+3*x)^5+363/9604/(2+3*x)^4-1089/16807/(2+3*x)^3-7260/117649/

(2+3*x)^2-45012/823543/(2+3*x)-17424/823543*ln(1-2*x)+17424/823543*ln(2+3*x)

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Rubi [A] time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, 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} \[ \frac {10648}{823543 (1-2 x)}-\frac {45012}{823543 (3 x+2)}-\frac {7260}{117649 (3 x+2)^2}-\frac {1089}{16807 (3 x+2)^3}+\frac {363}{9604 (3 x+2)^4}-\frac {101}{15435 (3 x+2)^5}+\frac {1}{2646 (3 x+2)^6}-\frac {17424 \log (1-2 x)}{823543}+\frac {17424 \log (3 x+2)}{823543} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

10648/(823543*(1 - 2*x)) + 1/(2646*(2 + 3*x)^6) - 101/(15435*(2 + 3*x)^5) + 363/(9604*(2 + 3*x)^4) - 1089/(168

07*(2 + 3*x)^3) - 7260/(117649*(2 + 3*x)^2) - 45012/(823543*(2 + 3*x)) - (17424*Log[1 - 2*x])/823543 + (17424*

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)^7} \, dx &=\int \left (\frac {21296}{823543 (-1+2 x)^2}-\frac {34848}{823543 (-1+2 x)}-\frac {1}{147 (2+3 x)^7}+\frac {101}{1029 (2+3 x)^6}-\frac {1089}{2401 (2+3 x)^5}+\frac {9801}{16807 (2+3 x)^4}+\frac {43560}{117649 (2+3 x)^3}+\frac {135036}{823543 (2+3 x)^2}+\frac {52272}{823543 (2+3 x)}\right ) \, dx\\ &=\frac {10648}{823543 (1-2 x)}+\frac {1}{2646 (2+3 x)^6}-\frac {101}{15435 (2+3 x)^5}+\frac {363}{9604 (2+3 x)^4}-\frac {1089}{16807 (2+3 x)^3}-\frac {7260}{117649 (2+3 x)^2}-\frac {45012}{823543 (2+3 x)}-\frac {17424 \log (1-2 x)}{823543}+\frac {17424 \log (2+3 x)}{823543}\\ \end {align*}

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Mathematica [A] time = 0.10, size = 69, normalized size = 0.70 \[ \frac {4 \left (-\frac {7 \left (2286377280 x^6+7811789040 x^5+10278112680 x^4+5935583610 x^3+887377581 x^2-461259404 x-145404842\right )}{16 (2 x-1) (3 x+2)^6}-588060 \log (1-2 x)+588060 \log (6 x+4)\right )}{111178305} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*((-7*(-145404842 - 461259404*x + 887377581*x^2 + 5935583610*x^3 + 10278112680*x^4 + 7811789040*x^5 + 228637

7280*x^6))/(16*(-1 + 2*x)*(2 + 3*x)^6) - 588060*Log[1 - 2*x] + 588060*Log[4 + 6*x]))/111178305

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fricas [A] time = 0.59, size = 140, normalized size = 1.43 \[ -\frac {16004640960 \, x^{6} + 54682523280 \, x^{5} + 71946788760 \, x^{4} + 41549085270 \, x^{3} + 6211643067 \, x^{2} - 9408960 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (3 \, x + 2\right ) + 9408960 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (2 \, x - 1\right ) - 3228815828 \, x - 1017833894}{444713220 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/444713220*(16004640960*x^6 + 54682523280*x^5 + 71946788760*x^4 + 41549085270*x^3 + 6211643067*x^2 - 9408960

*(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(3*x + 2) + 9408960*(1458*x^7 + 5103*x

^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(2*x - 1) - 3228815828*x - 1017833894)/(1458*x^7 + 5103*x

^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)

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giac [A] time = 0.89, size = 87, normalized size = 0.89 \[ -\frac {10648}{823543 \, {\left (2 \, x - 1\right )}} + \frac {4 \, {\left (\frac {1421066052}{2 \, x - 1} + \frac {7028898345}{{\left (2 \, x - 1\right )}^{2}} + \frac {17396565550}{{\left (2 \, x - 1\right )}^{3}} + \frac {21521363500}{{\left (2 \, x - 1\right )}^{4}} + \frac {10637822580}{{\left (2 \, x - 1\right )}^{5}} + 115177113\right )}}{28824005 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{6}} + \frac {17424}{823543} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10648/823543/(2*x - 1) + 4/28824005*(1421066052/(2*x - 1) + 7028898345/(2*x - 1)^2 + 17396565550/(2*x - 1)^3

+ 21521363500/(2*x - 1)^4 + 10637822580/(2*x - 1)^5 + 115177113)/(7/(2*x - 1) + 3)^6 + 17424/823543*log(abs(-7

/(2*x - 1) - 3))

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maple [A] time = 0.01, size = 81, normalized size = 0.83 \[ -\frac {17424 \ln \left (2 x -1\right )}{823543}+\frac {17424 \ln \left (3 x +2\right )}{823543}+\frac {1}{2646 \left (3 x +2\right )^{6}}-\frac {101}{15435 \left (3 x +2\right )^{5}}+\frac {363}{9604 \left (3 x +2\right )^{4}}-\frac {1089}{16807 \left (3 x +2\right )^{3}}-\frac {7260}{117649 \left (3 x +2\right )^{2}}-\frac {45012}{823543 \left (3 x +2\right )}-\frac {10648}{823543 \left (2 x -1\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2646/(3*x+2)^6-101/15435/(3*x+2)^5+363/9604/(3*x+2)^4-1089/16807/(3*x+2)^3-7260/117649/(3*x+2)^2-45012/82354

3/(3*x+2)+17424/823543*ln(3*x+2)-10648/823543/(2*x-1)-17424/823543*ln(2*x-1)

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maxima [A] time = 0.67, size = 81, normalized size = 0.83 \[ -\frac {2286377280 \, x^{6} + 7811789040 \, x^{5} + 10278112680 \, x^{4} + 5935583610 \, x^{3} + 887377581 \, x^{2} - 461259404 \, x - 145404842}{63530460 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} + \frac {17424}{823543} \, \log \left (3 \, x + 2\right ) - \frac {17424}{823543} \, \log \left (2 \, x - 1\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/63530460*(2286377280*x^6 + 7811789040*x^5 + 10278112680*x^4 + 5935583610*x^3 + 887377581*x^2 - 461259404*x

- 145404842)/(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64) + 17424/823543*log(3*x + 2) -

17424/823543*log(2*x - 1)

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mupad [B] time = 1.07, size = 71, normalized size = 0.72 \[ \frac {34848\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {2904\,x^6}{117649}+\frac {9922\,x^5}{117649}+\frac {117491\,x^4}{1058841}+\frac {271403\,x^3}{4235364}+\frac {98597509\,x^2}{10291934520}-\frac {115314851\,x}{23156852670}-\frac {72702421}{46313705340}}{x^7+\frac {7\,x^6}{2}+\frac {14\,x^5}{3}+\frac {70\,x^4}{27}-\frac {56\,x^2}{81}-\frac {224\,x}{729}-\frac {32}{729}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(34848*atanh((12*x)/7 + 1/7))/823543 - ((98597509*x^2)/10291934520 - (115314851*x)/23156852670 + (271403*x^3)/

4235364 + (117491*x^4)/1058841 + (9922*x^5)/117649 + (2904*x^6)/117649 - 72702421/46313705340)/((70*x^4)/27 -

(56*x^2)/81 - (224*x)/729 + (14*x^5)/3 + (7*x^6)/2 + x^7 - 32/729)

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sympy [A] time = 0.22, size = 80, normalized size = 0.82 \[ \frac {- 2286377280 x^{6} - 7811789040 x^{5} - 10278112680 x^{4} - 5935583610 x^{3} - 887377581 x^{2} + 461259404 x + 145404842}{92627410680 x^{7} + 324195937380 x^{6} + 432261249840 x^{5} + 240145138800 x^{4} - 64038703680 x^{2} - 28461646080 x - 4065949440} - \frac {17424 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {17424 \log {\left (x + \frac {2}{3} \right )}}{823543} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2286377280*x**6 - 7811789040*x**5 - 10278112680*x**4 - 5935583610*x**3 - 887377581*x**2 + 461259404*x + 1454

04842)/(92627410680*x**7 + 324195937380*x**6 + 432261249840*x**5 + 240145138800*x**4 - 64038703680*x**2 - 2846

1646080*x - 4065949440) - 17424*log(x - 1/2)/823543 + 17424*log(x + 2/3)/823543

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