3.14.11 \(\int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx\)

Optimal. Leaf size=98 \[ -\frac {352}{823543 (3 x+2)}-\frac {88}{117649 (3 x+2)^2}-\frac {88}{50421 (3 x+2)^3}-\frac {11}{2401 (3 x+2)^4}-\frac {22}{1715 (3 x+2)^5}-\frac {11}{294 (3 x+2)^6}+\frac {1}{147 (3 x+2)^7}-\frac {704 \log (1-2 x)}{5764801}+\frac {704 \log (3 x+2)}{5764801} \]

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Rubi [A]  time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} -\frac {352}{823543 (3 x+2)}-\frac {88}{117649 (3 x+2)^2}-\frac {88}{50421 (3 x+2)^3}-\frac {11}{2401 (3 x+2)^4}-\frac {22}{1715 (3 x+2)^5}-\frac {11}{294 (3 x+2)^6}+\frac {1}{147 (3 x+2)^7}-\frac {704 \log (1-2 x)}{5764801}+\frac {704 \log (3 x+2)}{5764801} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^8),x]

[Out]

1/(147*(2 + 3*x)^7) - 11/(294*(2 + 3*x)^6) - 22/(1715*(2 + 3*x)^5) - 11/(2401*(2 + 3*x)^4) - 88/(50421*(2 + 3*
x)^3) - 88/(117649*(2 + 3*x)^2) - 352/(823543*(2 + 3*x)) - (704*Log[1 - 2*x])/5764801 + (704*Log[2 + 3*x])/576
4801

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx &=\int \left (-\frac {1408}{5764801 (-1+2 x)}-\frac {1}{7 (2+3 x)^8}+\frac {33}{49 (2+3 x)^7}+\frac {66}{343 (2+3 x)^6}+\frac {132}{2401 (2+3 x)^5}+\frac {264}{16807 (2+3 x)^4}+\frac {528}{117649 (2+3 x)^3}+\frac {1056}{823543 (2+3 x)^2}+\frac {2112}{5764801 (2+3 x)}\right ) \, dx\\ &=\frac {1}{147 (2+3 x)^7}-\frac {11}{294 (2+3 x)^6}-\frac {22}{1715 (2+3 x)^5}-\frac {11}{2401 (2+3 x)^4}-\frac {88}{50421 (2+3 x)^3}-\frac {88}{117649 (2+3 x)^2}-\frac {352}{823543 (2+3 x)}-\frac {704 \log (1-2 x)}{5764801}+\frac {704 \log (2+3 x)}{5764801}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 60, normalized size = 0.61 \begin {gather*} \frac {-\frac {7 \left (7698240 x^6+35283600 x^5+69783120 x^4+77947650 x^3+54393768 x^2+25308459 x+5811068\right )}{(3 x+2)^7}-21120 \log (3-6 x)+21120 \log (3 x+2)}{172944030} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^8),x]

[Out]

((-7*(5811068 + 25308459*x + 54393768*x^2 + 77947650*x^3 + 69783120*x^4 + 35283600*x^5 + 7698240*x^6))/(2 + 3*
x)^7 - 21120*Log[3 - 6*x] + 21120*Log[2 + 3*x])/172944030

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^8),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^8), x]

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fricas [A]  time = 1.34, size = 155, normalized size = 1.58 \begin {gather*} -\frac {53887680 \, x^{6} + 246985200 \, x^{5} + 488481840 \, x^{4} + 545633550 \, x^{3} + 380756376 \, x^{2} - 21120 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 21120 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (2 \, x - 1\right ) + 177159213 \, x + 40677476}{172944030 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/172944030*(53887680*x^6 + 246985200*x^5 + 488481840*x^4 + 545633550*x^3 + 380756376*x^2 - 21120*(2187*x^7 +
10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(3*x + 2) + 21120*(2187*x^7 + 102
06*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(2*x - 1) + 177159213*x + 40677476)/(
2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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giac [A]  time = 1.10, size = 58, normalized size = 0.59 \begin {gather*} -\frac {7698240 \, x^{6} + 35283600 \, x^{5} + 69783120 \, x^{4} + 77947650 \, x^{3} + 54393768 \, x^{2} + 25308459 \, x + 5811068}{24706290 \, {\left (3 \, x + 2\right )}^{7}} + \frac {704}{5764801} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {704}{5764801} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)^8,x, algorithm="giac")

[Out]

-1/24706290*(7698240*x^6 + 35283600*x^5 + 69783120*x^4 + 77947650*x^3 + 54393768*x^2 + 25308459*x + 5811068)/(
3*x + 2)^7 + 704/5764801*log(abs(3*x + 2)) - 704/5764801*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 81, normalized size = 0.83 \begin {gather*} -\frac {704 \ln \left (2 x -1\right )}{5764801}+\frac {704 \ln \left (3 x +2\right )}{5764801}+\frac {1}{147 \left (3 x +2\right )^{7}}-\frac {11}{294 \left (3 x +2\right )^{6}}-\frac {22}{1715 \left (3 x +2\right )^{5}}-\frac {11}{2401 \left (3 x +2\right )^{4}}-\frac {88}{50421 \left (3 x +2\right )^{3}}-\frac {88}{117649 \left (3 x +2\right )^{2}}-\frac {352}{823543 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)/(1-2*x)/(3*x+2)^8,x)

[Out]

1/147/(3*x+2)^7-11/294/(3*x+2)^6-22/1715/(3*x+2)^5-11/2401/(3*x+2)^4-88/50421/(3*x+2)^3-88/117649/(3*x+2)^2-35
2/823543/(3*x+2)+704/5764801*ln(3*x+2)-704/5764801*ln(2*x-1)

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maxima [A]  time = 0.47, size = 86, normalized size = 0.88 \begin {gather*} -\frac {7698240 \, x^{6} + 35283600 \, x^{5} + 69783120 \, x^{4} + 77947650 \, x^{3} + 54393768 \, x^{2} + 25308459 \, x + 5811068}{24706290 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {704}{5764801} \, \log \left (3 \, x + 2\right ) - \frac {704}{5764801} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)^8,x, algorithm="maxima")

[Out]

-1/24706290*(7698240*x^6 + 35283600*x^5 + 69783120*x^4 + 77947650*x^3 + 54393768*x^2 + 25308459*x + 5811068)/(
2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 704/5764801*log(3*x + 2)
- 704/5764801*log(2*x - 1)

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mupad [B]  time = 0.05, size = 76, normalized size = 0.78 \begin {gather*} \frac {1408\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{5764801}-\frac {\frac {352\,x^6}{2470629}+\frac {4840\,x^5}{7411887}+\frac {86152\,x^4}{66706983}+\frac {288695\,x^3}{200120949}+\frac {335764\,x^2}{333534915}+\frac {2812051\,x}{6003628470}+\frac {2905534}{27016328115}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 3)/((2*x - 1)*(3*x + 2)^8),x)

[Out]

(1408*atanh((12*x)/7 + 1/7))/5764801 - ((2812051*x)/6003628470 + (335764*x^2)/333534915 + (288695*x^3)/2001209
49 + (86152*x^4)/66706983 + (4840*x^5)/7411887 + (352*x^6)/2470629 + 2905534/27016328115)/((448*x)/729 + (224*
x^2)/81 + (560*x^3)/81 + (280*x^4)/27 + (28*x^5)/3 + (14*x^6)/3 + x^7 + 128/2187)

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sympy [A]  time = 0.22, size = 85, normalized size = 0.87 \begin {gather*} - \frac {7698240 x^{6} + 35283600 x^{5} + 69783120 x^{4} + 77947650 x^{3} + 54393768 x^{2} + 25308459 x + 5811068}{54032656230 x^{7} + 252152395740 x^{6} + 504304791480 x^{5} + 560338657200 x^{4} + 373559104800 x^{3} + 149423641920 x^{2} + 33205253760 x + 3162405120} - \frac {704 \log {\left (x - \frac {1}{2} \right )}}{5764801} + \frac {704 \log {\left (x + \frac {2}{3} \right )}}{5764801} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)**8,x)

[Out]

-(7698240*x**6 + 35283600*x**5 + 69783120*x**4 + 77947650*x**3 + 54393768*x**2 + 25308459*x + 5811068)/(540326
56230*x**7 + 252152395740*x**6 + 504304791480*x**5 + 560338657200*x**4 + 373559104800*x**3 + 149423641920*x**2
+ 33205253760*x + 3162405120) - 704*log(x - 1/2)/5764801 + 704*log(x + 2/3)/5764801

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