3.1450 \(\int \frac{3+5 x}{(1-2 x) (2+3 x)^8} \, dx\)

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

[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

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Rubi [A]  time = 0.0335698, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{352}{823543 (3 x+2)}-\frac{88}{117649 (3 x+2)^2}-\frac{88}{50421 (3 x+2)^3}-\frac{11}{2401 (3 x+2)^4}-\frac{22}{1715 (3 x+2)^5}-\frac{11}{294 (3 x+2)^6}+\frac{1}{147 (3 x+2)^7}-\frac{704 \log (1-2 x)}{5764801}+\frac{704 \log (3 x+2)}{5764801} \]

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*}

Mathematica [A]  time = 0.0408961, size = 60, normalized size = 0.61 \[ \frac{-\frac{7 \left (7698240 x^6+35283600 x^5+69783120 x^4+77947650 x^3+54393768 x^2+25308459 x+5811068\right )}{(3 x+2)^7}-21120 \log (3-6 x)+21120 \log (3 x+2)}{172944030} \]

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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Maple [A]  time = 0.009, size = 81, normalized size = 0.8 \begin{align*} -{\frac{704\,\ln \left ( 2\,x-1 \right ) }{5764801}}+{\frac{1}{147\, \left ( 2+3\,x \right ) ^{7}}}-{\frac{11}{294\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{22}{1715\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{11}{2401\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{88}{50421\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{88}{117649\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{352}{1647086+2470629\,x}}+{\frac{704\,\ln \left ( 2+3\,x \right ) }{5764801}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-704/5764801*ln(2*x-1)+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)+704/5764801*ln(2+3*x)

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Maxima [A]  time = 1.7997, size = 116, normalized size = 1.18 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.60998, size = 560, normalized size = 5.71 \begin{align*} -\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{align*}

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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Sympy [A]  time = 0.210728, size = 85, normalized size = 0.87 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.72303, size = 78, normalized size = 0.8 \begin{align*} -\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{align*}

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))