3.1467 \(\int \frac{(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx\)

Optimal. Leaf size=98 \[ -\frac{1936}{823543 (3 x+2)}-\frac{484}{117649 (3 x+2)^2}-\frac{484}{50421 (3 x+2)^3}-\frac{121}{4802 (3 x+2)^4}-\frac{121}{1715 (3 x+2)^5}+\frac{34}{1323 (3 x+2)^6}-\frac{1}{441 (3 x+2)^7}-\frac{3872 \log (1-2 x)}{5764801}+\frac{3872 \log (3 x+2)}{5764801} \]

[Out]

-1/(441*(2 + 3*x)^7) + 34/(1323*(2 + 3*x)^6) - 121/(1715*(2 + 3*x)^5) - 121/(4802*(2 + 3*x)^4) - 484/(50421*(2
+ 3*x)^3) - 484/(117649*(2 + 3*x)^2) - 1936/(823543*(2 + 3*x)) - (3872*Log[1 - 2*x])/5764801 + (3872*Log[2 +
3*x])/5764801

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Rubi [A]  time = 0.0364104, 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} \[ -\frac{1936}{823543 (3 x+2)}-\frac{484}{117649 (3 x+2)^2}-\frac{484}{50421 (3 x+2)^3}-\frac{121}{4802 (3 x+2)^4}-\frac{121}{1715 (3 x+2)^5}+\frac{34}{1323 (3 x+2)^6}-\frac{1}{441 (3 x+2)^7}-\frac{3872 \log (1-2 x)}{5764801}+\frac{3872 \log (3 x+2)}{5764801} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(441*(2 + 3*x)^7) + 34/(1323*(2 + 3*x)^6) - 121/(1715*(2 + 3*x)^5) - 121/(4802*(2 + 3*x)^4) - 484/(50421*(2
+ 3*x)^3) - 484/(117649*(2 + 3*x)^2) - 1936/(823543*(2 + 3*x)) - (3872*Log[1 - 2*x])/5764801 + (3872*Log[2 +
3*x])/5764801

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+3 x)^8} \, dx &=\int \left (-\frac{7744}{5764801 (-1+2 x)}+\frac{1}{21 (2+3 x)^8}-\frac{68}{147 (2+3 x)^7}+\frac{363}{343 (2+3 x)^6}+\frac{726}{2401 (2+3 x)^5}+\frac{1452}{16807 (2+3 x)^4}+\frac{2904}{117649 (2+3 x)^3}+\frac{5808}{823543 (2+3 x)^2}+\frac{11616}{5764801 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{441 (2+3 x)^7}+\frac{34}{1323 (2+3 x)^6}-\frac{121}{1715 (2+3 x)^5}-\frac{121}{4802 (2+3 x)^4}-\frac{484}{50421 (2+3 x)^3}-\frac{484}{117649 (2+3 x)^2}-\frac{1936}{823543 (2+3 x)}-\frac{3872 \log (1-2 x)}{5764801}+\frac{3872 \log (2+3 x)}{5764801}\\ \end{align*}

Mathematica [A]  time = 0.0451267, size = 62, normalized size = 0.63 \[ \frac{8 \left (-\frac{7 \left (381062880 x^6+1746538200 x^5+3454264440 x^4+3858408675 x^3+2692491516 x^2+1098354408 x+193528666\right )}{16 (3 x+2)^7}-65340 \log (1-2 x)+65340 \log (6 x+4)\right )}{778248135} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*((-7*(193528666 + 1098354408*x + 2692491516*x^2 + 3858408675*x^3 + 3454264440*x^4 + 1746538200*x^5 + 381062
880*x^6))/(16*(2 + 3*x)^7) - 65340*Log[1 - 2*x] + 65340*Log[4 + 6*x]))/778248135

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Maple [A]  time = 0.007, size = 81, normalized size = 0.8 \begin{align*} -{\frac{3872\,\ln \left ( 2\,x-1 \right ) }{5764801}}-{\frac{1}{441\, \left ( 2+3\,x \right ) ^{7}}}+{\frac{34}{1323\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{121}{1715\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{121}{4802\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{484}{50421\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{484}{117649\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{1936}{1647086+2470629\,x}}+{\frac{3872\,\ln \left ( 2+3\,x \right ) }{5764801}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3872/5764801*ln(2*x-1)-1/441/(2+3*x)^7+34/1323/(2+3*x)^6-121/1715/(2+3*x)^5-121/4802/(2+3*x)^4-484/50421/(2+3
*x)^3-484/117649/(2+3*x)^2-1936/823543/(2+3*x)+3872/5764801*ln(2+3*x)

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Maxima [A]  time = 1.38679, size = 116, normalized size = 1.18 \begin{align*} -\frac{381062880 \, x^{6} + 1746538200 \, x^{5} + 3454264440 \, x^{4} + 3858408675 \, x^{3} + 2692491516 \, x^{2} + 1098354408 \, x + 193528666}{222356610 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac{3872}{5764801} \, \log \left (3 \, x + 2\right ) - \frac{3872}{5764801} \, \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+3*x)^8,x, algorithm="maxima")

[Out]

-1/222356610*(381062880*x^6 + 1746538200*x^5 + 3454264440*x^4 + 3858408675*x^3 + 2692491516*x^2 + 1098354408*x
+ 193528666)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 3872/5764
801*log(3*x + 2) - 3872/5764801*log(2*x - 1)

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Fricas [A]  time = 1.39409, size = 585, normalized size = 5.97 \begin{align*} -\frac{2667440160 \, x^{6} + 12225767400 \, x^{5} + 24179851080 \, x^{4} + 27008860725 \, x^{3} + 18847440612 \, x^{2} - 1045440 \,{\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 ) + 1045440 \,{\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 ) + 7688480856 \, x + 1354700662}{1556496270 \,{\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)^2/(1-2*x)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/1556496270*(2667440160*x^6 + 12225767400*x^5 + 24179851080*x^4 + 27008860725*x^3 + 18847440612*x^2 - 104544
0*(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) + 1045440*
(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(2*x - 1) + 7688480856
*x + 1354700662)/(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 = 0.21668, size = 85, normalized size = 0.87 \begin{align*} - \frac{381062880 x^{6} + 1746538200 x^{5} + 3454264440 x^{4} + 3858408675 x^{3} + 2692491516 x^{2} + 1098354408 x + 193528666}{486293906070 x^{7} + 2269371561660 x^{6} + 4538743123320 x^{5} + 5043047914800 x^{4} + 3362031943200 x^{3} + 1344812777280 x^{2} + 298847283840 x + 28461646080} - \frac{3872 \log{\left (x - \frac{1}{2} \right )}}{5764801} + \frac{3872 \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)**2/(1-2*x)/(2+3*x)**8,x)

[Out]

-(381062880*x**6 + 1746538200*x**5 + 3454264440*x**4 + 3858408675*x**3 + 2692491516*x**2 + 1098354408*x + 1935
28666)/(486293906070*x**7 + 2269371561660*x**6 + 4538743123320*x**5 + 5043047914800*x**4 + 3362031943200*x**3
+ 1344812777280*x**2 + 298847283840*x + 28461646080) - 3872*log(x - 1/2)/5764801 + 3872*log(x + 2/3)/5764801

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Giac [A]  time = 1.20977, size = 78, normalized size = 0.8 \begin{align*} -\frac{381062880 \, x^{6} + 1746538200 \, x^{5} + 3454264440 \, x^{4} + 3858408675 \, x^{3} + 2692491516 \, x^{2} + 1098354408 \, x + 193528666}{222356610 \,{\left (3 \, x + 2\right )}^{7}} + \frac{3872}{5764801} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{3872}{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)^2/(1-2*x)/(2+3*x)^8,x, algorithm="giac")

[Out]

-1/222356610*(381062880*x^6 + 1746538200*x^5 + 3454264440*x^4 + 3858408675*x^3 + 2692491516*x^2 + 1098354408*x
+ 193528666)/(3*x + 2)^7 + 3872/5764801*log(abs(3*x + 2)) - 3872/5764801*log(abs(2*x - 1))