3.1657 \(\int \frac{(3+5 x)^2}{(1-2 x)^3 (2+3 x)^5} \, dx\)

Optimal. Leaf size=87 \[ \frac{4180}{117649 (1-2 x)}-\frac{5750}{117649 (3 x+2)}+\frac{242}{16807 (1-2 x)^2}-\frac{829}{33614 (3 x+2)^2}+\frac{64}{7203 (3 x+2)^3}-\frac{1}{1372 (3 x+2)^4}-\frac{24040 \log (1-2 x)}{823543}+\frac{24040 \log (3 x+2)}{823543} \]

[Out]

242/(16807*(1 - 2*x)^2) + 4180/(117649*(1 - 2*x)) - 1/(1372*(2 + 3*x)^4) + 64/(7203*(2 + 3*x)^3) - 829/(33614*
(2 + 3*x)^2) - 5750/(117649*(2 + 3*x)) - (24040*Log[1 - 2*x])/823543 + (24040*Log[2 + 3*x])/823543

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Rubi [A]  time = 0.0441871, antiderivative size = 87, 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} \[ \frac{4180}{117649 (1-2 x)}-\frac{5750}{117649 (3 x+2)}+\frac{242}{16807 (1-2 x)^2}-\frac{829}{33614 (3 x+2)^2}+\frac{64}{7203 (3 x+2)^3}-\frac{1}{1372 (3 x+2)^4}-\frac{24040 \log (1-2 x)}{823543}+\frac{24040 \log (3 x+2)}{823543} \]

Antiderivative was successfully verified.

[In]

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

[Out]

242/(16807*(1 - 2*x)^2) + 4180/(117649*(1 - 2*x)) - 1/(1372*(2 + 3*x)^4) + 64/(7203*(2 + 3*x)^3) - 829/(33614*
(2 + 3*x)^2) - 5750/(117649*(2 + 3*x)) - (24040*Log[1 - 2*x])/823543 + (24040*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)^3 (2+3 x)^5} \, dx &=\int \left (-\frac{968}{16807 (-1+2 x)^3}+\frac{8360}{117649 (-1+2 x)^2}-\frac{48080}{823543 (-1+2 x)}+\frac{3}{343 (2+3 x)^5}-\frac{192}{2401 (2+3 x)^4}+\frac{2487}{16807 (2+3 x)^3}+\frac{17250}{117649 (2+3 x)^2}+\frac{72120}{823543 (2+3 x)}\right ) \, dx\\ &=\frac{242}{16807 (1-2 x)^2}+\frac{4180}{117649 (1-2 x)}-\frac{1}{1372 (2+3 x)^4}+\frac{64}{7203 (2+3 x)^3}-\frac{829}{33614 (2+3 x)^2}-\frac{5750}{117649 (2+3 x)}-\frac{24040 \log (1-2 x)}{823543}+\frac{24040 \log (2+3 x)}{823543}\\ \end{align*}

Mathematica [A]  time = 0.0503635, size = 64, normalized size = 0.74 \[ \frac{2 \left (-\frac{7 \left (15577920 x^5+24665040 x^4+3606000 x^3-10343210 x^2-4966396 x-460595\right )}{8 (1-2 x)^2 (3 x+2)^4}-36060 \log (1-2 x)+36060 \log (6 x+4)\right )}{2470629} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-7*(-460595 - 4966396*x - 10343210*x^2 + 3606000*x^3 + 24665040*x^4 + 15577920*x^5))/(8*(1 - 2*x)^2*(2 +
3*x)^4) - 36060*Log[1 - 2*x] + 36060*Log[4 + 6*x]))/2470629

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Maple [A]  time = 0.009, size = 72, normalized size = 0.8 \begin{align*}{\frac{242}{16807\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{4180}{235298\,x-117649}}-{\frac{24040\,\ln \left ( 2\,x-1 \right ) }{823543}}-{\frac{1}{1372\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{64}{7203\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{829}{33614\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{5750}{235298+352947\,x}}+{\frac{24040\,\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)^3/(2+3*x)^5,x)

[Out]

242/16807/(2*x-1)^2-4180/117649/(2*x-1)-24040/823543*ln(2*x-1)-1/1372/(2+3*x)^4+64/7203/(2+3*x)^3-829/33614/(2
+3*x)^2-5750/117649/(2+3*x)+24040/823543*ln(2+3*x)

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Maxima [A]  time = 2.42043, size = 103, normalized size = 1.18 \begin{align*} -\frac{15577920 \, x^{5} + 24665040 \, x^{4} + 3606000 \, x^{3} - 10343210 \, x^{2} - 4966396 \, x - 460595}{1411788 \,{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} + \frac{24040}{823543} \, \log \left (3 \, x + 2\right ) - \frac{24040}{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)^3/(2+3*x)^5,x, algorithm="maxima")

[Out]

-1/1411788*(15577920*x^5 + 24665040*x^4 + 3606000*x^3 - 10343210*x^2 - 4966396*x - 460595)/(324*x^6 + 540*x^5
+ 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16) + 24040/823543*log(3*x + 2) - 24040/823543*log(2*x - 1)

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Fricas [A]  time = 1.45426, size = 437, normalized size = 5.02 \begin{align*} -\frac{109045440 \, x^{5} + 172655280 \, x^{4} + 25242000 \, x^{3} - 72402470 \, x^{2} - 288480 \,{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 288480 \,{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (2 \, x - 1\right ) - 34764772 \, x - 3224165}{9882516 \,{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9882516*(109045440*x^5 + 172655280*x^4 + 25242000*x^3 - 72402470*x^2 - 288480*(324*x^6 + 540*x^5 + 81*x^4 -
264*x^3 - 104*x^2 + 32*x + 16)*log(3*x + 2) + 288480*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x +
16)*log(2*x - 1) - 34764772*x - 3224165)/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)

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Sympy [A]  time = 0.20235, size = 75, normalized size = 0.86 \begin{align*} - \frac{15577920 x^{5} + 24665040 x^{4} + 3606000 x^{3} - 10343210 x^{2} - 4966396 x - 460595}{457419312 x^{6} + 762365520 x^{5} + 114354828 x^{4} - 372712032 x^{3} - 146825952 x^{2} + 45177216 x + 22588608} - \frac{24040 \log{\left (x - \frac{1}{2} \right )}}{823543} + \frac{24040 \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)**3/(2+3*x)**5,x)

[Out]

-(15577920*x**5 + 24665040*x**4 + 3606000*x**3 - 10343210*x**2 - 4966396*x - 460595)/(457419312*x**6 + 7623655
20*x**5 + 114354828*x**4 - 372712032*x**3 - 146825952*x**2 + 45177216*x + 22588608) - 24040*log(x - 1/2)/82354
3 + 24040*log(x + 2/3)/823543

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Giac [A]  time = 2.64735, size = 105, normalized size = 1.21 \begin{align*} -\frac{5750}{117649 \,{\left (3 \, x + 2\right )}} + \frac{264 \,{\left (\frac{896}{3 \, x + 2} - 223\right )}}{823543 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}^{2}} - \frac{829}{33614 \,{\left (3 \, x + 2\right )}^{2}} + \frac{64}{7203 \,{\left (3 \, x + 2\right )}^{3}} - \frac{1}{1372 \,{\left (3 \, x + 2\right )}^{4}} - \frac{24040}{823543} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5750/117649/(3*x + 2) + 264/823543*(896/(3*x + 2) - 223)/(7/(3*x + 2) - 2)^2 - 829/33614/(3*x + 2)^2 + 64/720
3/(3*x + 2)^3 - 1/1372/(3*x + 2)^4 - 24040/823543*log(abs(-7/(3*x + 2) + 2))