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)-24040/823543*ln(1-2*x)+24040/823543*ln(2+3*x)

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Rubi [A]  time = 0.04, antiderivative size = 87, 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} \[ \frac {4180}{117649 (1-2 x)}-\frac {5750}{117649 (3 x+2)}+\frac {242}{16807 (1-2 x)^2}-\frac {829}{33614 (3 x+2)^2}+\frac {64}{7203 (3 x+2)^3}-\frac {1}{1372 (3 x+2)^4}-\frac {24040 \log (1-2 x)}{823543}+\frac {24040 \log (3 x+2)}{823543} \]

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*}

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Mathematica [A]  time = 0.08, 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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fricas [A]  time = 0.52, size = 135, normalized size = 1.55 \[ -\frac {109045440 \, x^{5} + 172655280 \, x^{4} + 25242000 \, x^{3} - 72402470 \, x^{2} - 288480 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 288480 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (2 \, x - 1\right ) - 34764772 \, x - 3224165}{9882516 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]

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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giac [A]  time = 1.21, size = 78, normalized size = 0.90 \[ -\frac {5750}{117649 \, {\left (3 \, x + 2\right )}} + \frac {264 \, {\left (\frac {896}{3 \, x + 2} - 223\right )}}{823543 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}^{2}} - \frac {829}{33614 \, {\left (3 \, x + 2\right )}^{2}} + \frac {64}{7203 \, {\left (3 \, x + 2\right )}^{3}} - \frac {1}{1372 \, {\left (3 \, x + 2\right )}^{4}} - \frac {24040}{823543} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 72, normalized size = 0.83 \[ -\frac {24040 \ln \left (2 x -1\right )}{823543}+\frac {24040 \ln \left (3 x +2\right )}{823543}-\frac {1}{1372 \left (3 x +2\right )^{4}}+\frac {64}{7203 \left (3 x +2\right )^{3}}-\frac {829}{33614 \left (3 x +2\right )^{2}}-\frac {5750}{117649 \left (3 x +2\right )}+\frac {242}{16807 \left (2 x -1\right )^{2}}-\frac {4180}{117649 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 76, normalized size = 0.87 \[ -\frac {15577920 \, x^{5} + 24665040 \, x^{4} + 3606000 \, x^{3} - 10343210 \, x^{2} - 4966396 \, x - 460595}{1411788 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} + \frac {24040}{823543} \, \log \left (3 \, x + 2\right ) - \frac {24040}{823543} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 0.04, size = 65, normalized size = 0.75 \[ \frac {48080\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}+\frac {-\frac {12020\,x^5}{352947}-\frac {57095\,x^4}{1058841}-\frac {75125\,x^3}{9529569}+\frac {5171605\,x^2}{228709656}+\frac {1241599\,x}{114354828}+\frac {460595}{457419312}}{x^6+\frac {5\,x^5}{3}+\frac {x^4}{4}-\frac {22\,x^3}{27}-\frac {26\,x^2}{81}+\frac {8\,x}{81}+\frac {4}{81}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(48080*atanh((12*x)/7 + 1/7))/823543 + ((1241599*x)/114354828 + (5171605*x^2)/228709656 - (75125*x^3)/9529569
- (57095*x^4)/1058841 - (12020*x^5)/352947 + 460595/457419312)/((8*x)/81 - (26*x^2)/81 - (22*x^3)/27 + x^4/4 +
(5*x^5)/3 + x^6 + 4/81)

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sympy [A]  time = 0.22, size = 75, normalized size = 0.86 \[ - \frac {15577920 x^{5} + 24665040 x^{4} + 3606000 x^{3} - 10343210 x^{2} - 4966396 x - 460595}{457419312 x^{6} + 762365520 x^{5} + 114354828 x^{4} - 372712032 x^{3} - 146825952 x^{2} + 45177216 x + 22588608} - \frac {24040 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {24040 \log {\left (x + \frac {2}{3} \right )}}{823543} \]

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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