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

Optimal. Leaf size=76 \[ \frac {484}{16807 (1-2 x)}-\frac {1364}{16807 (3 x+2)}-\frac {319}{4802 (3 x+2)^2}+\frac {22}{1029 (3 x+2)^3}-\frac {1}{588 (3 x+2)^4}-\frac {4180 \log (1-2 x)}{117649}+\frac {4180 \log (3 x+2)}{117649} \]

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Rubi [A]  time = 0.04, antiderivative size = 76, 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} \begin {gather*} \frac {484}{16807 (1-2 x)}-\frac {1364}{16807 (3 x+2)}-\frac {319}{4802 (3 x+2)^2}+\frac {22}{1029 (3 x+2)^3}-\frac {1}{588 (3 x+2)^4}-\frac {4180 \log (1-2 x)}{117649}+\frac {4180 \log (3 x+2)}{117649} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

484/(16807*(1 - 2*x)) - 1/(588*(2 + 3*x)^4) + 22/(1029*(2 + 3*x)^3) - 319/(4802*(2 + 3*x)^2) - 1364/(16807*(2
+ 3*x)) - (4180*Log[1 - 2*x])/117649 + (4180*Log[2 + 3*x])/117649

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 (2+3 x)^5} \, dx &=\int \left (\frac {968}{16807 (-1+2 x)^2}-\frac {8360}{117649 (-1+2 x)}+\frac {1}{49 (2+3 x)^5}-\frac {66}{343 (2+3 x)^4}+\frac {957}{2401 (2+3 x)^3}+\frac {4092}{16807 (2+3 x)^2}+\frac {12540}{117649 (2+3 x)}\right ) \, dx\\ &=\frac {484}{16807 (1-2 x)}-\frac {1}{588 (2+3 x)^4}+\frac {22}{1029 (2+3 x)^3}-\frac {319}{4802 (2+3 x)^2}-\frac {1364}{16807 (2+3 x)}-\frac {4180 \log (1-2 x)}{117649}+\frac {4180 \log (2+3 x)}{117649}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 59, normalized size = 0.78 \begin {gather*} \frac {2 \left (-\frac {7 \left (1354320 x^4+2821500 x^3+1724250 x^2+172990 x-83327\right )}{8 (2 x-1) (3 x+2)^4}-6270 \log (1-2 x)+6270 \log (6 x+4)\right )}{352947} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-7*(-83327 + 172990*x + 1724250*x^2 + 2821500*x^3 + 1354320*x^4))/(8*(-1 + 2*x)*(2 + 3*x)^4) - 6270*Log[1
- 2*x] + 6270*Log[4 + 6*x]))/352947

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.36, size = 115, normalized size = 1.51 \begin {gather*} -\frac {9480240 \, x^{4} + 19750500 \, x^{3} + 12069750 \, x^{2} - 50160 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) + 50160 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) + 1210930 \, x - 583289}{1411788 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1411788*(9480240*x^4 + 19750500*x^3 + 12069750*x^2 - 50160*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 1
6)*log(3*x + 2) + 50160*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(2*x - 1) + 1210930*x - 583289)/
(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

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giac [A]  time = 1.08, size = 67, normalized size = 0.88 \begin {gather*} -\frac {1364}{16807 \, {\left (3 \, x + 2\right )}} + \frac {2904}{117649 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}} - \frac {319}{4802 \, {\left (3 \, x + 2\right )}^{2}} + \frac {22}{1029 \, {\left (3 \, x + 2\right )}^{3}} - \frac {1}{588 \, {\left (3 \, x + 2\right )}^{4}} - \frac {4180}{117649} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1364/16807/(3*x + 2) + 2904/117649/(7/(3*x + 2) - 2) - 319/4802/(3*x + 2)^2 + 22/1029/(3*x + 2)^3 - 1/588/(3*
x + 2)^4 - 4180/117649*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 63, normalized size = 0.83 \begin {gather*} -\frac {4180 \ln \left (2 x -1\right )}{117649}+\frac {4180 \ln \left (3 x +2\right )}{117649}-\frac {1}{588 \left (3 x +2\right )^{4}}+\frac {22}{1029 \left (3 x +2\right )^{3}}-\frac {319}{4802 \left (3 x +2\right )^{2}}-\frac {1364}{16807 \left (3 x +2\right )}-\frac {484}{16807 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/588/(3*x+2)^4+22/1029/(3*x+2)^3-319/4802/(3*x+2)^2-1364/16807/(3*x+2)+4180/117649*ln(3*x+2)-484/16807/(2*x-
1)-4180/117649*ln(2*x-1)

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maxima [A]  time = 0.55, size = 66, normalized size = 0.87 \begin {gather*} -\frac {1354320 \, x^{4} + 2821500 \, x^{3} + 1724250 \, x^{2} + 172990 \, x - 83327}{201684 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac {4180}{117649} \, \log \left (3 \, x + 2\right ) - \frac {4180}{117649} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/201684*(1354320*x^4 + 2821500*x^3 + 1724250*x^2 + 172990*x - 83327)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 -
64*x - 16) + 4180/117649*log(3*x + 2) - 4180/117649*log(2*x - 1)

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mupad [B]  time = 1.13, size = 57, normalized size = 0.75 \begin {gather*} \frac {8360\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}+\frac {\frac {2090\,x^4}{50421}+\frac {26125\,x^3}{302526}+\frac {287375\,x^2}{5445468}+\frac {86495\,x}{16336404}-\frac {83327}{32672808}}{-x^5-\frac {13\,x^4}{6}-\frac {4\,x^3}{3}+\frac {4\,x^2}{27}+\frac {32\,x}{81}+\frac {8}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8360*atanh((12*x)/7 + 1/7))/117649 + ((86495*x)/16336404 + (287375*x^2)/5445468 + (26125*x^3)/302526 + (2090*
x^4)/50421 - 83327/32672808)/((32*x)/81 + (4*x^2)/27 - (4*x^3)/3 - (13*x^4)/6 - x^5 + 8/81)

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sympy [A]  time = 0.19, size = 65, normalized size = 0.86 \begin {gather*} \frac {- 1354320 x^{4} - 2821500 x^{3} - 1724250 x^{2} - 172990 x + 83327}{32672808 x^{5} + 70791084 x^{4} + 43563744 x^{3} - 4840416 x^{2} - 12907776 x - 3226944} - \frac {4180 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {4180 \log {\left (x + \frac {2}{3} \right )}}{117649} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1354320*x**4 - 2821500*x**3 - 1724250*x**2 - 172990*x + 83327)/(32672808*x**5 + 70791084*x**4 + 43563744*x**
3 - 4840416*x**2 - 12907776*x - 3226944) - 4180*log(x - 1/2)/117649 + 4180*log(x + 2/3)/117649

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