3.16.17 \(\int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx\)

Optimal. Leaf size=76 \[ \frac {1364}{16807 (1-2 x)}-\frac {829}{16807 (3 x+2)}+\frac {121}{2401 (1-2 x)^2}+\frac {32}{2401 (3 x+2)^2}-\frac {1}{1029 (3 x+2)^3}-\frac {5750 \log (1-2 x)}{117649}+\frac {5750 \log (3 x+2)}{117649} \]

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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 {1364}{16807 (1-2 x)}-\frac {829}{16807 (3 x+2)}+\frac {121}{2401 (1-2 x)^2}+\frac {32}{2401 (3 x+2)^2}-\frac {1}{1029 (3 x+2)^3}-\frac {5750 \log (1-2 x)}{117649}+\frac {5750 \log (3 x+2)}{117649} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/((1 - 2*x)^3*(2 + 3*x)^4),x]

[Out]

121/(2401*(1 - 2*x)^2) + 1364/(16807*(1 - 2*x)) - 1/(1029*(2 + 3*x)^3) + 32/(2401*(2 + 3*x)^2) - 829/(16807*(2
+ 3*x)) - (5750*Log[1 - 2*x])/117649 + (5750*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)^3 (2+3 x)^4} \, dx &=\int \left (-\frac {484}{2401 (-1+2 x)^3}+\frac {2728}{16807 (-1+2 x)^2}-\frac {11500}{117649 (-1+2 x)}+\frac {3}{343 (2+3 x)^4}-\frac {192}{2401 (2+3 x)^3}+\frac {2487}{16807 (2+3 x)^2}+\frac {17250}{117649 (2+3 x)}\right ) \, dx\\ &=\frac {121}{2401 (1-2 x)^2}+\frac {1364}{16807 (1-2 x)}-\frac {1}{1029 (2+3 x)^3}+\frac {32}{2401 (2+3 x)^2}-\frac {829}{16807 (2+3 x)}-\frac {5750 \log (1-2 x)}{117649}+\frac {5750 \log (2+3 x)}{117649}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 57, normalized size = 0.75 \begin {gather*} \frac {\frac {7 \left (-310500 x^4-284625 x^3+117875 x^2+180100 x+44411\right )}{(1-2 x)^2 (3 x+2)^3}-17250 \log (1-2 x)+17250 \log (6 x+4)}{352947} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^2/((1 - 2*x)^3*(2 + 3*x)^4),x]

[Out]

((7*(44411 + 180100*x + 117875*x^2 - 284625*x^3 - 310500*x^4))/((1 - 2*x)^2*(2 + 3*x)^3) - 17250*Log[1 - 2*x]
+ 17250*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)^3 (2+3 x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)^2/((1 - 2*x)^3*(2 + 3*x)^4),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)^2/((1 - 2*x)^3*(2 + 3*x)^4), x]

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fricas [A]  time = 1.50, size = 115, normalized size = 1.51 \begin {gather*} -\frac {2173500 \, x^{4} + 1992375 \, x^{3} - 825125 \, x^{2} - 17250 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 17250 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (2 \, x - 1\right ) - 1260700 \, x - 310877}{352947 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^3/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/352947*(2173500*x^4 + 1992375*x^3 - 825125*x^2 - 17250*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(
3*x + 2) + 17250*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(2*x - 1) - 1260700*x - 310877)/(108*x^5 +
108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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giac [A]  time = 1.22, size = 55, normalized size = 0.72 \begin {gather*} -\frac {310500 \, x^{4} + 284625 \, x^{3} - 117875 \, x^{2} - 180100 \, x - 44411}{50421 \, {\left (3 \, x + 2\right )}^{3} {\left (2 \, x - 1\right )}^{2}} + \frac {5750}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {5750}{117649} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^3/(2+3*x)^4,x, algorithm="giac")

[Out]

-1/50421*(310500*x^4 + 284625*x^3 - 117875*x^2 - 180100*x - 44411)/((3*x + 2)^3*(2*x - 1)^2) + 5750/117649*log
(abs(3*x + 2)) - 5750/117649*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 63, normalized size = 0.83 \begin {gather*} -\frac {5750 \ln \left (2 x -1\right )}{117649}+\frac {5750 \ln \left (3 x +2\right )}{117649}-\frac {1}{1029 \left (3 x +2\right )^{3}}+\frac {32}{2401 \left (3 x +2\right )^{2}}-\frac {829}{16807 \left (3 x +2\right )}+\frac {121}{2401 \left (2 x -1\right )^{2}}-\frac {1364}{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)^3/(3*x+2)^4,x)

[Out]

-1/1029/(3*x+2)^3+32/2401/(3*x+2)^2-829/16807/(3*x+2)+5750/117649*ln(3*x+2)+121/2401/(2*x-1)^2-1364/16807/(2*x
-1)-5750/117649*ln(2*x-1)

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maxima [A]  time = 0.59, size = 66, normalized size = 0.87 \begin {gather*} -\frac {310500 \, x^{4} + 284625 \, x^{3} - 117875 \, x^{2} - 180100 \, x - 44411}{50421 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} + \frac {5750}{117649} \, \log \left (3 \, x + 2\right ) - \frac {5750}{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)^3/(2+3*x)^4,x, algorithm="maxima")

[Out]

-1/50421*(310500*x^4 + 284625*x^3 - 117875*x^2 - 180100*x - 44411)/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x
+ 8) + 5750/117649*log(3*x + 2) - 5750/117649*log(2*x - 1)

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mupad [B]  time = 1.07, size = 53, normalized size = 0.70 \begin {gather*} \frac {11500\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}+\frac {-\frac {2875\,x^4}{50421}-\frac {31625\,x^3}{605052}+\frac {117875\,x^2}{5445468}+\frac {45025\,x}{1361367}+\frac {44411}{5445468}}{x^5+x^4-\frac {5\,x^3}{12}-\frac {29\,x^2}{54}+\frac {x}{27}+\frac {2}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 3)^2/((2*x - 1)^3*(3*x + 2)^4),x)

[Out]

(11500*atanh((12*x)/7 + 1/7))/117649 + ((45025*x)/1361367 + (117875*x^2)/5445468 - (31625*x^3)/605052 - (2875*
x^4)/50421 + 44411/5445468)/(x/27 - (29*x^2)/54 - (5*x^3)/12 + x^4 + x^5 + 2/27)

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sympy [A]  time = 0.20, size = 65, normalized size = 0.86 \begin {gather*} - \frac {310500 x^{4} + 284625 x^{3} - 117875 x^{2} - 180100 x - 44411}{5445468 x^{5} + 5445468 x^{4} - 2268945 x^{3} - 2924418 x^{2} + 201684 x + 403368} - \frac {5750 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {5750 \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)**3/(2+3*x)**4,x)

[Out]

-(310500*x**4 + 284625*x**3 - 117875*x**2 - 180100*x - 44411)/(5445468*x**5 + 5445468*x**4 - 2268945*x**3 - 29
24418*x**2 + 201684*x + 403368) - 5750*log(x - 1/2)/117649 + 5750*log(x + 2/3)/117649

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