\(\int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx\) [1656]

Optimal result

Integrand size = 22, antiderivative size = 76 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx=\frac {121}{2401 (1-2 x)^2}+\frac {1364}{16807 (1-2 x)}-\frac {1}{1029 (2+3 x)^3}+\frac {32}{2401 (2+3 x)^2}-\frac {829}{16807 (2+3 x)}-\frac {5750 \log (1-2 x)}{117649}+\frac {5750 \log (2+3 x)}{117649} \]

[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)-5750/117649*ln(1-2*
x)+5750/117649*ln(2+3*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx=\frac {1364}{16807 (1-2 x)}-\frac {829}{16807 (3 x+2)}+\frac {121}{2401 (1-2 x)^2}+\frac {32}{2401 (3 x+2)^2}-\frac {1}{1029 (3 x+2)^3}-\frac {5750 \log (1-2 x)}{117649}+\frac {5750 \log (3 x+2)}{117649} \]

[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 90

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx=\frac {\frac {7 \left (44411+180100 x+117875 x^2-284625 x^3-310500 x^4\right )}{(1-2 x)^2 (2+3 x)^3}-17250 \log (1-2 x)+17250 \log (4+6 x)}{352947} \]

[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

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70

method	result	size
norman	\(\frac {-\frac {103500}{16807} x^{4}-\frac {94875}{16807} x^{3}+\frac {117875}{50421} x^{2}+\frac {180100}{50421} x +\frac {44411}{50421}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{3}}-\frac {5750 \ln \left (-1+2 x \right )}{117649}+\frac {5750 \ln \left (2+3 x \right )}{117649}\)	\(53\)
risch	\(\frac {-\frac {103500}{16807} x^{4}-\frac {94875}{16807} x^{3}+\frac {117875}{50421} x^{2}+\frac {180100}{50421} x +\frac {44411}{50421}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{3}}-\frac {5750 \ln \left (-1+2 x \right )}{117649}+\frac {5750 \ln \left (2+3 x \right )}{117649}\)	\(54\)
default	\(\frac {121}{2401 \left (-1+2 x \right )^{2}}-\frac {1364}{16807 \left (-1+2 x \right )}-\frac {5750 \ln \left (-1+2 x \right )}{117649}-\frac {1}{1029 \left (2+3 x \right )^{3}}+\frac {32}{2401 \left (2+3 x \right )^{2}}-\frac {829}{16807 \left (2+3 x \right )}+\frac {5750 \ln \left (2+3 x \right )}{117649}\)	\(63\)
parallelrisch	\(\frac {2484000 \ln \left (\frac {2}{3}+x \right ) x^{5}-2484000 \ln \left (x -\frac {1}{2}\right ) x^{5}-2947364+2484000 \ln \left (\frac {2}{3}+x \right ) x^{4}-2484000 \ln \left (x -\frac {1}{2}\right ) x^{4}-45385200 x^{5}-1035000 \ln \left (\frac {2}{3}+x \right ) x^{3}+1035000 \ln \left (x -\frac {1}{2}\right ) x^{3}-48283200 x^{4}-1334000 \ln \left (\frac {2}{3}+x \right ) x^{2}+1334000 \ln \left (x -\frac {1}{2}\right ) x^{2}[phone] x^{3}+92000 \ln \left (\frac {2}{3}+x \right ) x -92000 \ln \left (x -\frac {1}{2}\right ) x [phone] x^{2}+184000 \ln \left (\frac {2}{3}+x \right )-184000 \ln \left (x -\frac {1}{2}\right )}{470596 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{3}}\)	\(137\)

[In]

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

[Out]

(-103500/16807*x^4-94875/16807*x^3+117875/50421*x^2+180100/50421*x+44411/50421)/(-1+2*x)^2/(2+3*x)^3-5750/1176
49*ln(-1+2*x)+5750/117649*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.51 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx=-\frac {2173500 \, x^{4} + 1992375 \, x^{3} - 825125 \, x^{2} - 17250 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 17250 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (2 \, x - 1\right ) - 1260700 \, x - 310877}{352947 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx=- \frac {310500 x^{4} + 284625 x^{3} - 117875 x^{2} - 180100 x - 44411}{5445468 x^{5} + 5445468 x^{4} - 2268945 x^{3} - 2924418 x^{2} + 201684 x + 403368} - \frac {5750 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {5750 \log {\left (x + \frac {2}{3} \right )}}{117649} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx=-\frac {310500 \, x^{4} + 284625 \, x^{3} - 117875 \, x^{2} - 180100 \, x - 44411}{50421 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} + \frac {5750}{117649} \, \log \left (3 \, x + 2\right ) - \frac {5750}{117649} \, \log \left (2 \, x - 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx=-\frac {310500 \, x^{4} + 284625 \, x^{3} - 117875 \, x^{2} - 180100 \, x - 44411}{50421 \, {\left (3 \, x + 2\right )}^{3} {\left (2 \, x - 1\right )}^{2}} + \frac {5750}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {5750}{117649} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^2}{(1-2 x)^3 (2+3 x)^4} \, dx=\frac {11500\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}+\frac {-\frac {2875\,x^4}{50421}-\frac {31625\,x^3}{605052}+\frac {117875\,x^2}{5445468}+\frac {45025\,x}{1361367}+\frac {44411}{5445468}}{x^5+x^4-\frac {5\,x^3}{12}-\frac {29\,x^2}{54}+\frac {x}{27}+\frac {2}{27}} \]

[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)