\(\int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx\) [1466]

Optimal result

Integrand size = 22, antiderivative size = 87 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx=-\frac {1}{378 (2+3 x)^6}+\frac {68}{2205 (2+3 x)^5}-\frac {121}{1372 (2+3 x)^4}-\frac {242}{7203 (2+3 x)^3}-\frac {242}{16807 (2+3 x)^2}-\frac {968}{117649 (2+3 x)}-\frac {1936 \log (1-2 x)}{823543}+\frac {1936 \log (2+3 x)}{823543} \]

[Out]

-1/378/(2+3*x)^6+68/2205/(2+3*x)^5-121/1372/(2+3*x)^4-242/7203/(2+3*x)^3-242/16807/(2+3*x)^2-968/117649/(2+3*x
)-1936/823543*ln(1-2*x)+1936/823543*ln(2+3*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 87, 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} \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx=-\frac {968}{117649 (3 x+2)}-\frac {242}{16807 (3 x+2)^2}-\frac {242}{7203 (3 x+2)^3}-\frac {121}{1372 (3 x+2)^4}+\frac {68}{2205 (3 x+2)^5}-\frac {1}{378 (3 x+2)^6}-\frac {1936 \log (1-2 x)}{823543}+\frac {1936 \log (3 x+2)}{823543} \]

[In]

Int[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^7),x]

[Out]

-1/378*1/(2 + 3*x)^6 + 68/(2205*(2 + 3*x)^5) - 121/(1372*(2 + 3*x)^4) - 242/(7203*(2 + 3*x)^3) - 242/(16807*(2
+ 3*x)^2) - 968/(117649*(2 + 3*x)) - (1936*Log[1 - 2*x])/823543 + (1936*Log[2 + 3*x])/823543

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 {3872}{823543 (-1+2 x)}+\frac {1}{21 (2+3 x)^7}-\frac {68}{147 (2+3 x)^6}+\frac {363}{343 (2+3 x)^5}+\frac {726}{2401 (2+3 x)^4}+\frac {1452}{16807 (2+3 x)^3}+\frac {2904}{117649 (2+3 x)^2}+\frac {5808}{823543 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{378 (2+3 x)^6}+\frac {68}{2205 (2+3 x)^5}-\frac {121}{1372 (2+3 x)^4}-\frac {242}{7203 (2+3 x)^3}-\frac {242}{16807 (2+3 x)^2}-\frac {968}{117649 (2+3 x)}-\frac {1936 \log (1-2 x)}{823543}+\frac {1936 \log (2+3 x)}{823543} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx=\frac {4 \left (-\frac {7 \left (67099978+351466812 x+739632465 x^2+819755640 x^3+497498760 x^4+127020960 x^5\right )}{16 (2+3 x)^6}-65340 \log (1-2 x)+65340 \log (4+6 x)\right )}{111178305} \]

[In]

Integrate[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^7),x]

[Out]

(4*((-7*(67099978 + 351466812*x + 739632465*x^2 + 819755640*x^3 + 497498760*x^4 + 127020960*x^5))/(16*(2 + 3*x
)^6) - 65340*Log[1 - 2*x] + 65340*Log[4 + 6*x]))/111178305

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.59

method	result	size
norman	\(\frac {-\frac {9762967}{1764735} x -\frac {5478759}{470596} x^{2}-\frac {1518066}{117649} x^{3}-\frac {921294}{117649} x^{4}-\frac {235224}{117649} x^{5}-\frac {33549989}{31765230}}{\left (2+3 x \right )^{6}}-\frac {1936 \ln \left (-1+2 x \right )}{823543}+\frac {1936 \ln \left (2+3 x \right )}{823543}\)	\(51\)
risch	\(\frac {-\frac {9762967}{1764735} x -\frac {5478759}{470596} x^{2}-\frac {1518066}{117649} x^{3}-\frac {921294}{117649} x^{4}-\frac {235224}{117649} x^{5}-\frac {33549989}{31765230}}{\left (2+3 x \right )^{6}}-\frac {1936 \ln \left (-1+2 x \right )}{823543}+\frac {1936 \ln \left (2+3 x \right )}{823543}\)	\(52\)
default	\(-\frac {1936 \ln \left (-1+2 x \right )}{823543}-\frac {1}{378 \left (2+3 x \right )^{6}}+\frac {68}{2205 \left (2+3 x \right )^{5}}-\frac {121}{1372 \left (2+3 x \right )^{4}}-\frac {242}{7203 \left (2+3 x \right )^{3}}-\frac {242}{16807 \left (2+3 x \right )^{2}}-\frac {968}{117649 \left (2+3 x \right )}+\frac {1936 \ln \left (2+3 x \right )}{823543}\)	\(72\)
parallelrisch	\(\frac {2094258880 x [phone] \ln \left (\frac {2}{3}+x \right ) x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{3}[phone] x^{4}[phone] x^{2}-6021734400 \ln \left (x -\frac {1}{2}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )-5352652800 \ln \left (x -\frac {1}{2}\right ) x^{3}-2676326400 \ln \left (x -\frac {1}{2}\right ) x^{2}-713687040 \ln \left (x -\frac {1}{2}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-79298560 \ln \left (x -\frac {1}{2}\right )-903260160 \ln \left (x -\frac {1}{2}\right ) x^{6}-3613040640 \ln \left (x -\frac {1}{2}\right ) x^{5}}{527067520 \left (2+3 x \right )^{6}}\)	\(155\)

[In]

int((3+5*x)^2/(1-2*x)/(2+3*x)^7,x,method=_RETURNVERBOSE)

[Out]

(-9762967/1764735*x-5478759/470596*x^2-1518066/117649*x^3-921294/117649*x^4-235224/117649*x^5-33549989/3176523
0)/(2+3*x)^6-1936/823543*ln(-1+2*x)+1936/823543*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.55 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx=-\frac {889146720 \, x^{5} + 3482491320 \, x^{4} + 5738289480 \, x^{3} + 5177427255 \, x^{2} - 1045440 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 1045440 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (2 \, x - 1\right ) + 2460267684 \, x + 469699846}{444713220 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/444713220*(889146720*x^5 + 3482491320*x^4 + 5738289480*x^3 + 5177427255*x^2 - 1045440*(729*x^6 + 2916*x^5 +
4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) + 1045440*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3
+ 2160*x^2 + 576*x + 64)*log(2*x - 1) + 2460267684*x + 469699846)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 +
2160*x^2 + 576*x + 64)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx=- \frac {127020960 x^{5} + 497498760 x^{4} + 819755640 x^{3} + 739632465 x^{2} + 351466812 x + 67099978}{46313705340 x^{6} + 185254821360 x^{5} + 308758035600 x^{4} + 274451587200 x^{3} + 137225793600 x^{2} + 36593544960 x + 4065949440} - \frac {1936 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {1936 \log {\left (x + \frac {2}{3} \right )}}{823543} \]

[In]

integrate((3+5*x)**2/(1-2*x)/(2+3*x)**7,x)

[Out]

-(127020960*x**5 + 497498760*x**4 + 819755640*x**3 + 739632465*x**2 + 351466812*x + 67099978)/(46313705340*x**
6 + 185254821360*x**5 + 308758035600*x**4 + 274451587200*x**3 + 137225793600*x**2 + 36593544960*x + 4065949440
) - 1936*log(x - 1/2)/823543 + 1936*log(x + 2/3)/823543

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx=-\frac {127020960 \, x^{5} + 497498760 \, x^{4} + 819755640 \, x^{3} + 739632465 \, x^{2} + 351466812 \, x + 67099978}{63530460 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {1936}{823543} \, \log \left (3 \, x + 2\right ) - \frac {1936}{823543} \, \log \left (2 \, x - 1\right ) \]

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/63530460*(127020960*x^5 + 497498760*x^4 + 819755640*x^3 + 739632465*x^2 + 351466812*x + 67099978)/(729*x^6
+ 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 1936/823543*log(3*x + 2) - 1936/823543*log(2*x - 1
)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.61 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx=-\frac {127020960 \, x^{5} + 497498760 \, x^{4} + 819755640 \, x^{3} + 739632465 \, x^{2} + 351466812 \, x + 67099978}{63530460 \, {\left (3 \, x + 2\right )}^{6}} + \frac {1936}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {1936}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^7,x, algorithm="giac")

[Out]

-1/63530460*(127020960*x^5 + 497498760*x^4 + 819755640*x^3 + 739632465*x^2 + 351466812*x + 67099978)/(3*x + 2)
^6 + 1936/823543*log(abs(3*x + 2)) - 1936/823543*log(abs(2*x - 1))

Mupad [B] (verification not implemented)

Time = 1.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx=\frac {3872\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {968\,x^5}{352947}+\frac {11374\,x^4}{1058841}+\frac {168674\,x^3}{9529569}+\frac {67639\,x^2}{4235364}+\frac {9762967\,x}{1286491815}+\frac {33549989}{23156852670}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}} \]

[In]

int(-(5*x + 3)^2/((2*x - 1)*(3*x + 2)^7),x)

[Out]

(3872*atanh((12*x)/7 + 1/7))/823543 - ((9762967*x)/1286491815 + (67639*x^2)/4235364 + (168674*x^3)/9529569 + (
11374*x^4)/1058841 + (968*x^5)/352947 + 33549989/23156852670)/((64*x)/81 + (80*x^2)/27 + (160*x^3)/27 + (20*x^
4)/3 + 4*x^5 + x^6 + 64/729)