\(\int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx\) [1465]

Optimal result

Integrand size = 22, antiderivative size = 76 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx=-\frac {1}{315 (2+3 x)^5}+\frac {17}{441 (2+3 x)^4}-\frac {121}{1029 (2+3 x)^3}-\frac {121}{2401 (2+3 x)^2}-\frac {484}{16807 (2+3 x)}-\frac {968 \log (1-2 x)}{117649}+\frac {968 \log (2+3 x)}{117649} \]

[Out]

-1/315/(2+3*x)^5+17/441/(2+3*x)^4-121/1029/(2+3*x)^3-121/2401/(2+3*x)^2-484/16807/(2+3*x)-968/117649*ln(1-2*x)
+968/117649*ln(2+3*x)

Rubi [A] (verified)

Time = 0.02 (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} \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx=-\frac {484}{16807 (3 x+2)}-\frac {121}{2401 (3 x+2)^2}-\frac {121}{1029 (3 x+2)^3}+\frac {17}{441 (3 x+2)^4}-\frac {1}{315 (3 x+2)^5}-\frac {968 \log (1-2 x)}{117649}+\frac {968 \log (3 x+2)}{117649} \]

[In]

Int[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^6),x]

[Out]

-1/315*1/(2 + 3*x)^5 + 17/(441*(2 + 3*x)^4) - 121/(1029*(2 + 3*x)^3) - 121/(2401*(2 + 3*x)^2) - 484/(16807*(2
+ 3*x)) - (968*Log[1 - 2*x])/117649 + (968*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 {1936}{117649 (-1+2 x)}+\frac {1}{21 (2+3 x)^6}-\frac {68}{147 (2+3 x)^5}+\frac {363}{343 (2+3 x)^4}+\frac {726}{2401 (2+3 x)^3}+\frac {1452}{16807 (2+3 x)^2}+\frac {2904}{117649 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{315 (2+3 x)^5}+\frac {17}{441 (2+3 x)^4}-\frac {121}{1029 (2+3 x)^3}-\frac {121}{2401 (2+3 x)^2}-\frac {484}{16807 (2+3 x)}-\frac {968 \log (1-2 x)}{117649}+\frac {968 \log (2+3 x)}{117649} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx=\frac {4 \left (-\frac {7 \left (953231+4442775 x+7563105 x^2+5733585 x^3+1764180 x^4\right )}{4 (2+3 x)^5}-10890 \log (1-2 x)+10890 \log (4+6 x)\right )}{5294205} \]

[In]

Integrate[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^6),x]

[Out]

(4*((-7*(953231 + 4442775*x + 7563105*x^2 + 5733585*x^3 + 1764180*x^4))/(4*(2 + 3*x)^5) - 10890*Log[1 - 2*x] +
10890*Log[4 + 6*x]))/5294205

Maple [A] (verified)

Time = 2.53 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.61

method	result	size
norman	\(\frac {-\frac {296185}{50421} x -\frac {168069}{16807} x^{2}-\frac {127413}{16807} x^{3}-\frac {39204}{16807} x^{4}-\frac {953231}{756315}}{\left (2+3 x \right )^{5}}-\frac {968 \ln \left (-1+2 x \right )}{117649}+\frac {968 \ln \left (2+3 x \right )}{117649}\)	\(46\)
risch	\(\frac {-\frac {296185}{50421} x -\frac {168069}{16807} x^{2}-\frac {127413}{16807} x^{3}-\frac {39204}{16807} x^{4}-\frac {953231}{756315}}{\left (2+3 x \right )^{5}}-\frac {968 \ln \left (-1+2 x \right )}{117649}+\frac {968 \ln \left (2+3 x \right )}{117649}\)	\(47\)
default	\(-\frac {968 \ln \left (-1+2 x \right )}{117649}-\frac {1}{315 \left (2+3 x \right )^{5}}+\frac {17}{441 \left (2+3 x \right )^{4}}-\frac {121}{1029 \left (2+3 x \right )^{3}}-\frac {121}{2401 \left (2+3 x \right )^{2}}-\frac {484}{16807 \left (2+3 x \right )}+\frac {968 \ln \left (2+3 x \right )}{117649}\)	\(63\)
parallelrisch	\(\frac {37635840 \ln \left (\frac {2}{3}+x \right ) x^{5}-37635840 \ln \left (x -\frac {1}{2}\right ) x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}-125452800 \ln \left (x -\frac {1}{2}\right ) x^{4}[phone] x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-167270400 \ln \left (x -\frac {1}{2}\right ) x^{3}[phone] x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}-111513600 \ln \left (x -\frac {1}{2}\right ) x^{2}[phone] x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x -37171200 \ln \left (x -\frac {1}{2}\right ) x [phone] x^{2}+4956160 \ln \left (\frac {2}{3}+x \right )-4956160 \ln \left (x -\frac {1}{2}\right )[phone] x}{18823840 \left (2+3 x \right )^{5}}\)	\(132\)

[In]

int((3+5*x)^2/(1-2*x)/(2+3*x)^6,x,method=_RETURNVERBOSE)

[Out]

(-296185/50421*x-168069/16807*x^2-127413/16807*x^3-39204/16807*x^4-953231/756315)/(2+3*x)^5-968/117649*ln(-1+2
*x)+968/117649*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.51 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx=-\frac {12349260 \, x^{4} + 40135095 \, x^{3} + 52941735 \, x^{2} - 43560 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 43560 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (2 \, x - 1\right ) + 31099425 \, x + 6672617}{5294205 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/5294205*(12349260*x^4 + 40135095*x^3 + 52941735*x^2 - 43560*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)*log(3*x + 2) + 43560*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(2*x - 1) + 31099425*x +
6672617)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx=- \frac {1764180 x^{4} + 5733585 x^{3} + 7563105 x^{2} + 4442775 x + 953231}{183784545 x^{5} + 612615150 x^{4} + 816820200 x^{3} + 544546800 x^{2} + 181515600 x + 24202080} - \frac {968 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {968 \log {\left (x + \frac {2}{3} \right )}}{117649} \]

[In]

integrate((3+5*x)**2/(1-2*x)/(2+3*x)**6,x)

[Out]

-(1764180*x**4 + 5733585*x**3 + 7563105*x**2 + 4442775*x + 953231)/(183784545*x**5 + 612615150*x**4 + 81682020
0*x**3 + 544546800*x**2 + 181515600*x + 24202080) - 968*log(x - 1/2)/117649 + 968*log(x + 2/3)/117649

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx=-\frac {1764180 \, x^{4} + 5733585 \, x^{3} + 7563105 \, x^{2} + 4442775 \, x + 953231}{756315 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {968}{117649} \, \log \left (3 \, x + 2\right ) - \frac {968}{117649} \, \log \left (2 \, x - 1\right ) \]

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^6,x, algorithm="maxima")

[Out]

-1/756315*(1764180*x^4 + 5733585*x^3 + 7563105*x^2 + 4442775*x + 953231)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x
^2 + 240*x + 32) + 968/117649*log(3*x + 2) - 968/117649*log(2*x - 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.63 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx=-\frac {1764180 \, x^{4} + 5733585 \, x^{3} + 7563105 \, x^{2} + 4442775 \, x + 953231}{756315 \, {\left (3 \, x + 2\right )}^{5}} + \frac {968}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {968}{117649} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^6,x, algorithm="giac")

[Out]

-1/756315*(1764180*x^4 + 5733585*x^3 + 7563105*x^2 + 4442775*x + 953231)/(3*x + 2)^5 + 968/117649*log(abs(3*x
+ 2)) - 968/117649*log(abs(2*x - 1))

Mupad [B] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx=\frac {1936\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}-\frac {\frac {484\,x^4}{50421}+\frac {1573\,x^3}{50421}+\frac {56023\,x^2}{1361367}+\frac {296185\,x}{12252303}+\frac {953231}{183784545}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}} \]

[In]

int(-(5*x + 3)^2/((2*x - 1)*(3*x + 2)^6),x)

[Out]

(1936*atanh((12*x)/7 + 1/7))/117649 - ((296185*x)/12252303 + (56023*x^2)/1361367 + (1573*x^3)/50421 + (484*x^4
)/50421 + 953231/183784545)/((80*x)/81 + (80*x^2)/27 + (40*x^3)/9 + (10*x^4)/3 + x^5 + 32/243)