\(\int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^2} \, dx\) [1219]

Optimal result

Integrand size = 20, antiderivative size = 68 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {7}{4 (2+3 x)^4}-\frac {68}{3 (2+3 x)^3}-\frac {505}{2 (2+3 x)^2}-\frac {3350}{2+3 x}-\frac {1375}{3+5 x}+20875 \log (2+3 x)-20875 \log (3+5 x) \]

[Out]

-7/4/(2+3*x)^4-68/3/(2+3*x)^3-505/2/(2+3*x)^2-3350/(2+3*x)-1375/(3+5*x)+20875*ln(2+3*x)-20875*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {3350}{3 x+2}-\frac {1375}{5 x+3}-\frac {505}{2 (3 x+2)^2}-\frac {68}{3 (3 x+2)^3}-\frac {7}{4 (3 x+2)^4}+20875 \log (3 x+2)-20875 \log (5 x+3) \]

[In]

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

[Out]

-7/(4*(2 + 3*x)^4) - 68/(3*(2 + 3*x)^3) - 505/(2*(2 + 3*x)^2) - 3350/(2 + 3*x) - 1375/(3 + 5*x) + 20875*Log[2
+ 3*x] - 20875*Log[3 + 5*x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {21}{(2+3 x)^5}+\frac {204}{(2+3 x)^4}+\frac {1515}{(2+3 x)^3}+\frac {10050}{(2+3 x)^2}+\frac {62625}{2+3 x}+\frac {6875}{(3+5 x)^2}-\frac {104375}{3+5 x}\right ) \, dx \\ & = -\frac {7}{4 (2+3 x)^4}-\frac {68}{3 (2+3 x)^3}-\frac {505}{2 (2+3 x)^2}-\frac {3350}{2+3 x}-\frac {1375}{3+5 x}+20875 \log (2+3 x)-20875 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {7}{4 (2+3 x)^4}-\frac {68}{3 (2+3 x)^3}-\frac {505}{2 (2+3 x)^2}-\frac {3350}{2+3 x}-\frac {1375}{3+5 x}+20875 \log (2+3 x)-20875 \log (-3 (3+5 x)) \]

[In]

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

[Out]

-7/(4*(2 + 3*x)^4) - 68/(3*(2 + 3*x)^3) - 505/(2*(2 + 3*x)^2) - 3350/(2 + 3*x) - 1375/(3 + 5*x) + 20875*Log[2
+ 3*x] - 20875*Log[-3*(3 + 5*x)]

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78

method	result	size
norman	\(\frac {-563625 x^{4}-\frac {7708553}{12} x -\frac {2968425}{2} x^{3}-\frac {2930015}{2} x^{2}-\frac {422285}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )}+20875 \ln \left (2+3 x \right )-20875 \ln \left (3+5 x \right )\)	\(53\)
risch	\(\frac {-563625 x^{4}-\frac {7708553}{12} x -\frac {2968425}{2} x^{3}-\frac {2930015}{2} x^{2}-\frac {422285}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )}+20875 \ln \left (2+3 x \right )-20875 \ln \left (3+5 x \right )\)	\(54\)
default	\(-\frac {7}{4 \left (2+3 x \right )^{4}}-\frac {68}{3 \left (2+3 x \right )^{3}}-\frac {505}{2 \left (2+3 x \right )^{2}}-\frac {3350}{2+3 x}-\frac {1375}{3+5 x}+20875 \ln \left (2+3 x \right )-20875 \ln \left (3+5 x \right )\)	\(63\)
parallelrisch	\(\frac {1623240000 \ln \left (\frac {2}{3}+x \right ) x^{5}-1623240000 \ln \left (x +\frac {3}{5}\right ) x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}-5302584000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-6925824000 \ln \left (x +\frac {3}{5}\right ) x^{3}[phone] x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}-4521024000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x -1474944000 \ln \left (x +\frac {3}{5}\right ) x [phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right )-192384000 \ln \left (x +\frac {3}{5}\right )[phone] x}{192 \left (2+3 x \right )^{4} \left (3+5 x \right )}\)	\(139\)

[In]

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

[Out]

(-563625*x^4-7708553/12*x-2968425/2*x^3-2930015/2*x^2-422285/4)/(2+3*x)^4/(3+5*x)+20875*ln(2+3*x)-20875*ln(3+5
*x)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.69 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {6763500 \, x^{4} + 17810550 \, x^{3} + 17580090 \, x^{2} + 250500 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (5 \, x + 3\right ) - 250500 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (3 \, x + 2\right ) + 7708553 \, x + 1266855}{12 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]

[In]

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

[Out]

-1/12*(6763500*x^4 + 17810550*x^3 + 17580090*x^2 + 250500*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x +
48)*log(5*x + 3) - 250500*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log(3*x + 2) + 7708553*x + 1
266855)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^2} \, dx=- \frac {6763500 x^{4} + 17810550 x^{3} + 17580090 x^{2} + 7708553 x + 1266855}{4860 x^{5} + 15876 x^{4} + 20736 x^{3} + 13536 x^{2} + 4416 x + 576} - 20875 \log {\left (x + \frac {3}{5} \right )} + 20875 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

-(6763500*x**4 + 17810550*x**3 + 17580090*x**2 + 7708553*x + 1266855)/(4860*x**5 + 15876*x**4 + 20736*x**3 + 1
3536*x**2 + 4416*x + 576) - 20875*log(x + 3/5) + 20875*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {6763500 \, x^{4} + 17810550 \, x^{3} + 17580090 \, x^{2} + 7708553 \, x + 1266855}{12 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} - 20875 \, \log \left (5 \, x + 3\right ) + 20875 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/12*(6763500*x^4 + 17810550*x^3 + 17580090*x^2 + 7708553*x + 1266855)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*
x^2 + 368*x + 48) - 20875*log(5*x + 3) + 20875*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.99 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {1375}{5 \, x + 3} + \frac {375 \, {\left (\frac {26268}{5 \, x + 3} + \frac {10116}{{\left (5 \, x + 3\right )}^{2}} + \frac {1352}{{\left (5 \, x + 3\right )}^{3}} + 23319\right )}}{4 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{4}} + 20875 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

[In]

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

[Out]

-1375/(5*x + 3) + 375/4*(26268/(5*x + 3) + 10116/(5*x + 3)^2 + 1352/(5*x + 3)^3 + 23319)/(1/(5*x + 3) + 3)^4 +
20875*log(abs(-1/(5*x + 3) - 3))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^2} \, dx=41750\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {4175\,x^4}{3}+\frac {65965\,x^3}{18}+\frac {586003\,x^2}{162}+\frac {7708553\,x}{4860}+\frac {84457}{324}}{x^5+\frac {49\,x^4}{15}+\frac {64\,x^3}{15}+\frac {376\,x^2}{135}+\frac {368\,x}{405}+\frac {16}{135}} \]

[In]

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

[Out]

41750*atanh(30*x + 19) - ((7708553*x)/4860 + (586003*x^2)/162 + (65965*x^3)/18 + (4175*x^4)/3 + 84457/324)/((3
68*x)/405 + (376*x^2)/135 + (64*x^3)/15 + (49*x^4)/15 + x^5 + 16/135)