\(\int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx\) [1220]

Optimal result

Integrand size = 20, antiderivative size = 75 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx=-\frac {7}{5 (2+3 x)^5}-\frac {17}{(2+3 x)^4}-\frac {505}{3 (2+3 x)^3}-\frac {1675}{(2+3 x)^2}-\frac {20875}{2+3 x}-\frac {6875}{3+5 x}+125000 \log (2+3 x)-125000 \log (3+5 x) \]

[Out]

-7/5/(2+3*x)^5-17/(2+3*x)^4-505/3/(2+3*x)^3-1675/(2+3*x)^2-20875/(2+3*x)-6875/(3+5*x)+125000*ln(2+3*x)-125000*
ln(3+5*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, 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} \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx=-\frac {20875}{3 x+2}-\frac {6875}{5 x+3}-\frac {1675}{(3 x+2)^2}-\frac {505}{3 (3 x+2)^3}-\frac {17}{(3 x+2)^4}-\frac {7}{5 (3 x+2)^5}+125000 \log (3 x+2)-125000 \log (5 x+3) \]

[In]

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

[Out]

-7/(5*(2 + 3*x)^5) - 17/(2 + 3*x)^4 - 505/(3*(2 + 3*x)^3) - 1675/(2 + 3*x)^2 - 20875/(2 + 3*x) - 6875/(3 + 5*x
) + 125000*Log[2 + 3*x] - 125000*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)^6}+\frac {204}{(2+3 x)^5}+\frac {1515}{(2+3 x)^4}+\frac {10050}{(2+3 x)^3}+\frac {62625}{(2+3 x)^2}+\frac {375000}{2+3 x}+\frac {34375}{(3+5 x)^2}-\frac {625000}{3+5 x}\right ) \, dx \\ & = -\frac {7}{5 (2+3 x)^5}-\frac {17}{(2+3 x)^4}-\frac {505}{3 (2+3 x)^3}-\frac {1675}{(2+3 x)^2}-\frac {20875}{2+3 x}-\frac {6875}{3+5 x}+125000 \log (2+3 x)-125000 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx=-\frac {7}{5 (2+3 x)^5}-\frac {17}{(2+3 x)^4}-\frac {505}{3 (2+3 x)^3}-\frac {1675}{(2+3 x)^2}-\frac {20875}{2+3 x}-\frac {6875}{3+5 x}+125000 \log (2+3 x)-125000 \log (-3 (3+5 x)) \]

[In]

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

[Out]

-7/(5*(2 + 3*x)^5) - 17/(2 + 3*x)^4 - 505/(3*(2 + 3*x)^3) - 1675/(2 + 3*x)^2 - 20875/(2 + 3*x) - 6875/(3 + 5*x
) + 125000*Log[2 + 3*x] - 125000*Log[-3*(3 + 5*x)]

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77

method	result	size
norman	\(\frac {-44092500 x^{3}-33412500 x^{4}-29084750 x^{2}-10125000 x^{5}-\frac {28768970}{3} x -\frac {6321631}{5}}{\left (2+3 x \right )^{5} \left (3+5 x \right )}+125000 \ln \left (2+3 x \right )-125000 \ln \left (3+5 x \right )\)	\(58\)
risch	\(\frac {-44092500 x^{3}-33412500 x^{4}-29084750 x^{2}-10125000 x^{5}-\frac {28768970}{3} x -\frac {6321631}{5}}{\left (2+3 x \right )^{5} \left (3+5 x \right )}+125000 \ln \left (2+3 x \right )-125000 \ln \left (3+5 x \right )\)	\(59\)
default	\(-\frac {7}{5 \left (2+3 x \right )^{5}}-\frac {17}{\left (2+3 x \right )^{4}}-\frac {505}{3 \left (2+3 x \right )^{3}}-\frac {1675}{\left (2+3 x \right )^{2}}-\frac {20875}{2+3 x}-\frac {6875}{3+5 x}+125000 \ln \left (2+3 x \right )-125000 \ln \left (3+5 x \right )\)	\(72\)
parallelrisch	\(\frac {960000080 x -201600000000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-52800000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )-5760000000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-410400000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-286740000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-469800000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-72900000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{480 \left (2+3 x \right )^{5} \left (3+5 x \right )}\)	\(162\)

[In]

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

[Out]

(-44092500*x^3-33412500*x^4-29084750*x^2-10125000*x^5-28768970/3*x-6321631/5)/(2+3*x)^5/(3+5*x)+125000*ln(2+3*
x)-125000*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.80 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx=-\frac {151875000 \, x^{5} + 501187500 \, x^{4} + 661387500 \, x^{3} + 436271250 \, x^{2} + 1875000 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (5 \, x + 3\right ) - 1875000 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (3 \, x + 2\right ) + 143844850 \, x + 18964893}{15 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \]

[In]

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

[Out]

-1/15*(151875000*x^5 + 501187500*x^4 + 661387500*x^3 + 436271250*x^2 + 1875000*(1215*x^6 + 4779*x^5 + 7830*x^4
+ 6840*x^3 + 3360*x^2 + 880*x + 96)*log(5*x + 3) - 1875000*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*
x^2 + 880*x + 96)*log(3*x + 2) + 143844850*x + 18964893)/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2
+ 880*x + 96)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx=- \frac {151875000 x^{5} + 501187500 x^{4} + 661387500 x^{3} + 436271250 x^{2} + 143844850 x + 18964893}{18225 x^{6} + 71685 x^{5} + 117450 x^{4} + 102600 x^{3} + 50400 x^{2} + 13200 x + 1440} - 125000 \log {\left (x + \frac {3}{5} \right )} + 125000 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

-(151875000*x**5 + 501187500*x**4 + 661387500*x**3 + 436271250*x**2 + 143844850*x + 18964893)/(18225*x**6 + 71
685*x**5 + 117450*x**4 + 102600*x**3 + 50400*x**2 + 13200*x + 1440) - 125000*log(x + 3/5) + 125000*log(x + 2/3
)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx=-\frac {151875000 \, x^{5} + 501187500 \, x^{4} + 661387500 \, x^{3} + 436271250 \, x^{2} + 143844850 \, x + 18964893}{15 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} - 125000 \, \log \left (5 \, x + 3\right ) + 125000 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/15*(151875000*x^5 + 501187500*x^4 + 661387500*x^3 + 436271250*x^2 + 143844850*x + 18964893)/(1215*x^6 + 477
9*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96) - 125000*log(5*x + 3) + 125000*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx=-\frac {6875}{5 \, x + 3} + \frac {1875 \, {\left (\frac {34866}{5 \, x + 3} + \frac {19635}{{\left (5 \, x + 3\right )}^{2}} + \frac {5040}{{\left (5 \, x + 3\right )}^{3}} + \frac {505}{{\left (5 \, x + 3\right )}^{4}} + 23625\right )}}{{\left (\frac {1}{5 \, x + 3} + 3\right )}^{5}} + 125000 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

[In]

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

[Out]

-6875/(5*x + 3) + 1875*(34866/(5*x + 3) + 19635/(5*x + 3)^2 + 5040/(5*x + 3)^3 + 505/(5*x + 3)^4 + 23625)/(1/(
5*x + 3) + 3)^5 + 125000*log(abs(-1/(5*x + 3) - 3))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx=250000\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {25000\,x^5}{3}+27500\,x^4+\frac {2939500\,x^3}{81}+\frac {5816950\,x^2}{243}+\frac {5753794\,x}{729}+\frac {6321631}{6075}}{x^6+\frac {59\,x^5}{15}+\frac {58\,x^4}{9}+\frac {152\,x^3}{27}+\frac {224\,x^2}{81}+\frac {176\,x}{243}+\frac {32}{405}} \]

[In]

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

[Out]

250000*atanh(30*x + 19) - ((5753794*x)/729 + (5816950*x^2)/243 + (2939500*x^3)/81 + 27500*x^4 + (25000*x^5)/3
+ 6321631/6075)/((176*x)/243 + (224*x^2)/81 + (152*x^3)/27 + (58*x^4)/9 + (59*x^5)/15 + x^6 + 32/405)