\(\int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx\) [1206]

Optimal result

Integrand size = 20, antiderivative size = 81 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {7}{18 (2+3 x)^6}+\frac {11}{5 (2+3 x)^5}+\frac {55}{4 (2+3 x)^4}+\frac {275}{3 (2+3 x)^3}+\frac {1375}{2 (2+3 x)^2}+\frac {6875}{2+3 x}-34375 \log (2+3 x)+34375 \log (3+5 x) \]

[Out]

7/18/(2+3*x)^6+11/5/(2+3*x)^5+55/4/(2+3*x)^4+275/3/(2+3*x)^3+1375/2/(2+3*x)^2+6875/(2+3*x)-34375*ln(2+3*x)+343
75*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 81, 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} \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {6875}{3 x+2}+\frac {1375}{2 (3 x+2)^2}+\frac {275}{3 (3 x+2)^3}+\frac {55}{4 (3 x+2)^4}+\frac {11}{5 (3 x+2)^5}+\frac {7}{18 (3 x+2)^6}-34375 \log (3 x+2)+34375 \log (5 x+3) \]

[In]

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

[Out]

7/(18*(2 + 3*x)^6) + 11/(5*(2 + 3*x)^5) + 55/(4*(2 + 3*x)^4) + 275/(3*(2 + 3*x)^3) + 1375/(2*(2 + 3*x)^2) + 68
75/(2 + 3*x) - 34375*Log[2 + 3*x] + 34375*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 {7}{(2+3 x)^7}-\frac {33}{(2+3 x)^6}-\frac {165}{(2+3 x)^5}-\frac {825}{(2+3 x)^4}-\frac {4125}{(2+3 x)^3}-\frac {20625}{(2+3 x)^2}-\frac {103125}{2+3 x}+\frac {171875}{3+5 x}\right ) \, dx \\ & = \frac {7}{18 (2+3 x)^6}+\frac {11}{5 (2+3 x)^5}+\frac {55}{4 (2+3 x)^4}+\frac {275}{3 (2+3 x)^3}+\frac {1375}{2 (2+3 x)^2}+\frac {6875}{2+3 x}-34375 \log (2+3 x)+34375 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {70+396 (2+3 x)+2475 (2+3 x)^2+16500 (2+3 x)^3+123750 (2+3 x)^4+1237500 (2+3 x)^5}{180 (2+3 x)^6}-34375 \log (2+3 x)+34375 \log (-3 (3+5 x)) \]

[In]

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

[Out]

(70 + 396*(2 + 3*x) + 2475*(2 + 3*x)^2 + 16500*(2 + 3*x)^3 + 123750*(2 + 3*x)^4 + 1237500*(2 + 3*x)^5)/(180*(2
+ 3*x)^6) - 34375*Log[2 + 3*x] + 34375*Log[-3*(3 + 5*x)]

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63

method	result	size
norman	\(\frac {1670625 x^{5}+7575975 x^{3}+\frac {8597358}{5} x +\frac {11248875}{2} x^{4}+\frac {20414295}{4} x^{2}+\frac {20861381}{90}}{\left (2+3 x \right )^{6}}-34375 \ln \left (2+3 x \right )+34375 \ln \left (3+5 x \right )\)	\(51\)
risch	\(\frac {1670625 x^{5}+7575975 x^{3}+\frac {8597358}{5} x +\frac {11248875}{2} x^{4}+\frac {20414295}{4} x^{2}+\frac {20861381}{90}}{\left (2+3 x \right )^{6}}-34375 \ln \left (2+3 x \right )+34375 \ln \left (3+5 x \right )\)	\(52\)
default	\(\frac {7}{18 \left (2+3 x \right )^{6}}+\frac {11}{5 \left (2+3 x \right )^{5}}+\frac {55}{4 \left (2+3 x \right )^{4}}+\frac {275}{3 \left (2+3 x \right )^{3}}+\frac {1375}{2 \left (2+3 x \right )^{2}}+\frac {6875}{2+3 x}-34375 \ln \left (2+3 x \right )+34375 \ln \left (3+5 x \right )\)	\(72\)
parallelrisch	\(-\frac {234666560 x -47520000000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-12672000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )-1408000000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-95040000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-64152000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-106920000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-16038000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{640 \left (2+3 x \right )^{6}}\)	\(155\)

[In]

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

[Out]

(1670625*x^5+7575975*x^3+8597358/5*x+11248875/2*x^4+20414295/4*x^2+20861381/90)/(2+3*x)^6-34375*ln(2+3*x)+3437
5*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.67 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 6187500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (5 \, x + 3\right ) - 6187500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 309504888 \, x + 41722762}{180 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

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

[Out]

1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 6187500*(729*x^6 + 2916*x^5 + 4860*x^
4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(5*x + 3) - 6187500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*
x^2 + 576*x + 64)*log(3*x + 2) + 309504888*x + 41722762)/(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 = 73, normalized size of antiderivative = 0.90 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=- \frac {- 300712500 x^{5} - 1012398750 x^{4} - 1363675500 x^{3} - 918643275 x^{2} - 309504888 x - 41722762}{131220 x^{6} + 524880 x^{5} + 874800 x^{4} + 777600 x^{3} + 388800 x^{2} + 103680 x + 11520} + 34375 \log {\left (x + \frac {3}{5} \right )} - 34375 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

-(-300712500*x**5 - 1012398750*x**4 - 1363675500*x**3 - 918643275*x**2 - 309504888*x - 41722762)/(131220*x**6
+ 524880*x**5 + 874800*x**4 + 777600*x**3 + 388800*x**2 + 103680*x + 11520) + 34375*log(x + 3/5) - 34375*log(x
+ 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 309504888 \, x + 41722762}{180 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + 34375 \, \log \left (5 \, x + 3\right ) - 34375 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 309504888*x + 41722762)/(729*x^6 + 29
16*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 34375*log(5*x + 3) - 34375*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.65 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 309504888 \, x + 41722762}{180 \, {\left (3 \, x + 2\right )}^{6}} + 34375 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 34375 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

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

[Out]

1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 309504888*x + 41722762)/(3*x + 2)^6 +
34375*log(abs(5*x + 3)) - 34375*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.80 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)} \, dx=\frac {\frac {6875\,x^5}{3}+\frac {138875\,x^4}{18}+\frac {841775\,x^3}{81}+\frac {756085\,x^2}{108}+\frac {955262\,x}{405}+\frac {20861381}{65610}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}}-68750\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

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

[Out]

((955262*x)/405 + (756085*x^2)/108 + (841775*x^3)/81 + (138875*x^4)/18 + (6875*x^5)/3 + 20861381/65610)/((64*x
)/81 + (80*x^2)/27 + (160*x^3)/27 + (20*x^4)/3 + 4*x^5 + x^6 + 64/729) - 68750*atanh(30*x + 19)