\(\int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx\) [1221]

Optimal result

Integrand size = 20, antiderivative size = 90 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {7}{6 (2+3 x)^6}-\frac {68}{5 (2+3 x)^5}-\frac {505}{4 (2+3 x)^4}-\frac {3350}{3 (2+3 x)^3}-\frac {20875}{2 (2+3 x)^2}-\frac {125000}{2+3 x}-\frac {34375}{3+5 x}+728125 \log (2+3 x)-728125 \log (3+5 x) \]

[Out]

-7/6/(2+3*x)^6-68/5/(2+3*x)^5-505/4/(2+3*x)^4-3350/3/(2+3*x)^3-20875/2/(2+3*x)^2-125000/(2+3*x)-34375/(3+5*x)+
728125*ln(2+3*x)-728125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 90, 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} \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {125000}{3 x+2}-\frac {34375}{5 x+3}-\frac {20875}{2 (3 x+2)^2}-\frac {3350}{3 (3 x+2)^3}-\frac {505}{4 (3 x+2)^4}-\frac {68}{5 (3 x+2)^5}-\frac {7}{6 (3 x+2)^6}+728125 \log (3 x+2)-728125 \log (5 x+3) \]

[In]

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

[Out]

-7/(6*(2 + 3*x)^6) - 68/(5*(2 + 3*x)^5) - 505/(4*(2 + 3*x)^4) - 3350/(3*(2 + 3*x)^3) - 20875/(2*(2 + 3*x)^2) -
125000/(2 + 3*x) - 34375/(3 + 5*x) + 728125*Log[2 + 3*x] - 728125*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)^7}+\frac {204}{(2+3 x)^6}+\frac {1515}{(2+3 x)^5}+\frac {10050}{(2+3 x)^4}+\frac {62625}{(2+3 x)^3}+\frac {375000}{(2+3 x)^2}+\frac {2184375}{2+3 x}+\frac {171875}{(3+5 x)^2}-\frac {3640625}{3+5 x}\right ) \, dx \\ & = -\frac {7}{6 (2+3 x)^6}-\frac {68}{5 (2+3 x)^5}-\frac {505}{4 (2+3 x)^4}-\frac {3350}{3 (2+3 x)^3}-\frac {20875}{2 (2+3 x)^2}-\frac {125000}{2+3 x}-\frac {34375}{3+5 x}+728125 \log (2+3 x)-728125 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {7}{6 (2+3 x)^6}-\frac {68}{5 (2+3 x)^5}-\frac {505}{4 (2+3 x)^4}-\frac {3350}{3 (2+3 x)^3}-\frac {20875}{2 (2+3 x)^2}-\frac {125000}{2+3 x}-\frac {34375}{3+5 x}+728125 \log (2+3 x)-728125 \log (-3 (3+5 x)) \]

[In]

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

[Out]

-7/(6*(2 + 3*x)^6) - 68/(5*(2 + 3*x)^5) - 505/(4*(2 + 3*x)^4) - 3350/(3*(2 + 3*x)^3) - 20875/(2*(2 + 3*x)^2) -
125000/(2 + 3*x) - 34375/(3 + 5*x) + 728125*Log[2 + 3*x] - 728125*Log[-3*(3 + 5*x)]

Maple [A] (verified)

Time = 2.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70

method	result	size
norman	\(\frac {-176934375 x^{6}-\frac {4087734525}{4} x^{3}-\frac {2319544125}{2} x^{4}-\frac {2025666351}{4} x^{2}-\frac {1403679375}{2} x^{5}-\frac {1338136009}{10} x -\frac {147294001}{10}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+728125 \ln \left (2+3 x \right )-728125 \ln \left (3+5 x \right )\)	\(63\)
risch	\(\frac {-176934375 x^{6}-\frac {4087734525}{4} x^{3}-\frac {2319544125}{2} x^{4}-\frac {2025666351}{4} x^{2}-\frac {1403679375}{2} x^{5}-\frac {1338136009}{10} x -\frac {147294001}{10}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+728125 \ln \left (2+3 x \right )-728125 \ln \left (3+5 x \right )\)	\(64\)
default	\(-\frac {7}{6 \left (2+3 x \right )^{6}}-\frac {68}{5 \left (2+3 x \right )^{5}}-\frac {505}{4 \left (2+3 x \right )^{4}}-\frac {3350}{3 \left (2+3 x \right )^{3}}-\frac {20875}{2 \left (2+3 x \right )^{2}}-\frac {125000}{2+3 x}-\frac {34375}{3+5 x}+728125 \ln \left (2+3 x \right )-728125 \ln \left (3+5 x \right )\)	\(81\)
parallelrisch	\(\frac {44736000320 x -13085280000000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-2863104000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}-5095710000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-268416000000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-33216480000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-46201104000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-50579640000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-23440266000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{1920 \left (2+3 x \right )^{6} \left (3+5 x \right )}\)	\(185\)

[In]

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

[Out]

(-176934375*x^6-4087734525/4*x^3-2319544125/2*x^4-2025666351/4*x^2-1403679375/2*x^5-1338136009/10*x-147294001/
10)/(2+3*x)^6/(3+5*x)+728125*ln(2+3*x)-728125*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.72 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {3538687500 \, x^{6} + 14036793750 \, x^{5} + 23195441250 \, x^{4} + 20438672625 \, x^{3} + 10128331755 \, x^{2} + 14562500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (5 \, x + 3\right ) - 14562500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (3 \, x + 2\right ) + 2676272018 \, x + 294588002}{20 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} \]

[In]

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

[Out]

-1/20*(3538687500*x^6 + 14036793750*x^5 + 23195441250*x^4 + 20438672625*x^3 + 10128331755*x^2 + 14562500*(3645
*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(5*x + 3) - 14562500*(3645*
x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(3*x + 2) + 2676272018*x + 2
94588002)/(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.91 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=- \frac {3538687500 x^{6} + 14036793750 x^{5} + 23195441250 x^{4} + 20438672625 x^{3} + 10128331755 x^{2} + 2676272018 x + 294588002}{72900 x^{7} + 335340 x^{6} + 660960 x^{5} + 723600 x^{4} + 475200 x^{3} + 187200 x^{2} + 40960 x + 3840} - 728125 \log {\left (x + \frac {3}{5} \right )} + 728125 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

-(3538687500*x**6 + 14036793750*x**5 + 23195441250*x**4 + 20438672625*x**3 + 10128331755*x**2 + 2676272018*x +
294588002)/(72900*x**7 + 335340*x**6 + 660960*x**5 + 723600*x**4 + 475200*x**3 + 187200*x**2 + 40960*x + 3840
) - 728125*log(x + 3/5) + 728125*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {3538687500 \, x^{6} + 14036793750 \, x^{5} + 23195441250 \, x^{4} + 20438672625 \, x^{3} + 10128331755 \, x^{2} + 2676272018 \, x + 294588002}{20 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} - 728125 \, \log \left (5 \, x + 3\right ) + 728125 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/20*(3538687500*x^6 + 14036793750*x^5 + 23195441250*x^4 + 20438672625*x^3 + 10128331755*x^2 + 2676272018*x +
294588002)/(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192) - 728125*log(
5*x + 3) + 728125*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {34375}{5 \, x + 3} + \frac {5625 \, {\left (\frac {1100034}{5 \, x + 3} + \frac {811665}{{\left (5 \, x + 3\right )}^{2}} + \frac {304700}{{\left (5 \, x + 3\right )}^{3}} + \frac {58650}{{\left (5 \, x + 3\right )}^{4}} + \frac {4700}{{\left (5 \, x + 3\right )}^{5}} + 604017\right )}}{4 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{6}} + 728125 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

[In]

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

[Out]

-34375/(5*x + 3) + 5625/4*(1100034/(5*x + 3) + 811665/(5*x + 3)^2 + 304700/(5*x + 3)^3 + 58650/(5*x + 3)^4 + 4
700/(5*x + 3)^5 + 604017)/(1/(5*x + 3) + 3)^6 + 728125*log(abs(-1/(5*x + 3) - 3))

Mupad [B] (verification not implemented)

Time = 1.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {1-2 x}{(2+3 x)^7 (3+5 x)^2} \, dx=1456250\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {145625\,x^6}{3}+\frac {3465875\,x^5}{18}+\frac {51545425\,x^4}{162}+\frac {30279515\,x^3}{108}+\frac {225074039\,x^2}{1620}+\frac {1338136009\,x}{36450}+\frac {147294001}{36450}}{x^7+\frac {23\,x^6}{5}+\frac {136\,x^5}{15}+\frac {268\,x^4}{27}+\frac {176\,x^3}{27}+\frac {208\,x^2}{81}+\frac {2048\,x}{3645}+\frac {64}{1215}} \]

[In]

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

[Out]

1456250*atanh(30*x + 19) - ((1338136009*x)/36450 + (225074039*x^2)/1620 + (30279515*x^3)/108 + (51545425*x^4)/
162 + (3465875*x^5)/18 + (145625*x^6)/3 + 147294001/36450)/((2048*x)/3645 + (208*x^2)/81 + (176*x^3)/27 + (268
*x^4)/27 + (136*x^5)/15 + (23*x^6)/5 + x^7 + 64/1215)