\(\int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx\) [1234]

Optimal result

Integrand size = 20, antiderivative size = 75 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {21}{4 (2+3 x)^4}+\frac {103}{(2+3 x)^3}+\frac {1530}{(2+3 x)^2}+\frac {25350}{2+3 x}-\frac {1375}{2 (3+5 x)^2}+\frac {20875}{3+5 x}-189375 \log (2+3 x)+189375 \log (3+5 x) \]

[Out]

21/4/(2+3*x)^4+103/(2+3*x)^3+1530/(2+3*x)^2+25350/(2+3*x)-1375/2/(3+5*x)^2+20875/(3+5*x)-189375*ln(2+3*x)+1893
75*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} \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {25350}{3 x+2}+\frac {20875}{5 x+3}+\frac {1530}{(3 x+2)^2}-\frac {1375}{2 (5 x+3)^2}+\frac {103}{(3 x+2)^3}+\frac {21}{4 (3 x+2)^4}-189375 \log (3 x+2)+189375 \log (5 x+3) \]

[In]

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

[Out]

21/(4*(2 + 3*x)^4) + 103/(2 + 3*x)^3 + 1530/(2 + 3*x)^2 + 25350/(2 + 3*x) - 1375/(2*(3 + 5*x)^2) + 20875/(3 +
5*x) - 189375*Log[2 + 3*x] + 189375*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 {63}{(2+3 x)^5}-\frac {927}{(2+3 x)^4}-\frac {9180}{(2+3 x)^3}-\frac {76050}{(2+3 x)^2}-\frac {568125}{2+3 x}+\frac {6875}{(3+5 x)^3}-\frac {104375}{(3+5 x)^2}+\frac {946875}{3+5 x}\right ) \, dx \\ & = \frac {21}{4 (2+3 x)^4}+\frac {103}{(2+3 x)^3}+\frac {1530}{(2+3 x)^2}+\frac {25350}{2+3 x}-\frac {1375}{2 (3+5 x)^2}+\frac {20875}{3+5 x}-189375 \log (2+3 x)+189375 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {21}{4 (2+3 x)^4}+\frac {103}{(2+3 x)^3}+\frac {1530}{(2+3 x)^2}+\frac {25350}{2+3 x}-\frac {1375}{2 (3+5 x)^2}+\frac {20875}{3+5 x}-189375 \log (2+3 x)+189375 \log (-3 (3+5 x)) \]

[In]

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

[Out]

21/(4*(2 + 3*x)^4) + 103/(2 + 3*x)^3 + 1530/(2 + 3*x)^2 + 25350/(2 + 3*x) - 1375/(2*(3 + 5*x)^2) + 20875/(3 +
5*x) - 189375*Log[2 + 3*x] + 189375*Log[-3*(3 + 5*x)]

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77

method	result	size
norman	\(\frac {25565625 x^{5}[phone] x^{3}+\frac {44542717}{2} x +\frac {165324375}{2} x^{4}+\frac {276035525}{4} x^{2}+\frac {11492725}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-189375 \ln \left (2+3 x \right )+189375 \ln \left (3+5 x \right )\)	\(58\)
risch	\(\frac {25565625 x^{5}[phone] x^{3}+\frac {44542717}{2} x +\frac {165324375}{2} x^{4}+\frac {276035525}{4} x^{2}+\frac {11492725}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-189375 \ln \left (2+3 x \right )+189375 \ln \left (3+5 x \right )\)	\(59\)
default	\(\frac {21}{4 \left (2+3 x \right )^{4}}+\frac {103}{\left (2+3 x \right )^{3}}+\frac {1530}{\left (2+3 x \right )^{2}}+\frac {25350}{2+3 x}-\frac {1375}{2 \left (3+5 x \right )^{2}}+\frac {20875}{3+5 x}-189375 \ln \left (2+3 x \right )+189375 \ln \left (3+5 x \right )\)	\(72\)
parallelrisch	\(-\frac {2617919904 x -569833920000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-146603520000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )-15707520000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-1180681920000 \ln \left (x +\frac {3}{5}\right ) x^{3}-854096400000 \ln \left (x +\frac {3}{5}\right ) x^{5}-1375389720000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-220887000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{576 \left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}\)	\(162\)

[In]

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

[Out]

(25565625*x^5+106845375*x^3+44542717/2*x+165324375/2*x^4+276035525/4*x^2+11492725/4)/(2+3*x)^4/(3+5*x)^2-18937
5*ln(2+3*x)+189375*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.80 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {102262500 \, x^{5} + 330648750 \, x^{4} + 427381500 \, x^{3} + 276035525 \, x^{2} + 757500 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (5 \, x + 3\right ) - 757500 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (3 \, x + 2\right ) + 89085434 \, x + 11492725}{4 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]

[In]

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

[Out]

1/4*(102262500*x^5 + 330648750*x^4 + 427381500*x^3 + 276035525*x^2 + 757500*(2025*x^6 + 7830*x^5 + 12609*x^4 +
10824*x^3 + 5224*x^2 + 1344*x + 144)*log(5*x + 3) - 757500*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 522
4*x^2 + 1344*x + 144)*log(3*x + 2) + 89085434*x + 11492725)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 522
4*x^2 + 1344*x + 144)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=- \frac {- 102262500 x^{5} - 330648750 x^{4} - 427381500 x^{3} - 276035525 x^{2} - 89085434 x - 11492725}{8100 x^{6} + 31320 x^{5} + 50436 x^{4} + 43296 x^{3} + 20896 x^{2} + 5376 x + 576} + 189375 \log {\left (x + \frac {3}{5} \right )} - 189375 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

-(-102262500*x**5 - 330648750*x**4 - 427381500*x**3 - 276035525*x**2 - 89085434*x - 11492725)/(8100*x**6 + 313
20*x**5 + 50436*x**4 + 43296*x**3 + 20896*x**2 + 5376*x + 576) + 189375*log(x + 3/5) - 189375*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {102262500 \, x^{5} + 330648750 \, x^{4} + 427381500 \, x^{3} + 276035525 \, x^{2} + 89085434 \, x + 11492725}{4 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} + 189375 \, \log \left (5 \, x + 3\right ) - 189375 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/4*(102262500*x^5 + 330648750*x^4 + 427381500*x^3 + 276035525*x^2 + 89085434*x + 11492725)/(2025*x^6 + 7830*x
^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144) + 189375*log(5*x + 3) - 189375*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {25350}{3 \, x + 2} - \frac {9375 \, {\left (\frac {80}{3 \, x + 2} - 367\right )}}{2 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {1530}{{\left (3 \, x + 2\right )}^{2}} + \frac {103}{{\left (3 \, x + 2\right )}^{3}} + \frac {21}{4 \, {\left (3 \, x + 2\right )}^{4}} + 189375 \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) \]

[In]

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

[Out]

25350/(3*x + 2) - 9375/2*(80/(3*x + 2) - 367)/(1/(3*x + 2) - 5)^2 + 1530/(3*x + 2)^2 + 103/(3*x + 2)^3 + 21/4/
(3*x + 2)^4 + 189375*log(abs(-1/(3*x + 2) + 5))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {12625\,x^5+\frac {244925\,x^4}{6}+\frac {1424605\,x^3}{27}+\frac {11041421\,x^2}{324}+\frac {44542717\,x}{4050}+\frac {459709}{324}}{x^6+\frac {58\,x^5}{15}+\frac {467\,x^4}{75}+\frac {3608\,x^3}{675}+\frac {5224\,x^2}{2025}+\frac {448\,x}{675}+\frac {16}{225}}-378750\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

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

[Out]

((44542717*x)/4050 + (11041421*x^2)/324 + (1424605*x^3)/27 + (244925*x^4)/6 + 12625*x^5 + 459709/324)/((448*x)
/675 + (5224*x^2)/2025 + (3608*x^3)/675 + (467*x^4)/75 + (58*x^5)/15 + x^6 + 16/225) - 378750*atanh(30*x + 19)