3.1232 \(\int \frac {1-2 x}{(2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=57 \[ \frac {309}{3 x+2}+\frac {505}{5 x+3}+\frac {21}{2 (3 x+2)^2}-\frac {55}{2 (5 x+3)^2}-3060 \log (3 x+2)+3060 \log (5 x+3) \]

[Out]

21/2/(2+3*x)^2+309/(2+3*x)-55/2/(3+5*x)^2+505/(3+5*x)-3060*ln(2+3*x)+3060*ln(3+5*x)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ \frac {309}{3 x+2}+\frac {505}{5 x+3}+\frac {21}{2 (3 x+2)^2}-\frac {55}{2 (5 x+3)^2}-3060 \log (3 x+2)+3060 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

21/(2*(2 + 3*x)^2) + 309/(2 + 3*x) - 55/(2*(3 + 5*x)^2) + 505/(3 + 5*x) - 3060*Log[2 + 3*x] + 3060*Log[3 + 5*x
]

Rule 77

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*} \int \frac {1-2 x}{(2+3 x)^3 (3+5 x)^3} \, dx &=\int \left (-\frac {63}{(2+3 x)^3}-\frac {927}{(2+3 x)^2}-\frac {9180}{2+3 x}+\frac {275}{(3+5 x)^3}-\frac {2525}{(3+5 x)^2}+\frac {15300}{3+5 x}\right ) \, dx\\ &=\frac {21}{2 (2+3 x)^2}+\frac {309}{2+3 x}-\frac {55}{2 (3+5 x)^2}+\frac {505}{3+5 x}-3060 \log (2+3 x)+3060 \log (3+5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 59, normalized size = 1.04 \[ \frac {309}{3 x+2}+\frac {505}{5 x+3}+\frac {21}{2 (3 x+2)^2}-\frac {55}{2 (5 x+3)^2}-3060 \log (3 x+2)+3060 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

21/(2*(2 + 3*x)^2) + 309/(2 + 3*x) - 55/(2*(3 + 5*x)^2) + 505/(3 + 5*x) - 3060*Log[2 + 3*x] + 3060*Log[-3*(3 +
5*x)]

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 95, normalized size = 1.67 \[ \frac {91800 \, x^{3} + 174420 \, x^{2} + 6120 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (5 \, x + 3\right ) - 6120 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (3 \, x + 2\right ) + 110296 \, x + 23213}{2 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(91800*x^3 + 174420*x^2 + 6120*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(5*x + 3) - 6120*(225*x^4 + 5
70*x^3 + 541*x^2 + 228*x + 36)*log(3*x + 2) + 110296*x + 23213)/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

________________________________________________________________________________________

giac [A]  time = 1.19, size = 48, normalized size = 0.84 \[ \frac {91800 \, x^{3} + 174420 \, x^{2} + 110296 \, x + 23213}{2 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}^{2}} + 3060 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 3060 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(91800*x^3 + 174420*x^2 + 110296*x + 23213)/(15*x^2 + 19*x + 6)^2 + 3060*log(abs(5*x + 3)) - 3060*log(abs(
3*x + 2))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 54, normalized size = 0.95 \[ -3060 \ln \left (3 x +2\right )+3060 \ln \left (5 x +3\right )+\frac {21}{2 \left (3 x +2\right )^{2}}+\frac {309}{3 x +2}-\frac {55}{2 \left (5 x +3\right )^{2}}+\frac {505}{5 x +3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21/2/(3*x+2)^2+309/(3*x+2)-55/2/(5*x+3)^2+505/(5*x+3)-3060*ln(3*x+2)+3060*ln(5*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 56, normalized size = 0.98 \[ \frac {91800 \, x^{3} + 174420 \, x^{2} + 110296 \, x + 23213}{2 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} + 3060 \, \log \left (5 \, x + 3\right ) - 3060 \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(91800*x^3 + 174420*x^2 + 110296*x + 23213)/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36) + 3060*log(5*x + 3)
- 3060*log(3*x + 2)

________________________________________________________________________________________

mupad [B]  time = 1.11, size = 45, normalized size = 0.79 \[ \frac {204\,x^3+\frac {1938\,x^2}{5}+\frac {55148\,x}{225}+\frac {23213}{450}}{x^4+\frac {38\,x^3}{15}+\frac {541\,x^2}{225}+\frac {76\,x}{75}+\frac {4}{25}}-6120\,\mathrm {atanh}\left (30\,x+19\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((55148*x)/225 + (1938*x^2)/5 + 204*x^3 + 23213/450)/((76*x)/75 + (541*x^2)/225 + (38*x^3)/15 + x^4 + 4/25) -
6120*atanh(30*x + 19)

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 53, normalized size = 0.93 \[ - \frac {- 91800 x^{3} - 174420 x^{2} - 110296 x - 23213}{450 x^{4} + 1140 x^{3} + 1082 x^{2} + 456 x + 72} + 3060 \log {\left (x + \frac {3}{5} \right )} - 3060 \log {\left (x + \frac {2}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-91800*x**3 - 174420*x**2 - 110296*x - 23213)/(450*x**4 + 1140*x**3 + 1082*x**2 + 456*x + 72) + 3060*log(x +
3/5) - 3060*log(x + 2/3)

________________________________________________________________________________________