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

Optimal. Leaf size=46 \[ -\frac {68}{3 x+2}-\frac {55}{5 x+3}-\frac {7}{2 (3 x+2)^2}+505 \log (3 x+2)-505 \log (5 x+3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 46, 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} \begin {gather*} -\frac {68}{3 x+2}-\frac {55}{5 x+3}-\frac {7}{2 (3 x+2)^2}+505 \log (3 x+2)-505 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(2*(2 + 3*x)^2) - 68/(2 + 3*x) - 55/(3 + 5*x) + 505*Log[2 + 3*x] - 505*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)^2} \, dx &=\int \left (\frac {21}{(2+3 x)^3}+\frac {204}{(2+3 x)^2}+\frac {1515}{2+3 x}+\frac {275}{(3+5 x)^2}-\frac {2525}{3+5 x}\right ) \, dx\\ &=-\frac {7}{2 (2+3 x)^2}-\frac {68}{2+3 x}-\frac {55}{3+5 x}+505 \log (2+3 x)-505 \log (3+5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 48, normalized size = 1.04 \begin {gather*} -\frac {68}{3 x+2}-\frac {55}{5 x+3}-\frac {7}{2 (3 x+2)^2}+505 \log (3 x+2)-505 \log (-3 (5 x+3)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(2*(2 + 3*x)^2) - 68/(2 + 3*x) - 55/(3 + 5*x) + 505*Log[2 + 3*x] - 505*Log[-3*(3 + 5*x)]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x}{(2+3 x)^3 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.33, size = 75, normalized size = 1.63 \begin {gather*} -\frac {3030 \, x^{2} + 1010 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 1010 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 3939 \, x + 1277}{2 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3030*x^2 + 1010*(45*x^3 + 87*x^2 + 56*x + 12)*log(5*x + 3) - 1010*(45*x^3 + 87*x^2 + 56*x + 12)*log(3*x
+ 2) + 3939*x + 1277)/(45*x^3 + 87*x^2 + 56*x + 12)

________________________________________________________________________________________

giac [A]  time = 1.31, size = 49, normalized size = 1.07 \begin {gather*} -\frac {55}{5 \, x + 3} + \frac {15 \, {\left (\frac {206}{5 \, x + 3} + 513\right )}}{2 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{2}} + 505 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-55/(5*x + 3) + 15/2*(206/(5*x + 3) + 513)/(1/(5*x + 3) + 3)^2 + 505*log(abs(-1/(5*x + 3) - 3))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 45, normalized size = 0.98 \begin {gather*} 505 \ln \left (3 x +2\right )-505 \ln \left (5 x +3\right )-\frac {7}{2 \left (3 x +2\right )^{2}}-\frac {68}{3 x +2}-\frac {55}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/2/(3*x+2)^2-68/(3*x+2)-55/(5*x+3)+505*ln(3*x+2)-505*ln(5*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.58, size = 46, normalized size = 1.00 \begin {gather*} -\frac {3030 \, x^{2} + 3939 \, x + 1277}{2 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} - 505 \, \log \left (5 \, x + 3\right ) + 505 \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3030*x^2 + 3939*x + 1277)/(45*x^3 + 87*x^2 + 56*x + 12) - 505*log(5*x + 3) + 505*log(3*x + 2)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 36, normalized size = 0.78 \begin {gather*} 1010\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {101\,x^2}{3}+\frac {1313\,x}{30}+\frac {1277}{90}}{x^3+\frac {29\,x^2}{15}+\frac {56\,x}{45}+\frac {4}{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1010*atanh(30*x + 19) - ((1313*x)/30 + (101*x^2)/3 + 1277/90)/((56*x)/45 + (29*x^2)/15 + x^3 + 4/15)

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 41, normalized size = 0.89 \begin {gather*} - \frac {3030 x^{2} + 3939 x + 1277}{90 x^{3} + 174 x^{2} + 112 x + 24} - 505 \log {\left (x + \frac {3}{5} \right )} + 505 \log {\left (x + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3030*x**2 + 3939*x + 1277)/(90*x**3 + 174*x**2 + 112*x + 24) - 505*log(x + 3/5) + 505*log(x + 2/3)

________________________________________________________________________________________