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

Optimal. Leaf size=50 \[ \frac {343}{3 (3 x+2)}+\frac {8712}{25 (5 x+3)}-\frac {1331}{50 (5 x+3)^2}-1617 \log (3 x+2)+1617 \log (5 x+3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \begin {gather*} \frac {343}{3 (3 x+2)}+\frac {8712}{25 (5 x+3)}-\frac {1331}{50 (5 x+3)^2}-1617 \log (3 x+2)+1617 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

343/(3*(2 + 3*x)) - 1331/(50*(3 + 5*x)^2) + 8712/(25*(3 + 5*x)) - 1617*Log[2 + 3*x] + 1617*Log[3 + 5*x]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(1-2 x)^3}{(2+3 x)^2 (3+5 x)^3} \, dx &=\int \left (-\frac {343}{(2+3 x)^2}-\frac {4851}{2+3 x}+\frac {1331}{5 (3+5 x)^3}-\frac {8712}{5 (3+5 x)^2}+\frac {8085}{3+5 x}\right ) \, dx\\ &=\frac {343}{3 (2+3 x)}-\frac {1331}{50 (3+5 x)^2}+\frac {8712}{25 (3+5 x)}-1617 \log (2+3 x)+1617 \log (3+5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 48, normalized size = 0.96 \begin {gather*} \frac {343}{9 x+6}+\frac {8712}{125 x+75}-\frac {1331}{50 (5 x+3)^2}-1617 \log (5 (3 x+2))+1617 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1331/(50*(3 + 5*x)^2) + 343/(6 + 9*x) + 8712/(75 + 125*x) - 1617*Log[5*(2 + 3*x)] + 1617*Log[3 + 5*x]

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 75, normalized size = 1.50 \begin {gather*} \frac {1212830 \, x^{2} + 242550 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (5 \, x + 3\right ) - 242550 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (3 \, x + 2\right ) + 1495689 \, x + 459996}{150 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/150*(1212830*x^2 + 242550*(75*x^3 + 140*x^2 + 87*x + 18)*log(5*x + 3) - 242550*(75*x^3 + 140*x^2 + 87*x + 18
)*log(3*x + 2) + 1495689*x + 459996)/(75*x^3 + 140*x^2 + 87*x + 18)

________________________________________________________________________________________

giac [A]  time = 1.16, size = 49, normalized size = 0.98 \begin {gather*} \frac {343}{3 \, {\left (3 \, x + 2\right )}} - \frac {1089 \, {\left (\frac {14}{3 \, x + 2} - 59\right )}}{2 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + 1617 \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/3/(3*x + 2) - 1089/2*(14/(3*x + 2) - 59)/(1/(3*x + 2) - 5)^2 + 1617*log(abs(-1/(3*x + 2) + 5))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 45, normalized size = 0.90 \begin {gather*} -1617 \ln \left (3 x +2\right )+1617 \ln \left (5 x +3\right )+\frac {343}{3 \left (3 x +2\right )}-\frac {1331}{50 \left (5 x +3\right )^{2}}+\frac {8712}{25 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/3/(3*x+2)-1331/50/(5*x+3)^2+8712/25/(5*x+3)-1617*ln(3*x+2)+1617*ln(5*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 46, normalized size = 0.92 \begin {gather*} \frac {1212830 \, x^{2} + 1495689 \, x + 459996}{150 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} + 1617 \, \log \left (5 \, x + 3\right ) - 1617 \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/150*(1212830*x^2 + 1495689*x + 459996)/(75*x^3 + 140*x^2 + 87*x + 18) + 1617*log(5*x + 3) - 1617*log(3*x + 2
)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 35, normalized size = 0.70 \begin {gather*} \frac {\frac {121283\,x^2}{1125}+\frac {498563\,x}{3750}+\frac {76666}{1875}}{x^3+\frac {28\,x^2}{15}+\frac {29\,x}{25}+\frac {6}{25}}-3234\,\mathrm {atanh}\left (30\,x+19\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((498563*x)/3750 + (121283*x^2)/1125 + 76666/1875)/((29*x)/25 + (28*x^2)/15 + x^3 + 6/25) - 3234*atanh(30*x +
19)

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 42, normalized size = 0.84 \begin {gather*} - \frac {- 1212830 x^{2} - 1495689 x - 459996}{11250 x^{3} + 21000 x^{2} + 13050 x + 2700} + 1617 \log {\left (x + \frac {3}{5} \right )} - 1617 \log {\left (x + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1212830*x**2 - 1495689*x - 459996)/(11250*x**3 + 21000*x**2 + 13050*x + 2700) + 1617*log(x + 3/5) - 1617*lo
g(x + 2/3)

________________________________________________________________________________________