3.1624 \(\int \frac{2+3 x}{(1-2 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{14}{1331 (1-2 x)}-\frac{37}{1331 (5 x+3)}-\frac{1}{242 (5 x+3)^2}-\frac{144 \log (1-2 x)}{14641}+\frac{144 \log (5 x+3)}{14641} \]

[Out]

14/(1331*(1 - 2*x)) - 1/(242*(3 + 5*x)^2) - 37/(1331*(3 + 5*x)) - (144*Log[1 - 2*x])/14641 + (144*Log[3 + 5*x]
)/14641

________________________________________________________________________________________

Rubi [A]  time = 0.0222102, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{14}{1331 (1-2 x)}-\frac{37}{1331 (5 x+3)}-\frac{1}{242 (5 x+3)^2}-\frac{144 \log (1-2 x)}{14641}+\frac{144 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

14/(1331*(1 - 2*x)) - 1/(242*(3 + 5*x)^2) - 37/(1331*(3 + 5*x)) - (144*Log[1 - 2*x])/14641 + (144*Log[3 + 5*x]
)/14641

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{2+3 x}{(1-2 x)^2 (3+5 x)^3} \, dx &=\int \left (\frac{28}{1331 (-1+2 x)^2}-\frac{288}{14641 (-1+2 x)}+\frac{5}{121 (3+5 x)^3}+\frac{185}{1331 (3+5 x)^2}+\frac{720}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{14}{1331 (1-2 x)}-\frac{1}{242 (3+5 x)^2}-\frac{37}{1331 (3+5 x)}-\frac{144 \log (1-2 x)}{14641}+\frac{144 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0242962, size = 47, normalized size = 0.87 \[ \frac{-\frac{11 \left (1440 x^2+936 x+19\right )}{(2 x-1) (5 x+3)^2}-288 \log (1-2 x)+288 \log (10 x+6)}{29282} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(19 + 936*x + 1440*x^2))/((-1 + 2*x)*(3 + 5*x)^2) - 288*Log[1 - 2*x] + 288*Log[6 + 10*x])/29282

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 45, normalized size = 0.8 \begin{align*} -{\frac{14}{2662\,x-1331}}-{\frac{144\,\ln \left ( 2\,x-1 \right ) }{14641}}-{\frac{1}{242\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{37}{3993+6655\,x}}+{\frac{144\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-14/1331/(2*x-1)-144/14641*ln(2*x-1)-1/242/(3+5*x)^2-37/1331/(3+5*x)+144/14641*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.40802, size = 62, normalized size = 1.15 \begin{align*} -\frac{1440 \, x^{2} + 936 \, x + 19}{2662 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac{144}{14641} \, \log \left (5 \, x + 3\right ) - \frac{144}{14641} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2662*(1440*x^2 + 936*x + 19)/(50*x^3 + 35*x^2 - 12*x - 9) + 144/14641*log(5*x + 3) - 144/14641*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.28893, size = 220, normalized size = 4.07 \begin{align*} -\frac{15840 \, x^{2} - 288 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 288 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) + 10296 \, x + 209}{29282 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29282*(15840*x^2 - 288*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) + 288*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x
- 1) + 10296*x + 209)/(50*x^3 + 35*x^2 - 12*x - 9)

________________________________________________________________________________________

Sympy [A]  time = 0.141584, size = 44, normalized size = 0.81 \begin{align*} - \frac{1440 x^{2} + 936 x + 19}{133100 x^{3} + 93170 x^{2} - 31944 x - 23958} - \frac{144 \log{\left (x - \frac{1}{2} \right )}}{14641} + \frac{144 \log{\left (x + \frac{3}{5} \right )}}{14641} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1440*x**2 + 936*x + 19)/(133100*x**3 + 93170*x**2 - 31944*x - 23958) - 144*log(x - 1/2)/14641 + 144*log(x +
3/5)/14641

________________________________________________________________________________________

Giac [A]  time = 2.64719, size = 69, normalized size = 1.28 \begin{align*} -\frac{14}{1331 \,{\left (2 \, x - 1\right )}} + \frac{10 \,{\left (\frac{429}{2 \, x - 1} + 190\right )}}{14641 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}} + \frac{144}{14641} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-14/1331/(2*x - 1) + 10/14641*(429/(2*x - 1) + 190)/(11/(2*x - 1) + 5)^2 + 144/14641*log(abs(-11/(2*x - 1) - 5
))