3.1626 \(\int \frac{1}{(1-2 x)^2 (2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{8}{9317 (1-2 x)}+\frac{725}{1331 (5 x+3)}-\frac{25}{242 (5 x+3)^2}-\frac{1104 \log (1-2 x)}{717409}-\frac{81}{49} \log (3 x+2)+\frac{24225 \log (5 x+3)}{14641} \]

[Out]

8/(9317*(1 - 2*x)) - 25/(242*(3 + 5*x)^2) + 725/(1331*(3 + 5*x)) - (1104*Log[1 - 2*x])/717409 - (81*Log[2 + 3*
x])/49 + (24225*Log[3 + 5*x])/14641

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Rubi [A]  time = 0.0304716, antiderivative size = 64, normalized size of antiderivative = 1., 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} \[ \frac{8}{9317 (1-2 x)}+\frac{725}{1331 (5 x+3)}-\frac{25}{242 (5 x+3)^2}-\frac{1104 \log (1-2 x)}{717409}-\frac{81}{49} \log (3 x+2)+\frac{24225 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

8/(9317*(1 - 2*x)) - 25/(242*(3 + 5*x)^2) + 725/(1331*(3 + 5*x)) - (1104*Log[1 - 2*x])/717409 - (81*Log[2 + 3*
x])/49 + (24225*Log[3 + 5*x])/14641

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}{(1-2 x)^2 (2+3 x) (3+5 x)^3} \, dx &=\int \left (\frac{16}{9317 (-1+2 x)^2}-\frac{2208}{717409 (-1+2 x)}-\frac{243}{49 (2+3 x)}+\frac{125}{121 (3+5 x)^3}-\frac{3625}{1331 (3+5 x)^2}+\frac{121125}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{8}{9317 (1-2 x)}-\frac{25}{242 (3+5 x)^2}+\frac{725}{1331 (3+5 x)}-\frac{1104 \log (1-2 x)}{717409}-\frac{81}{49} \log (2+3 x)+\frac{24225 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0307163, size = 60, normalized size = 0.94 \[ \frac{3 \left (\frac{1232}{3-6 x}+\frac{781550}{15 x+9}-\frac{148225}{3 (5 x+3)^2}-736 \log (3-6 x)-790614 \log (3 x+2)+791350 \log (-3 (5 x+3))\right )}{1434818} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1232/(3 - 6*x) - 148225/(3*(3 + 5*x)^2) + 781550/(9 + 15*x) - 736*Log[3 - 6*x] - 790614*Log[2 + 3*x] + 791
350*Log[-3*(3 + 5*x)]))/1434818

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Maple [A]  time = 0.01, size = 53, normalized size = 0.8 \begin{align*} -{\frac{8}{18634\,x-9317}}-{\frac{1104\,\ln \left ( 2\,x-1 \right ) }{717409}}-{\frac{81\,\ln \left ( 2+3\,x \right ) }{49}}-{\frac{25}{242\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{725}{3993+6655\,x}}+{\frac{24225\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/9317/(2*x-1)-1104/717409*ln(2*x-1)-81/49*ln(2+3*x)-25/242/(3+5*x)^2+725/1331/(3+5*x)+24225/14641*ln(3+5*x)

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Maxima [A]  time = 1.03863, size = 73, normalized size = 1.14 \begin{align*} \frac{101100 \, x^{2} + 5820 \, x - 28669}{18634 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac{24225}{14641} \, \log \left (5 \, x + 3\right ) - \frac{81}{49} \, \log \left (3 \, x + 2\right ) - \frac{1104}{717409} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18634*(101100*x^2 + 5820*x - 28669)/(50*x^3 + 35*x^2 - 12*x - 9) + 24225/14641*log(5*x + 3) - 81/49*log(3*x
+ 2) - 1104/717409*log(2*x - 1)

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Fricas [A]  time = 1.46296, size = 308, normalized size = 4.81 \begin{align*} \frac{7784700 \, x^{2} + 2374050 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) - 2371842 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (3 \, x + 2\right ) - 2208 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) + 448140 \, x - 2207513}{1434818 \,{\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(1/(1-2*x)^2/(2+3*x)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/1434818*(7784700*x^2 + 2374050*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) - 2371842*(50*x^3 + 35*x^2 - 12*x -
9)*log(3*x + 2) - 2208*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x - 1) + 448140*x - 2207513)/(50*x^3 + 35*x^2 - 12*
x - 9)

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Sympy [A]  time = 0.197315, size = 54, normalized size = 0.84 \begin{align*} \frac{101100 x^{2} + 5820 x - 28669}{931700 x^{3} + 652190 x^{2} - 223608 x - 167706} - \frac{1104 \log{\left (x - \frac{1}{2} \right )}}{717409} + \frac{24225 \log{\left (x + \frac{3}{5} \right )}}{14641} - \frac{81 \log{\left (x + \frac{2}{3} \right )}}{49} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(101100*x**2 + 5820*x - 28669)/(931700*x**3 + 652190*x**2 - 223608*x - 167706) - 1104*log(x - 1/2)/717409 + 24
225*log(x + 3/5)/14641 - 81*log(x + 2/3)/49

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Giac [A]  time = 3.10102, size = 89, normalized size = 1.39 \begin{align*} -\frac{8}{9317 \,{\left (2 \, x - 1\right )}} - \frac{250 \,{\left (\frac{297}{2 \, x - 1} + 140\right )}}{14641 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}} - \frac{81}{49} \, \log \left ({\left | -\frac{7}{2 \, x - 1} - 3 \right |}\right ) + \frac{24225}{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(1/(1-2*x)^2/(2+3*x)/(3+5*x)^3,x, algorithm="giac")

[Out]

-8/9317/(2*x - 1) - 250/14641*(297/(2*x - 1) + 140)/(11/(2*x - 1) + 5)^2 - 81/49*log(abs(-7/(2*x - 1) - 3)) +
24225/14641*log(abs(-11/(2*x - 1) - 5))