3.1678 \(\int \frac{2+3 x}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{121 (1-2 x)}+\frac{7}{44 (1-2 x)^2}-\frac{5 \log (1-2 x)}{1331}+\frac{5 \log (5 x+3)}{1331} \]

[Out]

7/(44*(1 - 2*x)^2) + 1/(121*(1 - 2*x)) - (5*Log[1 - 2*x])/1331 + (5*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.015923, antiderivative size = 43, 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} \[ \frac{1}{121 (1-2 x)}+\frac{7}{44 (1-2 x)^2}-\frac{5 \log (1-2 x)}{1331}+\frac{5 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(44*(1 - 2*x)^2) + 1/(121*(1 - 2*x)) - (5*Log[1 - 2*x])/1331 + (5*Log[3 + 5*x])/1331

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)^3 (3+5 x)} \, dx &=\int \left (-\frac{7}{11 (-1+2 x)^3}+\frac{2}{121 (-1+2 x)^2}-\frac{10}{1331 (-1+2 x)}+\frac{25}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{7}{44 (1-2 x)^2}+\frac{1}{121 (1-2 x)}-\frac{5 \log (1-2 x)}{1331}+\frac{5 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0148925, size = 46, normalized size = 1.07 \[ \frac{-88 x-20 (1-2 x)^2 \log (1-2 x)+20 (1-2 x)^2 \log (10 x+6)+891}{5324 (1-2 x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(891 - 88*x - 20*(1 - 2*x)^2*Log[1 - 2*x] + 20*(1 - 2*x)^2*Log[6 + 10*x])/(5324*(1 - 2*x)^2)

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*}{\frac{7}{44\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{1}{242\,x-121}}-{\frac{5\,\ln \left ( 2\,x-1 \right ) }{1331}}+{\frac{5\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/44/(2*x-1)^2-1/121/(2*x-1)-5/1331*ln(2*x-1)+5/1331*ln(3+5*x)

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Maxima [A]  time = 1.13342, size = 49, normalized size = 1.14 \begin{align*} -\frac{8 \, x - 81}{484 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{5}{1331} \, \log \left (5 \, x + 3\right ) - \frac{5}{1331} \, \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)^3/(3+5*x),x, algorithm="maxima")

[Out]

-1/484*(8*x - 81)/(4*x^2 - 4*x + 1) + 5/1331*log(5*x + 3) - 5/1331*log(2*x - 1)

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Fricas [A]  time = 1.60221, size = 150, normalized size = 3.49 \begin{align*} \frac{20 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) - 20 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 88 \, x + 891}{5324 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5324*(20*(4*x^2 - 4*x + 1)*log(5*x + 3) - 20*(4*x^2 - 4*x + 1)*log(2*x - 1) - 88*x + 891)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.130404, size = 34, normalized size = 0.79 \begin{align*} - \frac{8 x - 81}{1936 x^{2} - 1936 x + 484} - \frac{5 \log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{5 \log{\left (x + \frac{3}{5} \right )}}{1331} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(8*x - 81)/(1936*x**2 - 1936*x + 484) - 5*log(x - 1/2)/1331 + 5*log(x + 3/5)/1331

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Giac [A]  time = 3.10754, size = 45, normalized size = 1.05 \begin{align*} -\frac{8 \, x - 81}{484 \,{\left (2 \, x - 1\right )}^{2}} + \frac{5}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{5}{1331} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/484*(8*x - 81)/(2*x - 1)^2 + 5/1331*log(abs(5*x + 3)) - 5/1331*log(abs(2*x - 1))