3.1442 \(\int \frac{3+5 x}{1-2 x} \, dx\)

Optimal. Leaf size=16 \[ -\frac{5 x}{2}-\frac{11}{4} \log (1-2 x) \]

[Out]

(-5*x)/2 - (11*Log[1 - 2*x])/4

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Rubi [A]  time = 0.0070926, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ -\frac{5 x}{2}-\frac{11}{4} \log (1-2 x) \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/(1 - 2*x),x]

[Out]

(-5*x)/2 - (11*Log[1 - 2*x])/4

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{3+5 x}{1-2 x} \, dx &=\int \left (-\frac{5}{2}-\frac{11}{2 (-1+2 x)}\right ) \, dx\\ &=-\frac{5 x}{2}-\frac{11}{4} \log (1-2 x)\\ \end{align*}

Mathematica [A]  time = 0.0026952, size = 17, normalized size = 1.06 \[ \frac{1}{4} (-10 x-11 \log (1-2 x)+5) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/(1 - 2*x),x]

[Out]

(5 - 10*x - 11*Log[1 - 2*x])/4

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Maple [A]  time = 0.002, size = 13, normalized size = 0.8 \begin{align*} -{\frac{5\,x}{2}}-{\frac{11\,\ln \left ( 2\,x-1 \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2*x-11/4*ln(2*x-1)

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Maxima [A]  time = 1.02367, size = 16, normalized size = 1. \begin{align*} -\frac{5}{2} \, x - \frac{11}{4} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2*x - 11/4*log(2*x - 1)

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Fricas [A]  time = 1.23041, size = 38, normalized size = 2.38 \begin{align*} -\frac{5}{2} \, x - \frac{11}{4} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2*x - 11/4*log(2*x - 1)

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Sympy [A]  time = 0.076542, size = 15, normalized size = 0.94 \begin{align*} - \frac{5 x}{2} - \frac{11 \log{\left (2 x - 1 \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*x/2 - 11*log(2*x - 1)/4

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Giac [A]  time = 2.95619, size = 18, normalized size = 1.12 \begin{align*} -\frac{5}{2} \, x - \frac{11}{4} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2*x - 11/4*log(abs(2*x - 1))