3.1199 \(\int \frac{1-2 x}{3+5 x} \, dx\)

Optimal. Leaf size=16 \[ \frac{11}{25} \log (5 x+3)-\frac{2 x}{5} \]

[Out]

(-2*x)/5 + (11*Log[3 + 5*x])/25

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Rubi [A]  time = 0.0071116, 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} \[ \frac{11}{25} \log (5 x+3)-\frac{2 x}{5} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)/(3 + 5*x),x]

[Out]

(-2*x)/5 + (11*Log[3 + 5*x])/25

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

Mathematica [A]  time = 0.0027299, size = 17, normalized size = 1.06 \[ \frac{1}{25} (-10 x+11 \log (5 x+3)-6) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)/(3 + 5*x),x]

[Out]

(-6 - 10*x + 11*Log[3 + 5*x])/25

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*} -{\frac{2\,x}{5}}+{\frac{11\,\ln \left ( 3+5\,x \right ) }{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*x+11/25*ln(3+5*x)

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Maxima [A]  time = 1.25247, size = 16, normalized size = 1. \begin{align*} -\frac{2}{5} \, x + \frac{11}{25} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*x + 11/25*log(5*x + 3)

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Fricas [A]  time = 1.46048, size = 39, normalized size = 2.44 \begin{align*} -\frac{2}{5} \, x + \frac{11}{25} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*x + 11/25*log(5*x + 3)

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Sympy [A]  time = 0.07203, size = 14, normalized size = 0.88 \begin{align*} - \frac{2 x}{5} + \frac{11 \log{\left (5 x + 3 \right )}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x/5 + 11*log(5*x + 3)/25

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Giac [A]  time = 2.86108, size = 18, normalized size = 1.12 \begin{align*} -\frac{2}{5} \, x + \frac{11}{25} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*x + 11/25*log(abs(5*x + 3))