3.1214 \(\int \frac{1-2 x}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=22 \[ -\frac{11}{25 (5 x+3)}-\frac{2}{25} \log (5 x+3) \]

[Out]

-11/(25*(3 + 5*x)) - (2*Log[3 + 5*x])/25

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Rubi [A]  time = 0.0080255, antiderivative size = 22, 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} \[ -\frac{11}{25 (5 x+3)}-\frac{2}{25} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)/(3 + 5*x)^2,x]

[Out]

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

Mathematica [A]  time = 0.0041088, size = 22, normalized size = 1. \[ -\frac{11}{25 (5 x+3)}-\frac{2}{25} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)/(3 + 5*x)^2,x]

[Out]

-11/(25*(3 + 5*x)) - (2*Log[3 + 5*x])/25

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Maple [A]  time = 0.005, size = 19, normalized size = 0.9 \begin{align*} -{\frac{11}{75+125\,x}}-{\frac{2\,\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)^2,x)

[Out]

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

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Maxima [A]  time = 1.03629, size = 24, normalized size = 1.09 \begin{align*} -\frac{11}{25 \,{\left (5 \, x + 3\right )}} - \frac{2}{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)^2,x, algorithm="maxima")

[Out]

-11/25/(5*x + 3) - 2/25*log(5*x + 3)

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Fricas [A]  time = 1.43811, size = 66, normalized size = 3. \begin{align*} -\frac{2 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 11}{25 \,{\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)^2,x, algorithm="fricas")

[Out]

-1/25*(2*(5*x + 3)*log(5*x + 3) + 11)/(5*x + 3)

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Sympy [A]  time = 0.086074, size = 17, normalized size = 0.77 \begin{align*} - \frac{2 \log{\left (5 x + 3 \right )}}{25} - \frac{11}{125 x + 75} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(5*x + 3)/25 - 11/(125*x + 75)

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Giac [A]  time = 2.64249, size = 38, normalized size = 1.73 \begin{align*} -\frac{11}{25 \,{\left (5 \, x + 3\right )}} + \frac{2}{25} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/25/(5*x + 3) + 2/25*log(1/5*abs(5*x + 3)/(5*x + 3)^2)