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

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

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {11}{25} \log (5 x+3)-\frac {2 x}{5} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.06 \begin {gather*} \frac {1}{25} (-10 x+11 \log (5 x+3)-6) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x}{3+5 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.39, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{5} \, x + \frac {11}{25} \, \log \left (5 \, x + 3\right ) \end {gather*}

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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giac [A]  time = 1.24, size = 13, normalized size = 0.81 \begin {gather*} -\frac {2}{5} \, x + \frac {11}{25} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end {gather*}

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))

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \begin {gather*} -\frac {2 x}{5}+\frac {11 \ln \left (5 x +3\right )}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{5} \, x + \frac {11}{25} \, \log \left (5 \, x + 3\right ) \end {gather*}

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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mupad [B]  time = 0.05, size = 10, normalized size = 0.62 \begin {gather*} \frac {11\,\ln \left (x+\frac {3}{5}\right )}{25}-\frac {2\,x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)/(5*x + 3),x)

[Out]

(11*log(x + 3/5))/25 - (2*x)/5

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sympy [A]  time = 0.08, size = 14, normalized size = 0.88 \begin {gather*} - \frac {2 x}{5} + \frac {11 \log {\left (5 x + 3 \right )}}{25} \end {gather*}

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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