3.55.31 \(\int \frac {1+5 x}{-4+5 x} \, dx\)

Optimal. Leaf size=19 \[ x-\log \left (\frac {5-\log (4)}{4-5 x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 0.42, 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*} x+\log (4-5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 5*x)/(-4 + 5*x),x]

[Out]

x + Log[4 - 5*x]

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 {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {5}{-4+5 x}\right ) \, dx\\ &=x+\log (4-5 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.42 \begin {gather*} x+\log (-4+5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 5*x)/(-4 + 5*x),x]

[Out]

x + Log[-4 + 5*x]

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fricas [A]  time = 0.99, size = 8, normalized size = 0.42 \begin {gather*} x + \log \left (5 \, x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+5*x)/(5*x-4),x, algorithm="fricas")

[Out]

x + log(5*x - 4)

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giac [A]  time = 0.20, size = 9, normalized size = 0.47 \begin {gather*} x + \log \left ({\left | 5 \, x - 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+5*x)/(5*x-4),x, algorithm="giac")

[Out]

x + log(abs(5*x - 4))

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maple [A]  time = 0.21, size = 9, normalized size = 0.47

method	result	size
default	\(x +\ln \left (5 x -4\right )\)	\(9\)
norman	\(x +\ln \left (5 x -4\right )\)	\(9\)
meijerg	\(\ln \left (1-\frac {5 x}{4}\right )+x\)	\(9\)
risch	\(x +\ln \left (5 x -4\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+5*x)/(5*x-4),x,method=_RETURNVERBOSE)

[Out]

x+ln(5*x-4)

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maxima [A]  time = 0.38, size = 8, normalized size = 0.42 \begin {gather*} x + \log \left (5 \, x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+5*x)/(5*x-4),x, algorithm="maxima")

[Out]

x + log(5*x - 4)

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mupad [B]  time = 0.06, size = 6, normalized size = 0.32 \begin {gather*} x+\ln \left (x-\frac {4}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 1)/(5*x - 4),x)

[Out]

x + log(x - 4/5)

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sympy [A]  time = 0.06, size = 7, normalized size = 0.37 \begin {gather*} x + \log {\left (5 x - 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+5*x)/(5*x-4),x)

[Out]

x + log(5*x - 4)

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