3.86.97 \(\int \frac {-21+x+e^4 (-5+5 x)}{-5+5 x} \, dx\)

Optimal. Leaf size=19 \[ -2+\left (\frac {1}{5}+e^4\right ) x-\log \left ((-1+x)^4\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {186, 43} \begin {gather*} \frac {1}{5} \left (1+5 e^4\right ) x-4 \log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-21 + x + E^4*(-5 + 5*x))/(-5 + 5*x),x]

[Out]

((1 + 5*E^4)*x)/5 - 4*Log[1 - 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])

Rule 186

Int[(u_)^(m_.)*(v_)^(n_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n, x] /; FreeQ[{m, n}, x] &&
LinearQ[{u, v}, x] &&  !LinearMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-21-5 e^4+\left (1+5 e^4\right ) x}{-5+5 x} \, dx\\ &=\int \left (\frac {1}{5} \left (1+5 e^4\right )-\frac {4}{-1+x}\right ) \, dx\\ &=\frac {1}{5} \left (1+5 e^4\right ) x-4 \log (1-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.16 \begin {gather*} \frac {1}{5} \left (\left (1+5 e^4\right ) (-1+x)-20 \log (-1+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-21 + x + E^4*(-5 + 5*x))/(-5 + 5*x),x]

[Out]

((1 + 5*E^4)*(-1 + x) - 20*Log[-1 + x])/5

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fricas [A]  time = 0.61, size = 14, normalized size = 0.74 \begin {gather*} x e^{4} + \frac {1}{5} \, x - 4 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x-5)*exp(4)+x-21)/(5*x-5),x, algorithm="fricas")

[Out]

x*e^4 + 1/5*x - 4*log(x - 1)

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giac [A]  time = 0.13, size = 15, normalized size = 0.79 \begin {gather*} x e^{4} + \frac {1}{5} \, x - 4 \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x-5)*exp(4)+x-21)/(5*x-5),x, algorithm="giac")

[Out]

x*e^4 + 1/5*x - 4*log(abs(x - 1))

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maple [A]  time = 0.39, size = 15, normalized size = 0.79

method	result	size
default	\(x \,{\mathrm e}^{4}+\frac {x}{5}-4 \ln \left (x -1\right )\)	\(15\)
risch	\(x \,{\mathrm e}^{4}+\frac {x}{5}-4 \ln \left (x -1\right )\)	\(15\)
norman	\(x \left (\frac {1}{5}+{\mathrm e}^{4}\right )-4 \ln \left (5 x -5\right )\)	\(16\)
meijerg	\(-\frac {21 \ln \left (1-x \right )}{5}-\left (\frac {1}{5}+{\mathrm e}^{4}\right ) \left (-x -\ln \left (1-x \right )\right )-{\mathrm e}^{4} \ln \left (1-x \right )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x-5)*exp(4)+x-21)/(5*x-5),x,method=_RETURNVERBOSE)

[Out]

x*exp(4)+1/5*x-4*ln(x-1)

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maxima [A]  time = 0.36, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, x {\left (5 \, e^{4} + 1\right )} - 4 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x-5)*exp(4)+x-21)/(5*x-5),x, algorithm="maxima")

[Out]

1/5*x*(5*e^4 + 1) - 4*log(x - 1)

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mupad [B]  time = 0.05, size = 13, normalized size = 0.68 \begin {gather*} x\,\left ({\mathrm {e}}^4+\frac {1}{5}\right )-4\,\ln \left (x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + exp(4)*(5*x - 5) - 21)/(5*x - 5),x)

[Out]

x*(exp(4) + 1/5) - 4*log(x - 1)

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sympy [A]  time = 0.09, size = 14, normalized size = 0.74 \begin {gather*} x \left (\frac {1}{5} + e^{4}\right ) - 4 \log {\left (x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x-5)*exp(4)+x-21)/(5*x-5),x)

[Out]

x*(1/5 + exp(4)) - 4*log(x - 1)

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