3.101.56 \(\int \frac {677 x^2+x^3+e^{-1+x} (676-675 x-x^2)}{676 x^2+x^3} \, dx\)

Optimal. Leaf size=17 \[ 1-\frac {e^{-1+x}}{x}+x+\log (676+x) \]

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Rubi [A]  time = 0.33, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1593, 6742, 2197, 43} \begin {gather*} x-\frac {e^{x-1}}{x}+\log (x+676) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(677*x^2 + x^3 + E^(-1 + x)*(676 - 675*x - x^2))/(676*x^2 + x^3),x]

[Out]

-(E^(-1 + x)/x) + x + Log[676 + 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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {677 x^2+x^3+e^{-1+x} \left (676-675 x-x^2\right )}{x^2 (676+x)} \, dx\\ &=\int \left (-\frac {e^{-1+x} (-1+x)}{x^2}+\frac {677+x}{676+x}\right ) \, dx\\ &=-\int \frac {e^{-1+x} (-1+x)}{x^2} \, dx+\int \frac {677+x}{676+x} \, dx\\ &=-\frac {e^{-1+x}}{x}+\int \left (1+\frac {1}{676+x}\right ) \, dx\\ &=-\frac {e^{-1+x}}{x}+x+\log (676+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 22, normalized size = 1.29 \begin {gather*} \frac {-\frac {e^x}{x}+e x+e \log (676+x)}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(677*x^2 + x^3 + E^(-1 + x)*(676 - 675*x - x^2))/(676*x^2 + x^3),x]

[Out]

(-(E^x/x) + E*x + E*Log[676 + x])/E

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fricas [A]  time = 0.59, size = 20, normalized size = 1.18 \begin {gather*} \frac {x^{2} + x \log \left (x + 676\right ) - e^{\left (x - 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-675*x+676)*exp(x-1)+x^3+677*x^2)/(x^3+676*x^2),x, algorithm="fricas")

[Out]

(x^2 + x*log(x + 676) - e^(x - 1))/x

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giac [A]  time = 0.13, size = 25, normalized size = 1.47 \begin {gather*} \frac {{\left (x^{2} e + x e \log \left (x + 676\right ) - e^{x}\right )} e^{\left (-1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-675*x+676)*exp(x-1)+x^3+677*x^2)/(x^3+676*x^2),x, algorithm="giac")

[Out]

(x^2*e + x*e*log(x + 676) - e^x)*e^(-1)/x

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maple [A]  time = 0.12, size = 16, normalized size = 0.94

method	result	size
risch	\(x +\ln \left (676+x \right )-\frac {{\mathrm e}^{x -1}}{x}\)	\(16\)
derivativedivides	\(x -1+\ln \left (676+x \right )-\frac {{\mathrm e}^{x -1}}{x}\)	\(17\)
default	\(x -1+\ln \left (676+x \right )-\frac {{\mathrm e}^{x -1}}{x}\)	\(17\)
norman	\(\frac {x^{2}-{\mathrm e}^{x -1}}{x}+\ln \left (676+x \right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-675*x+676)*exp(x-1)+x^3+677*x^2)/(x^3+676*x^2),x,method=_RETURNVERBOSE)

[Out]

x+ln(676+x)-exp(x-1)/x

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maxima [A]  time = 0.42, size = 15, normalized size = 0.88 \begin {gather*} x - \frac {e^{\left (x - 1\right )}}{x} + \log \left (x + 676\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-675*x+676)*exp(x-1)+x^3+677*x^2)/(x^3+676*x^2),x, algorithm="maxima")

[Out]

x - e^(x - 1)/x + log(x + 676)

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mupad [B]  time = 0.14, size = 20, normalized size = 1.18 \begin {gather*} \ln \left (x+676\right )-\frac {{\mathrm {e}}^{x-1}-x^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((677*x^2 - exp(x - 1)*(675*x + x^2 - 676) + x^3)/(676*x^2 + x^3),x)

[Out]

log(x + 676) - (exp(x - 1) - x^2)/x

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sympy [A]  time = 0.12, size = 12, normalized size = 0.71 \begin {gather*} x + \log {\left (x + 676 \right )} - \frac {e^{x - 1}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-675*x+676)*exp(x-1)+x**3+677*x**2)/(x**3+676*x**2),x)

[Out]

x + log(x + 676) - exp(x - 1)/x

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