3.55.99 \(\int \frac {150+e^x (12-2 x)-60 x+6 x^2}{25-10 x+x^2} \, dx\)

Optimal. Leaf size=17 \[ 2 \left (-2-\frac {e^x}{-5+x}+3 x\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 6688, 2197} \begin {gather*} 6 x+\frac {2 e^x}{5-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(150 + E^x*(12 - 2*x) - 60*x + 6*x^2)/(25 - 10*x + x^2),x]

[Out]

(2*E^x)/(5 - x) + 6*x

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[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 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {150+e^x (12-2 x)-60 x+6 x^2}{(-5+x)^2} \, dx\\ &=\int \left (6-\frac {2 e^x (-6+x)}{(-5+x)^2}\right ) \, dx\\ &=6 x-2 \int \frac {e^x (-6+x)}{(-5+x)^2} \, dx\\ &=\frac {2 e^x}{5-x}+6 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 14, normalized size = 0.82 \begin {gather*} -\frac {2 e^x}{-5+x}+6 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(150 + E^x*(12 - 2*x) - 60*x + 6*x^2)/(25 - 10*x + x^2),x]

[Out]

(-2*E^x)/(-5 + x) + 6*x

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fricas [A]  time = 1.44, size = 20, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (3 \, x^{2} - 15 \, x - e^{x}\right )}}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+12)*exp(x)+6*x^2-60*x+150)/(x^2-10*x+25),x, algorithm="fricas")

[Out]

2*(3*x^2 - 15*x - e^x)/(x - 5)

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giac [A]  time = 0.16, size = 20, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (3 \, x^{2} - 15 \, x - e^{x}\right )}}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+12)*exp(x)+6*x^2-60*x+150)/(x^2-10*x+25),x, algorithm="giac")

[Out]

2*(3*x^2 - 15*x - e^x)/(x - 5)

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maple [A]  time = 0.30, size = 14, normalized size = 0.82

method	result	size
default	\(6 x -\frac {2 \,{\mathrm e}^{x}}{x -5}\)	\(14\)
risch	\(6 x -\frac {2 \,{\mathrm e}^{x}}{x -5}\)	\(14\)
norman	\(\frac {6 x^{2}-2 \,{\mathrm e}^{x}-150}{x -5}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x+12)*exp(x)+6*x^2-60*x+150)/(x^2-10*x+25),x,method=_RETURNVERBOSE)

[Out]

6*x-2*exp(x)/(x-5)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 6 \, x - \frac {2 \, x e^{x}}{x^{2} - 10 \, x + 25} - \frac {12 \, e^{5} E_{2}\left (-x + 5\right )}{x - 5} - 2 \, \int \frac {{\left (x + 5\right )} e^{x}}{x^{3} - 15 \, x^{2} + 75 \, x - 125}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+12)*exp(x)+6*x^2-60*x+150)/(x^2-10*x+25),x, algorithm="maxima")

[Out]

6*x - 2*x*e^x/(x^2 - 10*x + 25) - 12*e^5*exp_integral_e(2, -x + 5)/(x - 5) - 2*integrate((x + 5)*e^x/(x^3 - 15
*x^2 + 75*x - 125), x)

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mupad [B]  time = 3.49, size = 13, normalized size = 0.76 \begin {gather*} 6\,x-\frac {2\,{\mathrm {e}}^x}{x-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(60*x + exp(x)*(2*x - 12) - 6*x^2 - 150)/(x^2 - 10*x + 25),x)

[Out]

6*x - (2*exp(x))/(x - 5)

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sympy [A]  time = 0.09, size = 10, normalized size = 0.59 \begin {gather*} 6 x - \frac {2 e^{x}}{x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+12)*exp(x)+6*x**2-60*x+150)/(x**2-10*x+25),x)

[Out]

6*x - 2*exp(x)/(x - 5)

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