3.94.35 \(\int \frac {112+8 e^2+e (-60-14 x)+56 x+7 x^2}{64+4 e^2+e (-32-8 x)+32 x+4 x^2} \, dx\)

Optimal. Leaf size=23 \[ (4+x) \left (2-\frac {x}{4 (4-e+x)}\right )+\log (\log (5)) \]

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Rubi [A]  time = 0.06, antiderivative size = 24, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {1984, 27, 12, 683} \begin {gather*} \frac {7 x}{4}+\frac {(4-e) e}{4 (x-e+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(112 + 8*E^2 + E*(-60 - 14*x) + 56*x + 7*x^2)/(64 + 4*E^2 + E*(-32 - 8*x) + 32*x + 4*x^2),x]

[Out]

(7*x)/4 + ((4 - E)*E)/(4*(4 - E + x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 1984

Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] &&
QuadraticQ[{u, v}, x] &&  !QuadraticMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (28-15 e+2 e^2\right )+14 (4-e) x+7 x^2}{4 (4-e)^2+8 (4-e) x+4 x^2} \, dx\\ &=\int \frac {4 \left (28-15 e+2 e^2\right )+14 (4-e) x+7 x^2}{4 (-4+e-x)^2} \, dx\\ &=\frac {1}{4} \int \frac {4 \left (28-15 e+2 e^2\right )+14 (4-e) x+7 x^2}{(-4+e-x)^2} \, dx\\ &=\frac {1}{4} \int \left (7+\frac {(-4+e) e}{(-4+e-x)^2}\right ) \, dx\\ &=\frac {7 x}{4}+\frac {(4-e) e}{4 (4-e+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 1.13 \begin {gather*} \frac {1}{4} \left (\frac {(-4+e) e}{-4+e-x}+7 (4-e+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(112 + 8*E^2 + E*(-60 - 14*x) + 56*x + 7*x^2)/(64 + 4*E^2 + E*(-32 - 8*x) + 32*x + 4*x^2),x]

[Out]

(((-4 + E)*E)/(-4 + E - x) + 7*(4 - E + x))/4

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fricas [A]  time = 0.50, size = 33, normalized size = 1.43 \begin {gather*} \frac {7 \, x^{2} - {\left (7 \, x - 4\right )} e + 28 \, x - e^{2}}{4 \, {\left (x - e + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(1)^2+(-14*x-60)*exp(1)+7*x^2+56*x+112)/(4*exp(1)^2+(-8*x-32)*exp(1)+4*x^2+32*x+64),x, algorit
hm="fricas")

[Out]

1/4*(7*x^2 - (7*x - 4)*e + 28*x - e^2)/(x - e + 4)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(1)^2+(-14*x-60)*exp(1)+7*x^2+56*x+112)/(4*exp(1)^2+(-8*x-32)*exp(1)+4*x^2+32*x+64),x, algorit
hm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 1/4*(7*sageVARx+(-8*exp(1)+2*exp(2))*1/2
/sqrt(-exp(1)^2+exp(2))*atan((sageVARx-exp(1)+4)/sqrt(-exp(1)^2+exp(2))))

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maple [A]  time = 0.63, size = 28, normalized size = 1.22

method	result	size
norman	\(\frac {-\frac {7 x^{2}}{4}+2 \,{\mathrm e}^{2}-15 \,{\mathrm e}+28}{{\mathrm e}-x -4}\)	\(28\)
gosper	\(\frac {-7 x^{2}+112+8 \,{\mathrm e}^{2}-60 \,{\mathrm e}}{4 \,{\mathrm e}-4 x -16}\)	\(29\)
risch	\(\frac {7 x}{4}+\frac {{\mathrm e}^{2}}{4 \,{\mathrm e}-4 x -16}-\frac {{\mathrm e}}{{\mathrm e}-x -4}\)	\(31\)
meijerg	\(\frac {28 x}{\left (4-{\mathrm e}\right )^{2} \left (1+\frac {x}{4-{\mathrm e}}\right )}+\frac {\left (-14 \,{\mathrm e}+56\right ) \left (-\frac {x}{\left (1+\frac {x}{4-{\mathrm e}}\right ) \left (4-{\mathrm e}\right )}+\ln \left (1+\frac {x}{4-{\mathrm e}}\right )\right )}{4}+\frac {7 \left (4-{\mathrm e}\right ) \left (\frac {x \left (\frac {3 x}{4-{\mathrm e}}+6\right )}{3 \left (4-{\mathrm e}\right ) \left (1+\frac {x}{4-{\mathrm e}}\right )}-2 \ln \left (1+\frac {x}{4-{\mathrm e}}\right )\right )}{4}+\frac {2 \,{\mathrm e}^{2} x}{\left (4-{\mathrm e}\right )^{2} \left (1+\frac {x}{4-{\mathrm e}}\right )}-\frac {15 \,{\mathrm e} x}{\left (4-{\mathrm e}\right )^{2} \left (1+\frac {x}{4-{\mathrm e}}\right )}\)	\(190\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*exp(1)^2+(-14*x-60)*exp(1)+7*x^2+56*x+112)/(4*exp(1)^2+(-8*x-32)*exp(1)+4*x^2+32*x+64),x,method=_RETURN
VERBOSE)

[Out]

(-7/4*x^2+2*exp(1)^2-15*exp(1)+28)/(exp(1)-x-4)

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maxima [A]  time = 0.36, size = 22, normalized size = 0.96 \begin {gather*} \frac {7}{4} \, x - \frac {e^{2} - 4 \, e}{4 \, {\left (x - e + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(1)^2+(-14*x-60)*exp(1)+7*x^2+56*x+112)/(4*exp(1)^2+(-8*x-32)*exp(1)+4*x^2+32*x+64),x, algorit
hm="maxima")

[Out]

7/4*x - 1/4*(e^2 - 4*e)/(x - e + 4)

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mupad [B]  time = 0.19, size = 21, normalized size = 0.91 \begin {gather*} \frac {7\,x}{4}+\frac {\mathrm {e}-\frac {{\mathrm {e}}^2}{4}}{x-\mathrm {e}+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((56*x + 8*exp(2) + 7*x^2 - exp(1)*(14*x + 60) + 112)/(32*x + 4*exp(2) + 4*x^2 - exp(1)*(8*x + 32) + 64),x)

[Out]

(7*x)/4 + (exp(1) - exp(2)/4)/(x - exp(1) + 4)

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sympy [A]  time = 0.21, size = 22, normalized size = 0.96 \begin {gather*} \frac {7 x}{4} + \frac {- e^{2} + 4 e}{4 x - 4 e + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(1)**2+(-14*x-60)*exp(1)+7*x**2+56*x+112)/(4*exp(1)**2+(-8*x-32)*exp(1)+4*x**2+32*x+64),x)

[Out]

7*x/4 + (-exp(2) + 4*E)/(4*x - 4*E + 16)

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