3.96.61 \(\int \frac {e^{4 x} (16120 x^4+e (-1690 x-3380 x^2))}{169 e^2-1612 e x^2+3844 x^4} \, dx\)

Optimal. Leaf size=27 \[ \frac {e^{4 x} x}{x-\frac {e+\frac {3 x^2}{13}}{5 x}} \]

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Rubi [A]  time = 0.26, antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {28, 6741, 2288} \begin {gather*} -\frac {65 e^{4 x} x \left (13 e x-62 x^3\right )}{\left (13 e-62 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4*x)*(16120*x^4 + E*(-1690*x - 3380*x^2)))/(169*E^2 - 1612*E*x^2 + 3844*x^4),x]

[Out]

(-65*E^(4*x)*x*(13*E*x - 62*x^3))/(13*E - 62*x^2)^2

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=3844 \int \frac {e^{4 x} \left (16120 x^4+e \left (-1690 x-3380 x^2\right )\right )}{\left (-806 e+3844 x^2\right )^2} \, dx\\ &=3844 \int \frac {e^{4 x} x \left (-1690 e-3380 e x+16120 x^3\right )}{\left (806 e-3844 x^2\right )^2} \, dx\\ &=-\frac {65 e^{4 x} x \left (13 e x-62 x^3\right )}{\left (13 e-62 x^2\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 21, normalized size = 0.78 \begin {gather*} \frac {130 e^{4 x} x^2}{-26 e+124 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*x)*(16120*x^4 + E*(-1690*x - 3380*x^2)))/(169*E^2 - 1612*E*x^2 + 3844*x^4),x]

[Out]

(130*E^(4*x)*x^2)/(-26*E + 124*x^2)

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fricas [A]  time = 0.53, size = 21, normalized size = 0.78 \begin {gather*} \frac {65 \, x^{2} e^{\left (4 \, x\right )}}{62 \, x^{2} - 13 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3380*x^2-1690*x)*exp(1)+16120*x^4)*exp(4*x)/(169*exp(1)^2-1612*x^2*exp(1)+3844*x^4),x, algorithm=
"fricas")

[Out]

65*x^2*e^(4*x)/(62*x^2 - 13*e)

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giac [A]  time = 0.33, size = 32, normalized size = 1.19 \begin {gather*} \frac {65 \, {\left (62 \, x^{2} e^{\left (4 \, x\right )} + 13 \, e^{\left (4 \, x + 1\right )}\right )}}{62 \, {\left (62 \, x^{2} - 13 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3380*x^2-1690*x)*exp(1)+16120*x^4)*exp(4*x)/(169*exp(1)^2-1612*x^2*exp(1)+3844*x^4),x, algorithm=
"giac")

[Out]

65/62*(62*x^2*e^(4*x) + 13*e^(4*x + 1))/(62*x^2 - 13*e)

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maple [A]  time = 0.13, size = 22, normalized size = 0.81

method	result	size
gosper	\(-\frac {65 \,{\mathrm e}^{4 x} x^{2}}{-62 x^{2}+13 \,{\mathrm e}}\)	\(22\)
norman	\(-\frac {65 \,{\mathrm e}^{4 x} x^{2}}{-62 x^{2}+13 \,{\mathrm e}}\)	\(22\)
derivativedivides	\(\frac {65 \,{\mathrm e}^{4 x}}{62}+\frac {6760 \,{\mathrm e}^{4 x} {\mathrm e} x}{31 \left (-496 x^{2}+104 \,{\mathrm e}\right )}-\frac {65 \,{\mathrm e} \sqrt {806}\, \left (2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+93 \,{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-93 \,{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{119164}-3380 \,{\mathrm e} \left (\frac {2 \,{\mathrm e}^{4 x} x}{31 \left (-496 x^{2}+104 \,{\mathrm e}\right )}-\frac {\sqrt {806}\, \left (2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+31 \,{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-31 \,{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{6196528}\right )-6760 \,{\mathrm e} \left (\frac {{\mathrm e}^{4 x}}{-30752 x^{2}+6448 \,{\mathrm e}}-\frac {\sqrt {806}\, \left ({\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{199888}\right )\)	\(357\)
default	\(\frac {65 \,{\mathrm e}^{4 x}}{62}+\frac {6760 \,{\mathrm e}^{4 x} {\mathrm e} x}{31 \left (-496 x^{2}+104 \,{\mathrm e}\right )}-\frac {65 \,{\mathrm e} \sqrt {806}\, \left (2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+93 \,{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-93 \,{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{119164}-3380 \,{\mathrm e} \left (\frac {2 \,{\mathrm e}^{4 x} x}{31 \left (-496 x^{2}+104 \,{\mathrm e}\right )}-\frac {\sqrt {806}\, \left (2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+31 \,{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-31 \,{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{6196528}\right )-6760 \,{\mathrm e} \left (\frac {{\mathrm e}^{4 x}}{-30752 x^{2}+6448 \,{\mathrm e}}-\frac {\sqrt {806}\, \left ({\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \expIntegralEi \left (1, -4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{199888}\right )\)	\(357\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3380*x^2-1690*x)*exp(1)+16120*x^4)*exp(4*x)/(169*exp(1)^2-1612*x^2*exp(1)+3844*x^4),x,method=_RETURNVER
BOSE)

[Out]

-65*exp(4*x)*x^2/(-62*x^2+13*exp(1))

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maxima [A]  time = 0.42, size = 21, normalized size = 0.78 \begin {gather*} \frac {65 \, x^{2} e^{\left (4 \, x\right )}}{62 \, x^{2} - 13 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3380*x^2-1690*x)*exp(1)+16120*x^4)*exp(4*x)/(169*exp(1)^2-1612*x^2*exp(1)+3844*x^4),x, algorithm=
"maxima")

[Out]

65*x^2*e^(4*x)/(62*x^2 - 13*e)

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mupad [B]  time = 6.44, size = 21, normalized size = 0.78 \begin {gather*} -\frac {65\,x^2\,{\mathrm {e}}^{4\,x}}{62\,\left (\frac {13\,\mathrm {e}}{62}-x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4*x)*(exp(1)*(1690*x + 3380*x^2) - 16120*x^4))/(169*exp(2) - 1612*x^2*exp(1) + 3844*x^4),x)

[Out]

-(65*x^2*exp(4*x))/(62*((13*exp(1))/62 - x^2))

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sympy [A]  time = 0.13, size = 19, normalized size = 0.70 \begin {gather*} \frac {65 x^{2} e^{4 x}}{62 x^{2} - 13 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3380*x**2-1690*x)*exp(1)+16120*x**4)*exp(4*x)/(169*exp(1)**2-1612*x**2*exp(1)+3844*x**4),x)

[Out]

65*x**2*exp(4*x)/(62*x**2 - 13*E)

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