3.83.26 \(\int \frac {-3380-260 x-5 x^2-e^{16+\frac {e^{16}}{26+x}} x^2}{676 x^2+52 x^3+x^4} \, dx\)

Optimal. Leaf size=17 \[ e^{\frac {e^{16}}{26+x}}+\frac {5}{x} \]

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Rubi [A]  time = 0.48, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {1594, 27, 6742, 2209} \begin {gather*} e^{\frac {e^{16}}{x+26}}+\frac {5}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3380 - 260*x - 5*x^2 - E^(16 + E^16/(26 + x))*x^2)/(676*x^2 + 52*x^3 + x^4),x]

[Out]

E^(E^16/(26 + x)) + 5/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 1594

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

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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 {-3380-260 x-5 x^2-e^{16+\frac {e^{16}}{26+x}} x^2}{x^2 \left (676+52 x+x^2\right )} \, dx\\ &=\int \frac {-3380-260 x-5 x^2-e^{16+\frac {e^{16}}{26+x}} x^2}{x^2 (26+x)^2} \, dx\\ &=\int \left (-\frac {5}{x^2}-\frac {e^{16+\frac {e^{16}}{26+x}}}{(26+x)^2}\right ) \, dx\\ &=\frac {5}{x}-\int \frac {e^{16+\frac {e^{16}}{26+x}}}{(26+x)^2} \, dx\\ &=e^{\frac {e^{16}}{26+x}}+\frac {5}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 17, normalized size = 1.00 \begin {gather*} e^{\frac {e^{16}}{26+x}}+\frac {5}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3380 - 260*x - 5*x^2 - E^(16 + E^16/(26 + x))*x^2)/(676*x^2 + 52*x^3 + x^4),x]

[Out]

E^(E^16/(26 + x)) + 5/x

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fricas [A]  time = 0.85, size = 27, normalized size = 1.59 \begin {gather*} \frac {{\left (x e^{\left (\frac {16 \, x + e^{16} + 416}{x + 26}\right )} + 5 \, e^{16}\right )} e^{\left (-16\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(16)*exp(exp(16)/(x+26))-5*x^2-260*x-3380)/(x^4+52*x^3+676*x^2),x, algorithm="fricas")

[Out]

(x*e^((16*x + e^16 + 416)/(x + 26)) + 5*e^16)*e^(-16)/x

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giac [B]  time = 0.22, size = 103, normalized size = 6.06 \begin {gather*} \frac {{\left ({\left (e^{16} + 416\right )} e^{\left (-32\right )} - 416 \, e^{\left (-32\right )}\right )} {\left (\frac {676 \, {\left (16 \, x + e^{16} + 416\right )} e^{\left (\frac {16 \, x + e^{16} + 416}{x + 26}\right )}}{x + 26} - 5 \, e^{32} - 26 \, e^{\left (\frac {16 \, x + e^{16} + 416}{x + 26} + 16\right )} - 10816 \, e^{\left (\frac {16 \, x + e^{16} + 416}{x + 26}\right )}\right )}}{26 \, {\left (\frac {26 \, {\left (16 \, x + e^{16} + 416\right )}}{x + 26} - e^{16} - 416\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(16)*exp(exp(16)/(x+26))-5*x^2-260*x-3380)/(x^4+52*x^3+676*x^2),x, algorithm="giac")

[Out]

1/26*((e^16 + 416)*e^(-32) - 416*e^(-32))*(676*(16*x + e^16 + 416)*e^((16*x + e^16 + 416)/(x + 26))/(x + 26) -
5*e^32 - 26*e^((16*x + e^16 + 416)/(x + 26) + 16) - 10816*e^((16*x + e^16 + 416)/(x + 26)))/(26*(16*x + e^16
+ 416)/(x + 26) - e^16 - 416)

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maple [A]  time = 0.26, size = 16, normalized size = 0.94

method	result	size
risch	\({\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}}+\frac {5}{x}\)	\(16\)
norman	\(\frac {130+x^{2} {\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}}+5 x +26 x \,{\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}}}{x \left (x +26\right )}\)	\(40\)
derivativedivides	\(-{\mathrm e}^{16} \left (-\frac {5}{\left (x +26\right ) \left ({\mathrm e}^{16}-\frac {26 \,{\mathrm e}^{16}}{x +26}\right )}-{\mathrm e}^{48} \left (\frac {{\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}} {\mathrm e}^{-32}}{26 \,{\mathrm e}^{16}-\frac {676 \,{\mathrm e}^{16}}{x +26}}-\frac {{\mathrm e}^{-32} {\mathrm e}^{\frac {{\mathrm e}^{16}}{26}} \expIntegralEi \left (1, \frac {{\mathrm e}^{16}}{26}-\frac {{\mathrm e}^{16}}{x +26}\right )}{676}\right )+52 \,{\mathrm e}^{32} \left (\frac {{\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}} {\mathrm e}^{-16}}{676 \,{\mathrm e}^{16}-\frac {17576 \,{\mathrm e}^{16}}{x +26}}-\frac {\left ({\mathrm e}^{16}+26\right ) {\mathrm e}^{-32} {\mathrm e}^{\frac {{\mathrm e}^{16}}{26}} \expIntegralEi \left (1, \frac {{\mathrm e}^{16}}{26}-\frac {{\mathrm e}^{16}}{x +26}\right )}{17576}\right )-676 \,{\mathrm e}^{16} \left (\frac {{\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}} {\mathrm e}^{-32}}{676}+\frac {{\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}}}{17576 \,{\mathrm e}^{16}-\frac {456976 \,{\mathrm e}^{16}}{x +26}}-\frac {{\mathrm e}^{-16} \left ({\mathrm e}^{16}+52\right ) {\mathrm e}^{\frac {{\mathrm e}^{16}}{26}} \expIntegralEi \left (1, \frac {{\mathrm e}^{16}}{26}-\frac {{\mathrm e}^{16}}{x +26}\right )}{456976}\right )\right )\)	\(229\)
default	\(-{\mathrm e}^{16} \left (-\frac {5}{\left (x +26\right ) \left ({\mathrm e}^{16}-\frac {26 \,{\mathrm e}^{16}}{x +26}\right )}-{\mathrm e}^{48} \left (\frac {{\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}} {\mathrm e}^{-32}}{26 \,{\mathrm e}^{16}-\frac {676 \,{\mathrm e}^{16}}{x +26}}-\frac {{\mathrm e}^{-32} {\mathrm e}^{\frac {{\mathrm e}^{16}}{26}} \expIntegralEi \left (1, \frac {{\mathrm e}^{16}}{26}-\frac {{\mathrm e}^{16}}{x +26}\right )}{676}\right )+52 \,{\mathrm e}^{32} \left (\frac {{\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}} {\mathrm e}^{-16}}{676 \,{\mathrm e}^{16}-\frac {17576 \,{\mathrm e}^{16}}{x +26}}-\frac {\left ({\mathrm e}^{16}+26\right ) {\mathrm e}^{-32} {\mathrm e}^{\frac {{\mathrm e}^{16}}{26}} \expIntegralEi \left (1, \frac {{\mathrm e}^{16}}{26}-\frac {{\mathrm e}^{16}}{x +26}\right )}{17576}\right )-676 \,{\mathrm e}^{16} \left (\frac {{\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}} {\mathrm e}^{-32}}{676}+\frac {{\mathrm e}^{\frac {{\mathrm e}^{16}}{x +26}}}{17576 \,{\mathrm e}^{16}-\frac {456976 \,{\mathrm e}^{16}}{x +26}}-\frac {{\mathrm e}^{-16} \left ({\mathrm e}^{16}+52\right ) {\mathrm e}^{\frac {{\mathrm e}^{16}}{26}} \expIntegralEi \left (1, \frac {{\mathrm e}^{16}}{26}-\frac {{\mathrm e}^{16}}{x +26}\right )}{456976}\right )\right )\)	\(229\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2*exp(16)*exp(exp(16)/(x+26))-5*x^2-260*x-3380)/(x^4+52*x^3+676*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(exp(16)/(x+26))+5/x

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maxima [B]  time = 0.37, size = 31, normalized size = 1.82 \begin {gather*} \frac {10 \, {\left (x + 13\right )}}{x^{2} + 26 \, x} - \frac {5}{x + 26} + e^{\left (\frac {e^{16}}{x + 26}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(16)*exp(exp(16)/(x+26))-5*x^2-260*x-3380)/(x^4+52*x^3+676*x^2),x, algorithm="maxima")

[Out]

10*(x + 13)/(x^2 + 26*x) - 5/(x + 26) + e^(e^16/(x + 26))

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mupad [B]  time = 5.08, size = 15, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x+26}}+\frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(260*x + 5*x^2 + x^2*exp(exp(16)/(x + 26))*exp(16) + 3380)/(676*x^2 + 52*x^3 + x^4),x)

[Out]

exp(exp(16)/(x + 26)) + 5/x

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sympy [A]  time = 0.17, size = 10, normalized size = 0.59 \begin {gather*} e^{\frac {e^{16}}{x + 26}} + \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2*exp(16)*exp(exp(16)/(x+26))-5*x**2-260*x-3380)/(x**4+52*x**3+676*x**2),x)

[Out]

exp(exp(16)/(x + 26)) + 5/x

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