\(\int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 (36 x^2+12 x^3)}{x}} (48+12 x^2-24 x^3+e^5 (36 x^2+24 x^3))}{x^2} \, dx\) [1377]

Optimal result

Integrand size = 69, antiderivative size = 27 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=-1+e^{12 \left (-\frac {4}{x}+x-x^2+e^5 x (3+x)\right )}+x \]

[Out]

-1+exp(12*exp(5)*(3+x)*x-12*x^2-48/x+12*x)+x

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {14, 6838} \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=\exp \left (-12 \left (1-e^5\right ) x^2+12 \left (1+3 e^5\right ) x-\frac {48}{x}\right )+x \]

[In]

Int[(x^2 + E^((-48 + 12*x^2 - 12*x^3 + E^5*(36*x^2 + 12*x^3))/x)*(48 + 12*x^2 - 24*x^3 + E^5*(36*x^2 + 24*x^3)
))/x^2,x]

[Out]

E^(-48/x + 12*(1 + 3*E^5)*x - 12*(1 - E^5)*x^2) + x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {12 \exp \left (-\frac {48}{x}+12 \left (1+3 e^5\right ) x+12 \left (-1+e^5\right ) x^2\right ) \left (4+\left (1+3 e^5\right ) x^2-2 \left (1-e^5\right ) x^3\right )}{x^2}\right ) \, dx \\ & = x+12 \int \frac {\exp \left (-\frac {48}{x}+12 \left (1+3 e^5\right ) x+12 \left (-1+e^5\right ) x^2\right ) \left (4+\left (1+3 e^5\right ) x^2-2 \left (1-e^5\right ) x^3\right )}{x^2} \, dx \\ & = e^{-\frac {48}{x}+12 \left (1+3 e^5\right ) x-12 \left (1-e^5\right ) x^2}+x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=e^{-\frac {48}{x}+12 \left (1+3 e^5\right ) x+12 \left (-1+e^5\right ) x^2}+x \]

[In]

Integrate[(x^2 + E^((-48 + 12*x^2 - 12*x^3 + E^5*(36*x^2 + 12*x^3))/x)*(48 + 12*x^2 - 24*x^3 + E^5*(36*x^2 + 2
4*x^3)))/x^2,x]

[Out]

E^(-48/x + 12*(1 + 3*E^5)*x + 12*(-1 + E^5)*x^2) + x

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19

method	result	size
risch	\(x +{\mathrm e}^{\frac {12 x^{3} {\mathrm e}^{5}+36 x^{2} {\mathrm e}^{5}-12 x^{3}+12 x^{2}-48}{x}}\)	\(32\)
parallelrisch	\(x +{\mathrm e}^{\frac {\left (12 x^{3}+36 x^{2}\right ) {\mathrm e}^{5}-12 x^{3}+12 x^{2}-48}{x}}\)	\(34\)
parts	\(x +{\mathrm e}^{\frac {\left (12 x^{3}+36 x^{2}\right ) {\mathrm e}^{5}-12 x^{3}+12 x^{2}-48}{x}}\)	\(34\)
norman	\(\frac {x^{2}+x \,{\mathrm e}^{\frac {\left (12 x^{3}+36 x^{2}\right ) {\mathrm e}^{5}-12 x^{3}+12 x^{2}-48}{x}}}{x}\)	\(42\)

[In]

int((((24*x^3+36*x^2)*exp(5)-24*x^3+12*x^2+48)*exp(((12*x^3+36*x^2)*exp(5)-12*x^3+12*x^2-48)/x)+x^2)/x^2,x,met
hod=_RETURNVERBOSE)

[Out]

x+exp(12*(x^3*exp(5)+3*x^2*exp(5)-x^3+x^2-4)/x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x + e^{\left (-\frac {12 \, {\left (x^{3} - x^{2} - {\left (x^{3} + 3 \, x^{2}\right )} e^{5} + 4\right )}}{x}\right )} \]

[In]

integrate((((24*x^3+36*x^2)*exp(5)-24*x^3+12*x^2+48)*exp(((12*x^3+36*x^2)*exp(5)-12*x^3+12*x^2-48)/x)+x^2)/x^2
,x, algorithm="fricas")

[Out]

x + e^(-12*(x^3 - x^2 - (x^3 + 3*x^2)*e^5 + 4)/x)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x + e^{\frac {- 12 x^{3} + 12 x^{2} + \left (12 x^{3} + 36 x^{2}\right ) e^{5} - 48}{x}} \]

[In]

integrate((((24*x**3+36*x**2)*exp(5)-24*x**3+12*x**2+48)*exp(((12*x**3+36*x**2)*exp(5)-12*x**3+12*x**2-48)/x)+
x**2)/x**2,x)

[Out]

x + exp((-12*x**3 + 12*x**2 + (12*x**3 + 36*x**2)*exp(5) - 48)/x)

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x + e^{\left (12 \, x^{2} e^{5} - 12 \, x^{2} + 36 \, x e^{5} + 12 \, x - \frac {48}{x}\right )} \]

[In]

integrate((((24*x^3+36*x^2)*exp(5)-24*x^3+12*x^2+48)*exp(((12*x^3+36*x^2)*exp(5)-12*x^3+12*x^2-48)/x)+x^2)/x^2
,x, algorithm="maxima")

[Out]

x + e^(12*x^2*e^5 - 12*x^2 + 36*x*e^5 + 12*x - 48/x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x + e^{\left (\frac {12 \, {\left (x^{3} e^{5} - x^{3} + 3 \, x^{2} e^{5} + x^{2} - 4\right )}}{x}\right )} \]

[In]

integrate((((24*x^3+36*x^2)*exp(5)-24*x^3+12*x^2+48)*exp(((12*x^3+36*x^2)*exp(5)-12*x^3+12*x^2-48)/x)+x^2)/x^2
,x, algorithm="giac")

[Out]

x + e^(12*(x^3*e^5 - x^3 + 3*x^2*e^5 + x^2 - 4)/x)

Mupad [B] (verification not implemented)

Time = 8.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {x^2+e^{\frac {-48+12 x^2-12 x^3+e^5 \left (36 x^2+12 x^3\right )}{x}} \left (48+12 x^2-24 x^3+e^5 \left (36 x^2+24 x^3\right )\right )}{x^2} \, dx=x+{\mathrm {e}}^{12\,x^2\,{\mathrm {e}}^5}\,{\mathrm {e}}^{12\,x}\,{\mathrm {e}}^{-12\,x^2}\,{\mathrm {e}}^{-\frac {48}{x}}\,{\mathrm {e}}^{36\,x\,{\mathrm {e}}^5} \]

[In]

int((exp((exp(5)*(36*x^2 + 12*x^3) + 12*x^2 - 12*x^3 - 48)/x)*(exp(5)*(36*x^2 + 24*x^3) + 12*x^2 - 24*x^3 + 48
) + x^2)/x^2,x)

[Out]

x + exp(12*x^2*exp(5))*exp(12*x)*exp(-12*x^2)*exp(-48/x)*exp(36*x*exp(5))