3.70.48 \(\int \frac {-324+72 x^2-18 x^3-3 x^5-48 e^{2 x} x^5+2 x^6+e^x (36 x^2-18 x^3+2 x^5-2 x^6)}{x^5} \, dx\)

Optimal. Leaf size=25 \[ 2-25 e^{2 x}+\left (2+e^x-\frac {9}{x^2}-x\right )^2+x \]

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 48, normalized size of antiderivative = 1.92, number of steps used = 15, number of rules used = 6, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {14, 2194, 2199, 2177, 2178, 2176} \begin {gather*} \frac {81}{x^4}+x^2-\frac {18 e^x}{x^2}-\frac {36}{x^2}-2 e^x x-3 x+4 e^x-24 e^{2 x}+\frac {18}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-324 + 72*x^2 - 18*x^3 - 3*x^5 - 48*E^(2*x)*x^5 + 2*x^6 + E^x*(36*x^2 - 18*x^3 + 2*x^5 - 2*x^6))/x^5,x]

[Out]

4*E^x - 24*E^(2*x) + 81/x^4 - 36/x^2 - (18*E^x)/x^2 + 18/x - 3*x - 2*E^x*x + x^2

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 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-48 e^{2 x}-\frac {2 e^x \left (-18+9 x-x^3+x^4\right )}{x^3}+\frac {-324+72 x^2-18 x^3-3 x^5+2 x^6}{x^5}\right ) \, dx\\ &=-\left (2 \int \frac {e^x \left (-18+9 x-x^3+x^4\right )}{x^3} \, dx\right )-48 \int e^{2 x} \, dx+\int \frac {-324+72 x^2-18 x^3-3 x^5+2 x^6}{x^5} \, dx\\ &=-24 e^{2 x}-2 \int \left (-e^x-\frac {18 e^x}{x^3}+\frac {9 e^x}{x^2}+e^x x\right ) \, dx+\int \left (-3-\frac {324}{x^5}+\frac {72}{x^3}-\frac {18}{x^2}+2 x\right ) \, dx\\ &=-24 e^{2 x}+\frac {81}{x^4}-\frac {36}{x^2}+\frac {18}{x}-3 x+x^2+2 \int e^x \, dx-2 \int e^x x \, dx-18 \int \frac {e^x}{x^2} \, dx+36 \int \frac {e^x}{x^3} \, dx\\ &=2 e^x-24 e^{2 x}+\frac {81}{x^4}-\frac {36}{x^2}-\frac {18 e^x}{x^2}+\frac {18}{x}+\frac {18 e^x}{x}-3 x-2 e^x x+x^2+2 \int e^x \, dx+18 \int \frac {e^x}{x^2} \, dx-18 \int \frac {e^x}{x} \, dx\\ &=4 e^x-24 e^{2 x}+\frac {81}{x^4}-\frac {36}{x^2}-\frac {18 e^x}{x^2}+\frac {18}{x}-3 x-2 e^x x+x^2-18 \text {Ei}(x)+18 \int \frac {e^x}{x} \, dx\\ &=4 e^x-24 e^{2 x}+\frac {81}{x^4}-\frac {36}{x^2}-\frac {18 e^x}{x^2}+\frac {18}{x}-3 x-2 e^x x+x^2\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 46, normalized size = 1.84 \begin {gather*} -4 e^x \left (-1+6 e^x\right )+\frac {81}{x^4}-\frac {18 \left (2+e^x\right )}{x^2}+\frac {18}{x}-\left (3+2 e^x\right ) x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-324 + 72*x^2 - 18*x^3 - 3*x^5 - 48*E^(2*x)*x^5 + 2*x^6 + E^x*(36*x^2 - 18*x^3 + 2*x^5 - 2*x^6))/x^
5,x]

[Out]

-4*E^x*(-1 + 6*E^x) + 81/x^4 - (18*(2 + E^x))/x^2 + 18/x - (3 + 2*E^x)*x + x^2

________________________________________________________________________________________

fricas [B]  time = 1.22, size = 51, normalized size = 2.04 \begin {gather*} \frac {x^{6} - 3 \, x^{5} - 24 \, x^{4} e^{\left (2 \, x\right )} + 18 \, x^{3} - 36 \, x^{2} - 2 \, {\left (x^{5} - 2 \, x^{4} + 9 \, x^{2}\right )} e^{x} + 81}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*x^5*exp(x)^2+(-2*x^6+2*x^5-18*x^3+36*x^2)*exp(x)+2*x^6-3*x^5-18*x^3+72*x^2-324)/x^5,x, algorith
m="fricas")

[Out]

(x^6 - 3*x^5 - 24*x^4*e^(2*x) + 18*x^3 - 36*x^2 - 2*(x^5 - 2*x^4 + 9*x^2)*e^x + 81)/x^4

________________________________________________________________________________________

giac [B]  time = 0.20, size = 54, normalized size = 2.16 \begin {gather*} \frac {x^{6} - 2 \, x^{5} e^{x} - 3 \, x^{5} - 24 \, x^{4} e^{\left (2 \, x\right )} + 4 \, x^{4} e^{x} + 18 \, x^{3} - 18 \, x^{2} e^{x} - 36 \, x^{2} + 81}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*x^5*exp(x)^2+(-2*x^6+2*x^5-18*x^3+36*x^2)*exp(x)+2*x^6-3*x^5-18*x^3+72*x^2-324)/x^5,x, algorith
m="giac")

[Out]

(x^6 - 2*x^5*e^x - 3*x^5 - 24*x^4*e^(2*x) + 4*x^4*e^x + 18*x^3 - 18*x^2*e^x - 36*x^2 + 81)/x^4

________________________________________________________________________________________

maple [A]  time = 0.04, size = 45, normalized size = 1.80

method	result	size
default	\(x^{2}-3 x +\frac {81}{x^{4}}-\frac {36}{x^{2}}+\frac {18}{x}-24 \,{\mathrm e}^{2 x}-\frac {18 \,{\mathrm e}^{x}}{x^{2}}-2 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x}\)	\(45\)
risch	\(x^{2}-3 x +\frac {18 x^{3}-36 x^{2}+81}{x^{4}}-24 \,{\mathrm e}^{2 x}-\frac {2 \left (x^{3}-2 x^{2}+9\right ) {\mathrm e}^{x}}{x^{2}}\)	\(47\)
norman	\(\frac {81+x^{6}-36 x^{2}+18 x^{3}-3 x^{5}-2 x^{5} {\mathrm e}^{x}-18 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{x} x^{4}-24 \,{\mathrm e}^{2 x} x^{4}}{x^{4}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-48*x^5*exp(x)^2+(-2*x^6+2*x^5-18*x^3+36*x^2)*exp(x)+2*x^6-3*x^5-18*x^3+72*x^2-324)/x^5,x,method=_RETURNV
ERBOSE)

[Out]

x^2-3*x+81/x^4-36/x^2+18/x-24*exp(x)^2-18*exp(x)/x^2-2*exp(x)*x+4*exp(x)

________________________________________________________________________________________

maxima [C]  time = 0.45, size = 53, normalized size = 2.12 \begin {gather*} x^{2} - 2 \, {\left (x - 1\right )} e^{x} - 3 \, x + \frac {18}{x} - \frac {36}{x^{2}} + \frac {81}{x^{4}} - 24 \, e^{\left (2 \, x\right )} + 2 \, e^{x} - 18 \, \Gamma \left (-1, -x\right ) - 36 \, \Gamma \left (-2, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*x^5*exp(x)^2+(-2*x^6+2*x^5-18*x^3+36*x^2)*exp(x)+2*x^6-3*x^5-18*x^3+72*x^2-324)/x^5,x, algorith
m="maxima")

[Out]

x^2 - 2*(x - 1)*e^x - 3*x + 18/x - 36/x^2 + 81/x^4 - 24*e^(2*x) + 2*e^x - 18*gamma(-1, -x) - 36*gamma(-2, -x)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 45, normalized size = 1.80 \begin {gather*} 4\,{\mathrm {e}}^x-24\,{\mathrm {e}}^{2\,x}-x\,\left (2\,{\mathrm {e}}^x+3\right )+\frac {18\,x^3-x^2\,\left (18\,{\mathrm {e}}^x+36\right )+81}{x^4}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48*x^5*exp(2*x) - exp(x)*(36*x^2 - 18*x^3 + 2*x^5 - 2*x^6) - 72*x^2 + 18*x^3 + 3*x^5 - 2*x^6 + 324)/x^5,
x)

[Out]

4*exp(x) - 24*exp(2*x) - x*(2*exp(x) + 3) + (18*x^3 - x^2*(18*exp(x) + 36) + 81)/x^4 + x^2

________________________________________________________________________________________

sympy [B]  time = 0.15, size = 49, normalized size = 1.96 \begin {gather*} x^{2} - 3 x + \frac {- 24 x^{2} e^{2 x} + \left (- 2 x^{3} + 4 x^{2} - 18\right ) e^{x}}{x^{2}} + \frac {18 x^{3} - 36 x^{2} + 81}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*x**5*exp(x)**2+(-2*x**6+2*x**5-18*x**3+36*x**2)*exp(x)+2*x**6-3*x**5-18*x**3+72*x**2-324)/x**5,
x)

[Out]

x**2 - 3*x + (-24*x**2*exp(2*x) + (-2*x**3 + 4*x**2 - 18)*exp(x))/x**2 + (18*x**3 - 36*x**2 + 81)/x**4

________________________________________________________________________________________