3.73.77 \(\int \frac {e^{8-4 x} (2 e^{-8+4 x} x^4+e^4 (-15-20 x-24 x^4))}{2 x^4} \, dx\)

Optimal. Leaf size=21 \[ e^{4-4 (-2+x)} \left (3+\frac {5}{2 x^3}\right )+x \]

________________________________________________________________________________________

Rubi [A]  time = 0.40, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {12, 6688, 2199, 2194, 2177, 2178} \begin {gather*} \frac {5 e^{12-4 x}}{2 x^3}+x+3 e^{12-4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(8 - 4*x)*(2*E^(-8 + 4*x)*x^4 + E^4*(-15 - 20*x - 24*x^4)))/(2*x^4),x]

[Out]

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

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{x^4} \, dx\\ &=\frac {1}{2} \int \left (2-\frac {e^{12-4 x} \left (15+20 x+24 x^4\right )}{x^4}\right ) \, dx\\ &=x-\frac {1}{2} \int \frac {e^{12-4 x} \left (15+20 x+24 x^4\right )}{x^4} \, dx\\ &=x-\frac {1}{2} \int \left (24 e^{12-4 x}+\frac {15 e^{12-4 x}}{x^4}+\frac {20 e^{12-4 x}}{x^3}\right ) \, dx\\ &=x-\frac {15}{2} \int \frac {e^{12-4 x}}{x^4} \, dx-10 \int \frac {e^{12-4 x}}{x^3} \, dx-12 \int e^{12-4 x} \, dx\\ &=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+\frac {5 e^{12-4 x}}{x^2}+x+10 \int \frac {e^{12-4 x}}{x^3} \, dx+20 \int \frac {e^{12-4 x}}{x^2} \, dx\\ &=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}-\frac {20 e^{12-4 x}}{x}+x-20 \int \frac {e^{12-4 x}}{x^2} \, dx-80 \int \frac {e^{12-4 x}}{x} \, dx\\ &=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x-80 e^{12} \text {Ei}(-4 x)+80 \int \frac {e^{12-4 x}}{x} \, dx\\ &=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 25, normalized size = 1.19 \begin {gather*} 3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(8 - 4*x)*(2*E^(-8 + 4*x)*x^4 + E^4*(-15 - 20*x - 24*x^4)))/(2*x^4),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 33, normalized size = 1.57 \begin {gather*} \frac {{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} + {\left (6 \, x^{3} + 5\right )} e^{4}\right )} e^{\left (-4 \, x + 8\right )}}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^4*exp(4*x-8)+(-24*x^4-20*x-15)*exp(4))/x^4/exp(4*x-8),x, algorithm="fricas")

[Out]

1/2*(2*x^4*e^(4*x - 8) + (6*x^3 + 5)*e^4)*e^(-4*x + 8)/x^3

________________________________________________________________________________________

giac [A]  time = 0.20, size = 30, normalized size = 1.43 \begin {gather*} \frac {2 \, x^{4} + 6 \, x^{3} e^{\left (-4 \, x + 12\right )} + 5 \, e^{\left (-4 \, x + 12\right )}}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^4*exp(4*x-8)+(-24*x^4-20*x-15)*exp(4))/x^4/exp(4*x-8),x, algorithm="giac")

[Out]

1/2*(2*x^4 + 6*x^3*e^(-4*x + 12) + 5*e^(-4*x + 12))/x^3

________________________________________________________________________________________

maple [A]  time = 0.12, size = 21, normalized size = 1.00

method	result	size
risch	\(x +\frac {\left (6 x^{3}+5\right ) {\mathrm e}^{-4 x +12}}{2 x^{3}}\)	\(21\)
norman	\(\frac {\left (x^{4} {\mathrm e}^{4 x -8}+3 x^{3} {\mathrm e}^{4}+\frac {5 \,{\mathrm e}^{4}}{2}\right ) {\mathrm e}^{-4 x +8}}{x^{3}}\)	\(35\)
derivativedivides	\(x -2-14048 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{384 x^{3}}+\frac {{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{6}\right )-6304 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{384 x^{3}}-\frac {11 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{6}\right )-1152 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{192 x^{3}}+\frac {59 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )-96 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{24 x^{3}}-\frac {619 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )-3 \,{\mathrm e}^{4} \left (-{\mathrm e}^{-4 x +8}-\frac {2 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{3 x^{3}}+\frac {6368 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )\)	\(205\)
default	\(x -2-14048 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{384 x^{3}}+\frac {{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{6}\right )-6304 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{384 x^{3}}-\frac {11 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{6}\right )-1152 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{192 x^{3}}+\frac {59 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )-96 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{24 x^{3}}-\frac {619 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )-3 \,{\mathrm e}^{4} \left (-{\mathrm e}^{-4 x +8}-\frac {2 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{3 x^{3}}+\frac {6368 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )\)	\(205\)
meijerg	\(-\frac {{\mathrm e}^{-4 x +4 x \,{\mathrm e}^{8}} \left (1-{\mathrm e}^{4 x \left (-{\mathrm e}^{8}+1\right )}\right )}{4 \left (-{\mathrm e}^{8}+1\right )}-3 \,{\mathrm e}^{-4 x +4+4 x \,{\mathrm e}^{8}} \left (1-{\mathrm e}^{-4 x \,{\mathrm e}^{8}}\right )-160 \,{\mathrm e}^{-4 x +28+4 x \,{\mathrm e}^{8}} \left (-\frac {{\mathrm e}^{-16}}{32 x^{2}}+\frac {{\mathrm e}^{-8}}{4 x}+\frac {13}{4}+\frac {\ln \relax (x )}{2}+\ln \relax (2)+\frac {{\mathrm e}^{-16} \left (144 x^{2} {\mathrm e}^{16}-48 x \,{\mathrm e}^{8}+6\right )}{192 x^{2}}-\frac {{\mathrm e}^{-16-4 x \,{\mathrm e}^{8}} \left (-12 x \,{\mathrm e}^{8}+3\right )}{96 x^{2}}-\frac {\ln \left (4 x \,{\mathrm e}^{8}\right )}{2}-\frac {\expIntegralEi \left (1, 4 x \,{\mathrm e}^{8}\right )}{2}\right )-480 \,{\mathrm e}^{4 x \,{\mathrm e}^{8}+36-4 x} \left (-\frac {{\mathrm e}^{-24}}{192 x^{3}}+\frac {{\mathrm e}^{-16}}{32 x^{2}}-\frac {{\mathrm e}^{-8}}{8 x}-\frac {37}{36}-\frac {\ln \relax (x )}{6}-\frac {\ln \relax (2)}{3}+\frac {{\mathrm e}^{-24} \left (-1408 x^{3} {\mathrm e}^{24}+576 x^{2} {\mathrm e}^{16}-144 x \,{\mathrm e}^{8}+24\right )}{4608 x^{3}}-\frac {{\mathrm e}^{-24-4 x \,{\mathrm e}^{8}} \left (64 x^{2} {\mathrm e}^{16}-16 x \,{\mathrm e}^{8}+8\right )}{1536 x^{3}}+\frac {\ln \left (4 x \,{\mathrm e}^{8}\right )}{6}+\frac {\expIntegralEi \left (1, 4 x \,{\mathrm e}^{8}\right )}{6}\right )\)	\(268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(2*x^4*exp(4*x-8)+(-24*x^4-20*x-15)*exp(4))/x^4/exp(4*x-8),x,method=_RETURNVERBOSE)

[Out]

x+1/2/x^3*(6*x^3+5)*exp(-4*x+12)

________________________________________________________________________________________

maxima [C]  time = 0.39, size = 28, normalized size = 1.33 \begin {gather*} 160 \, e^{12} \Gamma \left (-2, 4 \, x\right ) + 480 \, e^{12} \Gamma \left (-3, 4 \, x\right ) + x + 3 \, e^{\left (-4 \, x + 12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^4*exp(4*x-8)+(-24*x^4-20*x-15)*exp(4))/x^4/exp(4*x-8),x, algorithm="maxima")

[Out]

160*e^12*gamma(-2, 4*x) + 480*e^12*gamma(-3, 4*x) + x + 3*e^(-4*x + 12)

________________________________________________________________________________________

mupad [B]  time = 4.24, size = 21, normalized size = 1.00 \begin {gather*} x+3\,{\mathrm {e}}^{12-4\,x}+\frac {5\,{\mathrm {e}}^{12-4\,x}}{2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(8 - 4*x)*((exp(4)*(20*x + 24*x^4 + 15))/2 - x^4*exp(4*x - 8)))/x^4,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 26, normalized size = 1.24 \begin {gather*} x + \frac {\left (6 x^{3} e^{4} + 5 e^{4}\right ) e^{8 - 4 x}}{2 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x**4*exp(4*x-8)+(-24*x**4-20*x-15)*exp(4))/x**4/exp(4*x-8),x)

[Out]

x + (6*x**3*exp(4) + 5*exp(4))*exp(8 - 4*x)/(2*x**3)

________________________________________________________________________________________