3.6.36 \(\int \frac {e^x x^5+e^{-6 x+56 x^2} (-4-6 x+112 x^2)+e^{-6 x+29 x^2} (6 x+12 x^2-116 x^3)+e^{-6 x+2 x^2} (-2 x^2-6 x^3+4 x^4)}{x^5} \, dx\)

Optimal. Leaf size=29 \[ e^x+\frac {e^{2 (-3+x) x} \left (-e^{27 x^2}+x\right )^2}{x^4} \]

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Rubi [B]  time = 0.24, antiderivative size = 96, normalized size of antiderivative = 3.31, number of steps used = 6, number of rules used = 3, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {14, 2194, 2288} \begin {gather*} \frac {e^{-2 (3-28 x) x} \left (3 x-56 x^2\right )}{(3-56 x) x^5}-\frac {2 e^{29 x^2-6 x} \left (3 x-29 x^2\right )}{(3-29 x) x^4}+\frac {e^{-2 (3-x) x} \left (3 x-2 x^2\right )}{(3-2 x) x^3}+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*x^5 + E^(-6*x + 56*x^2)*(-4 - 6*x + 112*x^2) + E^(-6*x + 29*x^2)*(6*x + 12*x^2 - 116*x^3) + E^(-6*x +
2*x^2)*(-2*x^2 - 6*x^3 + 4*x^4))/x^5,x]

[Out]

E^x + (3*x - 56*x^2)/(E^(2*(3 - 28*x)*x)*(3 - 56*x)*x^5) - (2*E^(-6*x + 29*x^2)*(3*x - 29*x^2))/((3 - 29*x)*x^
4) + (3*x - 2*x^2)/(E^(2*(3 - x)*x)*(3 - 2*x)*x^3)

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 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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x+\frac {2 e^{-6 x+29 x^2} \left (3+6 x-58 x^2\right )}{x^4}+\frac {2 e^{2 (-3+x) x} \left (-1-3 x+2 x^2\right )}{x^3}+\frac {2 e^{2 x (-3+28 x)} \left (-2-3 x+56 x^2\right )}{x^5}\right ) \, dx\\ &=2 \int \frac {e^{-6 x+29 x^2} \left (3+6 x-58 x^2\right )}{x^4} \, dx+2 \int \frac {e^{2 (-3+x) x} \left (-1-3 x+2 x^2\right )}{x^3} \, dx+2 \int \frac {e^{2 x (-3+28 x)} \left (-2-3 x+56 x^2\right )}{x^5} \, dx+\int e^x \, dx\\ &=e^x+\frac {e^{-2 (3-28 x) x} \left (3 x-56 x^2\right )}{(3-56 x) x^5}-\frac {2 e^{-6 x+29 x^2} \left (3 x-29 x^2\right )}{(3-29 x) x^4}+\frac {e^{-2 (3-x) x} \left (3 x-2 x^2\right )}{(3-2 x) x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 47, normalized size = 1.62 \begin {gather*} \frac {e^{-6 x} \left (e^{56 x^2}-2 e^{29 x^2} x+e^{2 x^2} x^2+e^{7 x} x^4\right )}{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x^5 + E^(-6*x + 56*x^2)*(-4 - 6*x + 112*x^2) + E^(-6*x + 29*x^2)*(6*x + 12*x^2 - 116*x^3) + E^(
-6*x + 2*x^2)*(-2*x^2 - 6*x^3 + 4*x^4))/x^5,x]

[Out]

(E^(56*x^2) - 2*E^(29*x^2)*x + E^(2*x^2)*x^2 + E^(7*x)*x^4)/(E^(6*x)*x^4)

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fricas [A]  time = 0.65, size = 48, normalized size = 1.66 \begin {gather*} \frac {x^{4} e^{x} + x^{2} e^{\left (2 \, x^{2} - 6 \, x\right )} - 2 \, x e^{\left (29 \, x^{2} - 6 \, x\right )} + e^{\left (56 \, x^{2} - 6 \, x\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((112*x^2-6*x-4)*exp(x^2-3*x)^2*exp(27*x^2)^2+(-116*x^3+12*x^2+6*x)*exp(x^2-3*x)^2*exp(27*x^2)+(4*x^
4-6*x^3-2*x^2)*exp(x^2-3*x)^2+x^5*exp(x))/x^5,x, algorithm="fricas")

[Out]

(x^4*e^x + x^2*e^(2*x^2 - 6*x) - 2*x*e^(29*x^2 - 6*x) + e^(56*x^2 - 6*x))/x^4

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giac [A]  time = 0.30, size = 48, normalized size = 1.66 \begin {gather*} \frac {x^{4} e^{x} + x^{2} e^{\left (2 \, x^{2} - 6 \, x\right )} - 2 \, x e^{\left (29 \, x^{2} - 6 \, x\right )} + e^{\left (56 \, x^{2} - 6 \, x\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((112*x^2-6*x-4)*exp(x^2-3*x)^2*exp(27*x^2)^2+(-116*x^3+12*x^2+6*x)*exp(x^2-3*x)^2*exp(27*x^2)+(4*x^
4-6*x^3-2*x^2)*exp(x^2-3*x)^2+x^5*exp(x))/x^5,x, algorithm="giac")

[Out]

(x^4*e^x + x^2*e^(2*x^2 - 6*x) - 2*x*e^(29*x^2 - 6*x) + e^(56*x^2 - 6*x))/x^4

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maple [A]  time = 0.12, size = 41, normalized size = 1.41

method	result	size
risch	\({\mathrm e}^{x}+\frac {{\mathrm e}^{2 x \left (28 x -3\right )}}{x^{4}}-\frac {2 \,{\mathrm e}^{x \left (29 x -6\right )}}{x^{3}}+\frac {{\mathrm e}^{2 x \left (x -3\right )}}{x^{2}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((112*x^2-6*x-4)*exp(x^2-3*x)^2*exp(27*x^2)^2+(-116*x^3+12*x^2+6*x)*exp(x^2-3*x)^2*exp(27*x^2)+(4*x^4-6*x^
3-2*x^2)*exp(x^2-3*x)^2+x^5*exp(x))/x^5,x,method=_RETURNVERBOSE)

[Out]

exp(x)+1/x^4*exp(2*x*(28*x-3))-2/x^3*exp(x*(29*x-6))+1/x^2*exp(2*x*(x-3))

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maxima [A]  time = 0.61, size = 37, normalized size = 1.28 \begin {gather*} \frac {{\left (x^{2} e^{\left (2 \, x^{2}\right )} - 2 \, x e^{\left (29 \, x^{2}\right )} + e^{\left (56 \, x^{2}\right )}\right )} e^{\left (-6 \, x\right )}}{x^{4}} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((112*x^2-6*x-4)*exp(x^2-3*x)^2*exp(27*x^2)^2+(-116*x^3+12*x^2+6*x)*exp(x^2-3*x)^2*exp(27*x^2)+(4*x^
4-6*x^3-2*x^2)*exp(x^2-3*x)^2+x^5*exp(x))/x^5,x, algorithm="maxima")

[Out]

(x^2*e^(2*x^2) - 2*x*e^(29*x^2) + e^(56*x^2))*e^(-6*x)/x^4 + e^x

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mupad [B]  time = 0.67, size = 46, normalized size = 1.59 \begin {gather*} {\mathrm {e}}^x+\frac {{\mathrm {e}}^{2\,x^2-6\,x}}{x^2}-\frac {2\,{\mathrm {e}}^{29\,x^2-6\,x}}{x^3}+\frac {{\mathrm {e}}^{56\,x^2-6\,x}}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*exp(x) - exp(2*x^2 - 6*x)*(2*x^2 + 6*x^3 - 4*x^4) - exp(54*x^2)*exp(2*x^2 - 6*x)*(6*x - 112*x^2 + 4)
+ exp(27*x^2)*exp(2*x^2 - 6*x)*(6*x + 12*x^2 - 116*x^3))/x^5,x)

[Out]

exp(x) + exp(2*x^2 - 6*x)/x^2 - (2*exp(29*x^2 - 6*x))/x^3 + exp(56*x^2 - 6*x)/x^4

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sympy [B]  time = 1.14, size = 68, normalized size = 2.34 \begin {gather*} e^{x} + \frac {\left (x^{7} e^{12 x} \left (e^{27 x^{2}}\right )^{\frac {2}{27}} - 2 x^{6} e^{12 x} \left (e^{27 x^{2}}\right )^{\frac {29}{27}} + x^{5} e^{12 x} \left (e^{27 x^{2}}\right )^{\frac {56}{27}}\right ) e^{- 18 x}}{x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((112*x**2-6*x-4)*exp(x**2-3*x)**2*exp(27*x**2)**2+(-116*x**3+12*x**2+6*x)*exp(x**2-3*x)**2*exp(27*x
**2)+(4*x**4-6*x**3-2*x**2)*exp(x**2-3*x)**2+x**5*exp(x))/x**5,x)

[Out]

exp(x) + (x**7*exp(12*x)*exp(27*x**2)**(2/27) - 2*x**6*exp(12*x)*exp(27*x**2)**(29/27) + x**5*exp(12*x)*exp(27
*x**2)**(56/27))*exp(-18*x)/x**9

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