3.94.37 \(\int \frac {-64+16 e^5 x^2+e^{-1-x+x^2} (-x^5+2 x^6)}{x^5} \, dx\)

Optimal. Leaf size=22 \[ 1+e^{-1+(-1+x) x}+\left (e^5-\frac {4}{x^2}\right )^2 \]

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14, 2236} \begin {gather*} \frac {16}{x^4}+e^{x^2-x-1}-\frac {8 e^5}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-64 + 16*E^5*x^2 + E^(-1 - x + x^2)*(-x^5 + 2*x^6))/x^5,x]

[Out]

E^(-1 - x + x^2) + 16/x^4 - (8*E^5)/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 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{-1-x+x^2} (-1+2 x)+\frac {16 \left (-4+e^5 x^2\right )}{x^5}\right ) \, dx\\ &=16 \int \frac {-4+e^5 x^2}{x^5} \, dx+\int e^{-1-x+x^2} (-1+2 x) \, dx\\ &=e^{-1-x+x^2}+16 \int \left (-\frac {4}{x^5}+\frac {e^5}{x^3}\right ) \, dx\\ &=e^{-1-x+x^2}+\frac {16}{x^4}-\frac {8 e^5}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 24, normalized size = 1.09 \begin {gather*} e^{-1-x+x^2}+\frac {16}{x^4}-\frac {8 e^5}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-64 + 16*E^5*x^2 + E^(-1 - x + x^2)*(-x^5 + 2*x^6))/x^5,x]

[Out]

E^(-1 - x + x^2) + 16/x^4 - (8*E^5)/x^2

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fricas [A]  time = 0.67, size = 26, normalized size = 1.18 \begin {gather*} \frac {x^{4} e^{\left (x^{2} - x - 1\right )} - 8 \, x^{2} e^{5} + 16}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-x^5)*exp(x^2-x-1)+16*x^2*exp(5)-64)/x^5,x, algorithm="fricas")

[Out]

(x^4*e^(x^2 - x - 1) - 8*x^2*e^5 + 16)/x^4

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giac [A]  time = 0.16, size = 30, normalized size = 1.36 \begin {gather*} \frac {{\left (x^{4} e^{\left (x^{2} - x\right )} - 8 \, x^{2} e^{6} + 16 \, e\right )} e^{\left (-1\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-x^5)*exp(x^2-x-1)+16*x^2*exp(5)-64)/x^5,x, algorithm="giac")

[Out]

(x^4*e^(x^2 - x) - 8*x^2*e^6 + 16*e)*e^(-1)/x^4

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maple [A]  time = 0.04, size = 23, normalized size = 1.05

method	result	size
default	\(\frac {16}{x^{4}}-\frac {8 \,{\mathrm e}^{5}}{x^{2}}+{\mathrm e}^{x^{2}-x -1}\)	\(23\)
risch	\(\frac {-8 x^{2} {\mathrm e}^{5}+16}{x^{4}}+{\mathrm e}^{x^{2}-x -1}\)	\(24\)
norman	\(\frac {16+x^{4} {\mathrm e}^{x^{2}-x -1}-8 x^{2} {\mathrm e}^{5}}{x^{4}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^6-x^5)*exp(x^2-x-1)+16*x^2*exp(5)-64)/x^5,x,method=_RETURNVERBOSE)

[Out]

16/x^4-8*exp(5)/x^2+exp(x^2-x-1)

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maxima [C]  time = 0.40, size = 79, normalized size = 3.59 \begin {gather*} \frac {1}{2} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x - \frac {1}{2} i\right ) e^{\left (-\frac {5}{4}\right )} + \frac {1}{2} \, {\left (\frac {\sqrt {\pi } {\left (2 \, x - 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (2 \, x - 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x - 1\right )}^{2}}} + 2 \, e^{\left (\frac {1}{4} \, {\left (2 \, x - 1\right )}^{2}\right )}\right )} e^{\left (-\frac {5}{4}\right )} - \frac {8 \, e^{5}}{x^{2}} + \frac {16}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-x^5)*exp(x^2-x-1)+16*x^2*exp(5)-64)/x^5,x, algorithm="maxima")

[Out]

1/2*I*sqrt(pi)*erf(I*x - 1/2*I)*e^(-5/4) + 1/2*(sqrt(pi)*(2*x - 1)*(erf(1/2*sqrt(-(2*x - 1)^2)) - 1)/sqrt(-(2*
x - 1)^2) + 2*e^(1/4*(2*x - 1)^2))*e^(-5/4) - 8*e^5/x^2 + 16/x^4

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mupad [B]  time = 8.38, size = 24, normalized size = 1.09 \begin {gather*} {\mathrm {e}}^{x^2-x-1}-\frac {8\,x^2\,{\mathrm {e}}^5-16}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2 - x - 1)*(x^5 - 2*x^6) - 16*x^2*exp(5) + 64)/x^5,x)

[Out]

exp(x^2 - x - 1) - (8*x^2*exp(5) - 16)/x^4

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sympy [A]  time = 0.16, size = 20, normalized size = 0.91 \begin {gather*} e^{x^{2} - x - 1} + \frac {- 8 x^{2} e^{5} + 16}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**6-x**5)*exp(x**2-x-1)+16*x**2*exp(5)-64)/x**5,x)

[Out]

exp(x**2 - x - 1) + (-8*x**2*exp(5) + 16)/x**4

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