3.43.68 \(\int \frac {5 x^2-40 x^9+e^{-23-x} (1+x)}{5 x^2} \, dx\)

Optimal. Leaf size=22 \[ 3-\frac {e^{-23-x}}{5 x}+x-x^8 \]

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Rubi [A]  time = 0.05, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 14, 2197} \begin {gather*} -x^8+x-\frac {e^{-x-23}}{5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5*x^2 - 40*x^9 + E^(-23 - x)*(1 + x))/(5*x^2),x]

[Out]

-1/5*E^(-23 - x)/x + x - x^8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {5 x^2-40 x^9+e^{-23-x} (1+x)}{x^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {e^{-23-x} (1+x)}{x^2}-5 \left (-1+8 x^7\right )\right ) \, dx\\ &=\frac {1}{5} \int \frac {e^{-23-x} (1+x)}{x^2} \, dx-\int \left (-1+8 x^7\right ) \, dx\\ &=-\frac {e^{-23-x}}{5 x}+x-x^8\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 0.95 \begin {gather*} -\frac {e^{-23-x}}{5 x}+x-x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*x^2 - 40*x^9 + E^(-23 - x)*(1 + x))/(5*x^2),x]

[Out]

-1/5*E^(-23 - x)/x + x - x^8

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fricas [A]  time = 0.63, size = 22, normalized size = 1.00 \begin {gather*} -\frac {5 \, x^{9} - 5 \, x^{2} + e^{\left (-x - 23\right )}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((x+1)*exp(-x-23)-40*x^9+5*x^2)/x^2,x, algorithm="fricas")

[Out]

-1/5*(5*x^9 - 5*x^2 + e^(-x - 23))/x

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giac [A]  time = 0.23, size = 26, normalized size = 1.18 \begin {gather*} -\frac {{\left (5 \, x^{9} e^{23} - 5 \, x^{2} e^{23} + e^{\left (-x\right )}\right )} e^{\left (-23\right )}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((x+1)*exp(-x-23)-40*x^9+5*x^2)/x^2,x, algorithm="giac")

[Out]

-1/5*(5*x^9*e^23 - 5*x^2*e^23 + e^(-x))*e^(-23)/x

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maple [A]  time = 0.09, size = 19, normalized size = 0.86

method	result	size
risch	\(-x^{8}+x -\frac {{\mathrm e}^{-x -23}}{5 x}\)	\(19\)
norman	\(\frac {x^{2}-x^{9}-\frac {{\mathrm e}^{-x -23}}{5}}{x}\)	\(22\)
derivativedivides	\(27238603577 x [phone]-360435208 \left (-x -23\right )^{3}-19588870 \left (-x -23\right )^{4}-681352 \left (-x -23\right )^{5}-14812 \left (-x -23\right )^{6}-184 \left (-x -23\right )^{7}-\left (-x -23\right )^{8}-4145004892 \left (-x -23\right )^{2}-\frac {{\mathrm e}^{-x -23}}{5 x}\)	\(80\)
default	\(27238603577 x [phone]-360435208 \left (-x -23\right )^{3}-19588870 \left (-x -23\right )^{4}-681352 \left (-x -23\right )^{5}-14812 \left (-x -23\right )^{6}-184 \left (-x -23\right )^{7}-\left (-x -23\right )^{8}-4145004892 \left (-x -23\right )^{2}-\frac {{\mathrm e}^{-x -23}}{5 x}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((x+1)*exp(-x-23)-40*x^9+5*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x^8+x-1/5*exp(-x-23)/x

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maxima [C]  time = 0.37, size = 22, normalized size = 1.00 \begin {gather*} -x^{8} + \frac {1}{5} \, {\rm Ei}\left (-x\right ) e^{\left (-23\right )} - \frac {1}{5} \, e^{\left (-23\right )} \Gamma \left (-1, x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((x+1)*exp(-x-23)-40*x^9+5*x^2)/x^2,x, algorithm="maxima")

[Out]

-x^8 + 1/5*Ei(-x)*e^(-23) - 1/5*e^(-23)*gamma(-1, x) + x

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mupad [B]  time = 3.13, size = 18, normalized size = 0.82 \begin {gather*} x-\frac {{\mathrm {e}}^{-x-23}}{5\,x}-x^8 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(- x - 23)*(x + 1))/5 + x^2 - 8*x^9)/x^2,x)

[Out]

x - exp(- x - 23)/(5*x) - x^8

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sympy [A]  time = 0.09, size = 14, normalized size = 0.64 \begin {gather*} - x^{8} + x - \frac {e^{- x - 23}}{5 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((x+1)*exp(-x-23)-40*x**9+5*x**2)/x**2,x)

[Out]

-x**8 + x - exp(-x - 23)/(5*x)

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