3.21.65 \(\int \frac {x^2+e^{e^{12}+\frac {17+x}{x}} (425+170 x+7 x^2-2 x^3)}{x^2} \, dx\) [2065]

Optimal. Leaf size=22 \[ x-e^{e^{12}+\frac {17+x}{x}} (5+x)^2 \]

[Out]

x-(5+x)^2*exp((x+17)/x)*exp(exp(12))

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Rubi [A]
time = 0.06, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {14, 2326} \begin {gather*} x-e^{\frac {17}{x}+e^{12}+1} (x+5)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2 + E^(E^12 + (17 + x)/x)*(425 + 170*x + 7*x^2 - 2*x^3))/x^2,x]

[Out]

x - E^(1 + E^12 + 17/x)*(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 2326

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 (1-\frac {e^{1+e^{12}+\frac {17}{x}} (5+x) \left (-85-17 x+2 x^2\right )}{x^2}\right ) \, dx\\ &=x-\int \frac {e^{1+e^{12}+\frac {17}{x}} (5+x) \left (-85-17 x+2 x^2\right )}{x^2} \, dx\\ &=x-e^{1+e^{12}+\frac {17}{x}} (5+x)^2\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.04, size = 21, normalized size = 0.95 \begin {gather*} x-e^{1+e^{12}+\frac {17}{x}} (5+x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 + E^(E^12 + (17 + x)/x)*(425 + 170*x + 7*x^2 - 2*x^3))/x^2,x]

[Out]

x - E^(1 + E^12 + 17/x)*(5 + x)^2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(20)=40\).
time = 0.57, size = 42, normalized size = 1.91

method	result	size
risch	\(x +\left (-{\mathrm e}^{{\mathrm e}^{12}} x^{2}-10 \,{\mathrm e}^{{\mathrm e}^{12}} x -25 \,{\mathrm e}^{{\mathrm e}^{12}}\right ) {\mathrm e}^{\frac {x +17}{x}}\)	\(32\)
derivativedivides	\(x -{\mathrm e}^{{\mathrm e}^{12}} \left ({\mathrm e}^{1+\frac {17}{x}} x^{2}+10 \,{\mathrm e}^{1+\frac {17}{x}} x +25 \,{\mathrm e}^{1+\frac {17}{x}}\right )\)	\(42\)
default	\(x -{\mathrm e}^{{\mathrm e}^{12}} \left ({\mathrm e}^{1+\frac {17}{x}} x^{2}+10 \,{\mathrm e}^{1+\frac {17}{x}} x +25 \,{\mathrm e}^{1+\frac {17}{x}}\right )\)	\(42\)
norman	\(\frac {x^{2}-25 \,{\mathrm e}^{\frac {x +17}{x}} {\mathrm e}^{{\mathrm e}^{12}} x -10 \,{\mathrm e}^{\frac {x +17}{x}} {\mathrm e}^{{\mathrm e}^{12}} x^{2}-{\mathrm e}^{\frac {x +17}{x}} {\mathrm e}^{{\mathrm e}^{12}} x^{3}}{x}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+7*x^2+170*x+425)*exp((x+17)/x)*exp(exp(12))+x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x-exp(exp(12))*(exp(1+17/x)*x^2+10*exp(1+17/x)*x+25*exp(1+17/x))

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.50, size = 82, normalized size = 3.73 \begin {gather*} -170 \, {\rm Ei}\left (\frac {17}{x}\right ) e^{\left ({\left (e^{8} - e^{4} + 1\right )} {\left (e^{4} + 1\right )}\right )} - 119 \, e^{\left ({\left (e^{8} - e^{4} + 1\right )} {\left (e^{4} + 1\right )}\right )} \Gamma \left (-1, -\frac {17}{x}\right ) - 578 \, e^{\left ({\left (e^{8} - e^{4} + 1\right )} {\left (e^{4} + 1\right )}\right )} \Gamma \left (-2, -\frac {17}{x}\right ) + x - 25 \, e^{\left (\frac {17}{x} + e^{12} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+7*x^2+170*x+425)*exp((x+17)/x)*exp(exp(12))+x^2)/x^2,x, algorithm="maxima")

[Out]

-170*Ei(17/x)*e^((e^8 - e^4 + 1)*(e^4 + 1)) - 119*e^((e^8 - e^4 + 1)*(e^4 + 1))*gamma(-1, -17/x) - 578*e^((e^8
- e^4 + 1)*(e^4 + 1))*gamma(-2, -17/x) + x - 25*e^(17/x + e^12 + 1)

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Fricas [A]
time = 0.42, size = 24, normalized size = 1.09 \begin {gather*} -{\left (x^{2} + 10 \, x + 25\right )} e^{\left (\frac {x e^{12} + x + 17}{x}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+7*x^2+170*x+425)*exp((x+17)/x)*exp(exp(12))+x^2)/x^2,x, algorithm="fricas")

[Out]

-(x^2 + 10*x + 25)*e^((x*e^12 + x + 17)/x) + x

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Sympy [A]
time = 0.10, size = 32, normalized size = 1.45 \begin {gather*} x + \left (- x^{2} e^{e^{12}} - 10 x e^{e^{12}} - 25 e^{e^{12}}\right ) e^{\frac {x + 17}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+7*x**2+170*x+425)*exp((x+17)/x)*exp(exp(12))+x**2)/x**2,x)

[Out]

x + (-x**2*exp(exp(12)) - 10*x*exp(exp(12)) - 25*exp(exp(12)))*exp((x + 17)/x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
time = 0.40, size = 48, normalized size = 2.18 \begin {gather*} -x^{2} e^{\left (\frac {x e^{12} + x + 17}{x}\right )} - 10 \, x e^{\left (\frac {x e^{12} + x + 17}{x}\right )} + x - 25 \, e^{\left (\frac {x e^{12} + x + 17}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+7*x^2+170*x+425)*exp((x+17)/x)*exp(exp(12))+x^2)/x^2,x, algorithm="giac")

[Out]

-x^2*e^((x*e^12 + x + 17)/x) - 10*x*e^((x*e^12 + x + 17)/x) + x - 25*e^((x*e^12 + x + 17)/x)

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Mupad [B]
time = 1.19, size = 42, normalized size = 1.91 \begin {gather*} x-25\,{\mathrm {e}}^{{\mathrm {e}}^{12}+\frac {17}{x}+1}-10\,x\,{\mathrm {e}}^{{\mathrm {e}}^{12}+\frac {17}{x}+1}-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{12}+\frac {17}{x}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + exp((x + 17)/x)*exp(exp(12))*(170*x + 7*x^2 - 2*x^3 + 425))/x^2,x)

[Out]

x - 25*exp(exp(12) + 17/x + 1) - 10*x*exp(exp(12) + 17/x + 1) - x^2*exp(exp(12) + 17/x + 1)

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