3.17.26 \(\int \frac {1}{45} (1020+90 e^5+46 e^{\frac {2}{5} (5+23 x)}+90 x+e^{\frac {1}{5} (5+23 x)} (1594+138 e^5+138 x)) \, dx\)

Optimal. Leaf size=23 \[ \left (11+e^5+\frac {1}{3} \left (1+e^{1+\frac {23 x}{5}}\right )+x\right )^2 \]

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Rubi [B]  time = 0.05, antiderivative size = 71, normalized size of antiderivative = 3.09, number of steps used = 5, number of rules used = 3, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {12, 2194, 2176} \begin {gather*} x^2+\frac {2}{3} \left (34+3 e^5\right ) x-\frac {10}{69} e^{\frac {1}{5} (23 x+5)}+\frac {1}{9} e^{\frac {2}{5} (23 x+5)}+\frac {2}{207} e^{\frac {1}{5} (23 x+5)} \left (69 x+69 e^5+797\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1020 + 90*E^5 + 46*E^((2*(5 + 23*x))/5) + 90*x + E^((5 + 23*x)/5)*(1594 + 138*E^5 + 138*x))/45,x]

[Out]

(-10*E^((5 + 23*x)/5))/69 + E^((2*(5 + 23*x))/5)/9 + (2*(34 + 3*E^5)*x)/3 + x^2 + (2*E^((5 + 23*x)/5)*(797 + 6
9*E^5 + 69*x))/207

Rule 12

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !$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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{45} \int \left (1020+90 e^5+46 e^{\frac {2}{5} (5+23 x)}+90 x+e^{\frac {1}{5} (5+23 x)} \left (1594+138 e^5+138 x\right )\right ) \, dx\\ &=\frac {2}{3} \left (34+3 e^5\right ) x+x^2+\frac {1}{45} \int e^{\frac {1}{5} (5+23 x)} \left (1594+138 e^5+138 x\right ) \, dx+\frac {46}{45} \int e^{\frac {2}{5} (5+23 x)} \, dx\\ &=\frac {1}{9} e^{\frac {2}{5} (5+23 x)}+\frac {2}{3} \left (34+3 e^5\right ) x+x^2+\frac {2}{207} e^{\frac {1}{5} (5+23 x)} \left (797+69 e^5+69 x\right )-\frac {2}{3} \int e^{\frac {1}{5} (5+23 x)} \, dx\\ &=-\frac {10}{69} e^{\frac {1}{5} (5+23 x)}+\frac {1}{9} e^{\frac {2}{5} (5+23 x)}+\frac {2}{3} \left (34+3 e^5\right ) x+x^2+\frac {2}{207} e^{\frac {1}{5} (5+23 x)} \left (797+69 e^5+69 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{9} \left (34+3 e^5+e^{1+\frac {23 x}{5}}+3 x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1020 + 90*E^5 + 46*E^((2*(5 + 23*x))/5) + 90*x + E^((5 + 23*x)/5)*(1594 + 138*E^5 + 138*x))/45,x]

[Out]

(34 + 3*E^5 + E^(1 + (23*x)/5) + 3*x)^2/9

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fricas [A]  time = 0.78, size = 37, normalized size = 1.61 \begin {gather*} x^{2} + 2 \, x e^{5} + \frac {2}{9} \, {\left (3 \, x + 3 \, e^{5} + 34\right )} e^{\left (\frac {23}{5} \, x + 1\right )} + \frac {68}{3} \, x + \frac {1}{9} \, e^{\left (\frac {46}{5} \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(46/45*exp(23/5*x+1)^2+1/45*(138*exp(5)+138*x+1594)*exp(23/5*x+1)+2*exp(5)+2*x+68/3,x, algorithm="fri
cas")

[Out]

x^2 + 2*x*e^5 + 2/9*(3*x + 3*e^5 + 34)*e^(23/5*x + 1) + 68/3*x + 1/9*e^(46/5*x + 2)

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giac [B]  time = 0.23, size = 41, normalized size = 1.78 \begin {gather*} x^{2} + 2 \, x e^{5} + \frac {2}{9} \, {\left (3 \, x + 34\right )} e^{\left (\frac {23}{5} \, x + 1\right )} + \frac {68}{3} \, x + \frac {1}{9} \, e^{\left (\frac {46}{5} \, x + 2\right )} + \frac {2}{3} \, e^{\left (\frac {23}{5} \, x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(46/45*exp(23/5*x+1)^2+1/45*(138*exp(5)+138*x+1594)*exp(23/5*x+1)+2*exp(5)+2*x+68/3,x, algorithm="gia
c")

[Out]

x^2 + 2*x*e^5 + 2/9*(3*x + 34)*e^(23/5*x + 1) + 68/3*x + 1/9*e^(46/5*x + 2) + 2/3*e^(23/5*x + 6)

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maple [B]  time = 0.05, size = 38, normalized size = 1.65

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {46 x}{5}+2}}{9}+\frac {\left (30 \,{\mathrm e}^{5}+340+30 x \right ) {\mathrm e}^{\frac {23 x}{5}+1}}{45}+2 x \,{\mathrm e}^{5}+x^{2}+\frac {68 x}{3}\)	\(38\)
norman	\(x^{2}+\left (\frac {68}{9}+\frac {2 \,{\mathrm e}^{5}}{3}\right ) {\mathrm e}^{\frac {23 x}{5}+1}+\left (2 \,{\mathrm e}^{5}+\frac {68}{3}\right ) x +\frac {{\mathrm e}^{\frac {46 x}{5}+2}}{9}+\frac {2 \,{\mathrm e}^{\frac {23 x}{5}+1} x}{3}\)	\(45\)
default	\(\frac {68 x}{3}+x^{2}+\frac {{\mathrm e}^{\frac {46 x}{5}+2}}{9}+\frac {10 \,{\mathrm e}^{\frac {23 x}{5}+1} \left (\frac {23 x}{5}+1\right )}{69}+\frac {1534 \,{\mathrm e}^{\frac {23 x}{5}+1}}{207}+\frac {2 \,{\mathrm e}^{\frac {23 x}{5}+1} {\mathrm e}^{5}}{3}+2 x \,{\mathrm e}^{5}\)	\(54\)
derivativedivides	\(\frac {1534 x}{69}+\frac {7670}{1587}+\frac {25 \left (\frac {23 x}{5}+1\right )^{2}}{529}+\frac {{\mathrm e}^{\frac {46 x}{5}+2}}{9}+\frac {10 \,{\mathrm e}^{\frac {23 x}{5}+1} \left (\frac {23 x}{5}+1\right )}{69}+\frac {1534 \,{\mathrm e}^{\frac {23 x}{5}+1}}{207}+\frac {2 \,{\mathrm e}^{\frac {23 x}{5}+1} {\mathrm e}^{5}}{3}+\frac {10 \,{\mathrm e}^{5} \left (\frac {23 x}{5}+1\right )}{23}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(46/45*exp(23/5*x+1)^2+1/45*(138*exp(5)+138*x+1594)*exp(23/5*x+1)+2*exp(5)+2*x+68/3,x,method=_RETURNVERBOSE
)

[Out]

1/9*exp(46/5*x+2)+1/45*(30*exp(5)+340+30*x)*exp(23/5*x+1)+2*x*exp(5)+x^2+68/3*x

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maxima [B]  time = 0.38, size = 40, normalized size = 1.74 \begin {gather*} x^{2} + 2 \, x e^{5} + \frac {2}{9} \, {\left (3 \, x e + 3 \, e^{6} + 34 \, e\right )} e^{\left (\frac {23}{5} \, x\right )} + \frac {68}{3} \, x + \frac {1}{9} \, e^{\left (\frac {46}{5} \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(46/45*exp(23/5*x+1)^2+1/45*(138*exp(5)+138*x+1594)*exp(23/5*x+1)+2*exp(5)+2*x+68/3,x, algorithm="max
ima")

[Out]

x^2 + 2*x*e^5 + 2/9*(3*x*e + 3*e^6 + 34*e)*e^(23/5*x) + 68/3*x + 1/9*e^(46/5*x + 2)

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mupad [B]  time = 0.08, size = 27, normalized size = 1.17 \begin {gather*} \frac {\left (3\,x+{\mathrm {e}}^{\frac {23\,x}{5}+1}\right )\,\left (3\,x+6\,{\mathrm {e}}^5+{\mathrm {e}}^{\frac {23\,x}{5}+1}+68\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + 2*exp(5) + (46*exp((46*x)/5 + 2))/45 + (exp((23*x)/5 + 1)*(138*x + 138*exp(5) + 1594))/45 + 68/3,x)

[Out]

((3*x + exp((23*x)/5 + 1))*(3*x + 6*exp(5) + exp((23*x)/5 + 1) + 68))/9

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sympy [B]  time = 0.14, size = 42, normalized size = 1.83 \begin {gather*} x^{2} + x \left (\frac {68}{3} + 2 e^{5}\right ) + \frac {\left (54 x + 612 + 54 e^{5}\right ) e^{\frac {23 x}{5} + 1}}{81} + \frac {e^{\frac {46 x}{5} + 2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(46/45*exp(23/5*x+1)**2+1/45*(138*exp(5)+138*x+1594)*exp(23/5*x+1)+2*exp(5)+2*x+68/3,x)

[Out]

x**2 + x*(68/3 + 2*exp(5)) + (54*x + 612 + 54*exp(5))*exp(23*x/5 + 1)/81 + exp(46*x/5 + 2)/9

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