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

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

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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} $$\begin{array}{c}{x}^2+\frac{2}{3}(34+3e^5)x-\frac{10}{69}{e}^{\frac{1}{5}(23x+5)}+\frac{1}{9}{e}^{\frac{2}{5}(23x+5)}+\frac{2}{207}{e}^{\frac{1}{5}(23x+5)}(69x+69e^5+797)\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{45}\int (1020+90e^5+46e^{\frac{2}{5}(5+23x)}+90x+e^{\frac{1}{5}(5+23x)}(1594+138e^5+138x))dx\\ & =\frac{2}{3}(34+3e^5)x+x^2+\frac{1}{45}\int e^{\frac{1}{5}(5+23x)}(1594+138e^5+138x)dx+\frac{46}{45}\int e^{\frac{2}{5}(5+23x)}dx\\ & =\frac{1}{9}{e}^{\frac{2}{5}(5+23x)}+\frac{2}{3}(34+3e^5)x+x^2+\frac{2}{207}{e}^{\frac{1}{5}(5+23x)}(797+69e^5+69x)-\frac{2}{3}\int e^{\frac{1}{5}(5+23x)}dx\\ & =-\frac{10}{69}{e}^{\frac{1}{5}(5+23x)}+\frac{1}{9}{e}^{\frac{2}{5}(5+23x)}+\frac{2}{3}(34+3e^5)x+x^2+\frac{2}{207}{e}^{\frac{1}{5}(5+23x)}(797+69e^5+69x)\end{array}\end{array}$$

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

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 $$\begin{array}{c}{x}^2+2x{e}^5+\frac{2}{9}(3x+3e^5+34)e^{(\frac{23}{5}x+1)}+\frac{68}{3}x+\frac{1}{9}{e}^{(\frac{46}{5}x+2)}\end{array}$$

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 $$\begin{array}{c}{x}^2+2x{e}^5+\frac{2}{9}(3x+34)e^{(\frac{23}{5}x+1)}+\frac{68}{3}x+\frac{1}{9}{e}^{(\frac{46}{5}x+2)}+\frac{2}{3}{e}^{(\frac{23}{5}x+6)}\end{array}$$

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

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 $$\begin{array}{c}{x}^2+2x{e}^5+\frac{2}{9}(3xe+3e^6+34e)e^{\left(\frac{23}{5}x\right)}+\frac{68}{3}x+\frac{1}{9}{e}^{(\frac{46}{5}x+2)}\end{array}$$

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 $$\begin{array}{c}\frac{(3x+{\mathrm{e}}^{\frac{23x}{5}+1})(3x+6{\mathrm{e}}^5+{\mathrm{e}}^{\frac{23x}{5}+1}+68)}{9}\end{array}$$

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 $$\begin{array}{c}{x}^2+x(\frac{68}{3}+2e^5)+\frac{(54x+612+54e^5)e^{\frac{23x}{5}+1}}{81}+\frac{e^{\frac{46x}{5}+2}}{9}\end{array}$$

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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