3.6.26 \(\int \frac {1}{100} (-100+e^{\frac {1}{5} (e^{\frac {1}{20} (-20+3 x)}+10 x)} (200+3 e^{\frac {1}{20} (-20+3 x)})) \, dx\)

Optimal. Leaf size=24 \[ 3+e^{\frac {1}{5} e^{-1+\frac {3 x}{20}}+2 x}-x \]

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Rubi [A]  time = 0.09, antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {12, 6706} \begin {gather*} e^{\frac {1}{5} \left (10 x+e^{\frac {1}{20} (3 x-20)}\right )}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-100 + E^((E^((-20 + 3*x)/20) + 10*x)/5)*(200 + 3*E^((-20 + 3*x)/20)))/100,x]

[Out]

E^((E^((-20 + 3*x)/20) + 10*x)/5) - x

Rule 12

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{100} \int \left (-100+e^{\frac {1}{5} \left (e^{\frac {1}{20} (-20+3 x)}+10 x\right )} \left (200+3 e^{\frac {1}{20} (-20+3 x)}\right )\right ) \, dx\\ &=-x+\frac {1}{100} \int e^{\frac {1}{5} \left (e^{\frac {1}{20} (-20+3 x)}+10 x\right )} \left (200+3 e^{\frac {1}{20} (-20+3 x)}\right ) \, dx\\ &=e^{\frac {1}{5} \left (e^{\frac {1}{20} (-20+3 x)}+10 x\right )}-x\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.14, size = 102, normalized size = 4.25 \begin {gather*} -x-\frac {48828125000}{3} \sqrt [3]{5} e^{\frac {40}{3}-\frac {x}{10}} \left (-e^{3 x/20}\right )^{2/3} \Gamma \left (\frac {40}{3},-\frac {1}{5} e^{-1+\frac {3 x}{20}}\right )[phone] \sqrt [3]{5} e^{\frac {40}{3}-\frac {x}{10}} \left (-e^{3 x/20}\right )^{2/3} \Gamma \left (\frac {43}{3},-\frac {1}{5} e^{-1+\frac {3 x}{20}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-100 + E^((E^((-20 + 3*x)/20) + 10*x)/5)*(200 + 3*E^((-20 + 3*x)/20)))/100,x]

[Out]

-x - (48828125000*5^(1/3)*E^(40/3 - x/10)*(-E^((3*x)/20))^(2/3)*Gamma[40/3, -1/5*E^(-1 + (3*x)/20)])/3 + 12207
03125*5^(1/3)*E^(40/3 - x/10)*(-E^((3*x)/20))^(2/3)*Gamma[43/3, -1/5*E^(-1 + (3*x)/20)]

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fricas [A]  time = 0.55, size = 17, normalized size = 0.71 \begin {gather*} -x + e^{\left (2 \, x + \frac {1}{5} \, e^{\left (\frac {3}{20} \, x - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/100*(3*exp(3/20*x-1)+200)*exp(1/5*exp(3/20*x-1)+2*x)-1,x, algorithm="fricas")

[Out]

-x + e^(2*x + 1/5*e^(3/20*x - 1))

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giac [A]  time = 0.54, size = 17, normalized size = 0.71 \begin {gather*} -x + e^{\left (2 \, x + \frac {1}{5} \, e^{\left (\frac {3}{20} \, x - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/100*(3*exp(3/20*x-1)+200)*exp(1/5*exp(3/20*x-1)+2*x)-1,x, algorithm="giac")

[Out]

-x + e^(2*x + 1/5*e^(3/20*x - 1))

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maple [A]  time = 0.05, size = 18, normalized size = 0.75

method	result	size
default	\(-x +{\mathrm e}^{\frac {{\mathrm e}^{\frac {3 x}{20}-1}}{5}+2 x}\)	\(18\)
norman	\(-x +{\mathrm e}^{\frac {{\mathrm e}^{\frac {3 x}{20}-1}}{5}+2 x}\)	\(18\)
risch	\(-x +{\mathrm e}^{\frac {{\mathrm e}^{\frac {3 x}{20}-1}}{5}+2 x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/100*(3*exp(3/20*x-1)+200)*exp(1/5*exp(3/20*x-1)+2*x)-1,x,method=_RETURNVERBOSE)

[Out]

-x+exp(1/5*exp(3/20*x-1)+2*x)

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maxima [A]  time = 0.67, size = 17, normalized size = 0.71 \begin {gather*} -x + e^{\left (2 \, x + \frac {1}{5} \, e^{\left (\frac {3}{20} \, x - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/100*(3*exp(3/20*x-1)+200)*exp(1/5*exp(3/20*x-1)+2*x)-1,x, algorithm="maxima")

[Out]

-x + e^(2*x + 1/5*e^(3/20*x - 1))

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mupad [B]  time = 0.48, size = 17, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^{2\,x+\frac {{\mathrm {e}}^{-1}\,{\left ({\mathrm {e}}^x\right )}^{3/20}}{5}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x + exp((3*x)/20 - 1)/5)*(3*exp((3*x)/20 - 1) + 200))/100 - 1,x)

[Out]

exp(2*x + (exp(-1)*exp(x)^(3/20))/5) - x

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sympy [A]  time = 0.16, size = 15, normalized size = 0.62 \begin {gather*} - x + e^{2 x + \frac {e^{\frac {3 x}{20} - 1}}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/100*(3*exp(3/20*x-1)+200)*exp(1/5*exp(3/20*x-1)+2*x)-1,x)

[Out]

-x + exp(2*x + exp(3*x/20 - 1)/5)

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