3.1.61 \(\int \frac {1}{81} (-9720-81 e^x+e (3240-900 x)+e^5 (3240-900 x)+15660 x-5400 x^2+500 x^3) \, dx\)

Optimal. Leaf size=35 \[ 1-e^x+5 \left (3-e-e^5+x+5 \left (-x+\frac {x^2}{9}\right )\right )^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.34, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6, 12, 2194} \begin {gather*} \frac {125 x^4}{81}-\frac {200 x^3}{9}+\frac {290 x^2}{3}-120 x-e^x-\frac {2}{9} e \left (1+e^4\right ) (18-5 x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9720 - 81*E^x + E*(3240 - 900*x) + E^5*(3240 - 900*x) + 15660*x - 5400*x^2 + 500*x^3)/81,x]

[Out]

-E^x - (2*E*(1 + E^4)*(18 - 5*x)^2)/9 - 120*x + (290*x^2)/3 - (200*x^3)/9 + (125*x^4)/81

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 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} &=\int \frac {1}{81} \left (-9720-81 e^x+\left (e+e^5\right ) (3240-900 x)+15660 x-5400 x^2+500 x^3\right ) \, dx\\ &=\frac {1}{81} \int \left (-9720-81 e^x+\left (e+e^5\right ) (3240-900 x)+15660 x-5400 x^2+500 x^3\right ) \, dx\\ &=-\frac {2}{9} e \left (1+e^4\right ) (18-5 x)^2-120 x+\frac {290 x^2}{3}-\frac {200 x^3}{9}+\frac {125 x^4}{81}-\int e^x \, dx\\ &=-e^x-\frac {2}{9} e \left (1+e^4\right ) (18-5 x)^2-120 x+\frac {290 x^2}{3}-\frac {200 x^3}{9}+\frac {125 x^4}{81}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 58, normalized size = 1.66 \begin {gather*} -e^x-120 x+40 e x+40 e^5 x+\frac {290 x^2}{3}-\frac {50 e x^2}{9}-\frac {50 e^5 x^2}{9}-\frac {200 x^3}{9}+\frac {125 x^4}{81} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9720 - 81*E^x + E*(3240 - 900*x) + E^5*(3240 - 900*x) + 15660*x - 5400*x^2 + 500*x^3)/81,x]

[Out]

-E^x - 120*x + 40*E*x + 40*E^5*x + (290*x^2)/3 - (50*E*x^2)/9 - (50*E^5*x^2)/9 - (200*x^3)/9 + (125*x^4)/81

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fricas [A]  time = 0.69, size = 49, normalized size = 1.40 \begin {gather*} \frac {125}{81} \, x^{4} - \frac {200}{9} \, x^{3} + \frac {290}{3} \, x^{2} - \frac {10}{9} \, {\left (5 \, x^{2} - 36 \, x\right )} e^{5} - \frac {10}{9} \, {\left (5 \, x^{2} - 36 \, x\right )} e - 120 \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)+1/81*(-900*x+3240)*exp(5)+1/81*(-900*x+3240)*exp(1)+500/81*x^3-200/3*x^2+580/3*x-120,x, algo
rithm="fricas")

[Out]

125/81*x^4 - 200/9*x^3 + 290/3*x^2 - 10/9*(5*x^2 - 36*x)*e^5 - 10/9*(5*x^2 - 36*x)*e - 120*x - e^x

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giac [A]  time = 0.30, size = 49, normalized size = 1.40 \begin {gather*} \frac {125}{81} \, x^{4} - \frac {200}{9} \, x^{3} + \frac {290}{3} \, x^{2} - \frac {10}{9} \, {\left (5 \, x^{2} - 36 \, x\right )} e^{5} - \frac {10}{9} \, {\left (5 \, x^{2} - 36 \, x\right )} e - 120 \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)+1/81*(-900*x+3240)*exp(5)+1/81*(-900*x+3240)*exp(1)+500/81*x^3-200/3*x^2+580/3*x-120,x, algo
rithm="giac")

[Out]

125/81*x^4 - 200/9*x^3 + 290/3*x^2 - 10/9*(5*x^2 - 36*x)*e^5 - 10/9*(5*x^2 - 36*x)*e - 120*x - e^x

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maple [A]  time = 0.03, size = 42, normalized size = 1.20

method	result	size
norman	\(\left (40 \,{\mathrm e}^{5}+40 \,{\mathrm e}-120\right ) x +\left (-\frac {50 \,{\mathrm e}^{5}}{9}-\frac {50 \,{\mathrm e}}{9}+\frac {290}{3}\right ) x^{2}-\frac {200 x^{3}}{9}+\frac {125 x^{4}}{81}-{\mathrm e}^{x}\)	\(42\)
risch	\(-{\mathrm e}^{x}-\frac {50 x^{2} {\mathrm e}^{5}}{9}+40 x \,{\mathrm e}^{5}-\frac {50 x^{2} {\mathrm e}}{9}+40 x \,{\mathrm e}+\frac {125 x^{4}}{81}-\frac {200 x^{3}}{9}+\frac {290 x^{2}}{3}-120 x\)	\(48\)
default	\(-120 x +\frac {290 x^{2}}{3}-\frac {200 x^{3}}{9}+\frac {125 x^{4}}{81}+\frac {{\mathrm e} \left (-450 x^{2}+3240 x \right )}{81}+\frac {{\mathrm e}^{5} \left (-450 x^{2}+3240 x \right )}{81}-{\mathrm e}^{x}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x)+1/81*(-900*x+3240)*exp(5)+1/81*(-900*x+3240)*exp(1)+500/81*x^3-200/3*x^2+580/3*x-120,x,method=_RET
URNVERBOSE)

[Out]

(40*exp(5)+40*exp(1)-120)*x+(-50/9*exp(5)-50/9*exp(1)+290/3)*x^2-200/9*x^3+125/81*x^4-exp(x)

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maxima [A]  time = 0.34, size = 49, normalized size = 1.40 \begin {gather*} \frac {125}{81} \, x^{4} - \frac {200}{9} \, x^{3} + \frac {290}{3} \, x^{2} - \frac {10}{9} \, {\left (5 \, x^{2} - 36 \, x\right )} e^{5} - \frac {10}{9} \, {\left (5 \, x^{2} - 36 \, x\right )} e - 120 \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)+1/81*(-900*x+3240)*exp(5)+1/81*(-900*x+3240)*exp(1)+500/81*x^3-200/3*x^2+580/3*x-120,x, algo
rithm="maxima")

[Out]

125/81*x^4 - 200/9*x^3 + 290/3*x^2 - 10/9*(5*x^2 - 36*x)*e^5 - 10/9*(5*x^2 - 36*x)*e - 120*x - e^x

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mupad [B]  time = 0.07, size = 42, normalized size = 1.20 \begin {gather*} x\,\left (40\,\mathrm {e}+40\,{\mathrm {e}}^5-120\right )-x^2\,\left (\frac {50\,\mathrm {e}}{9}+\frac {50\,{\mathrm {e}}^5}{9}-\frac {290}{3}\right )-{\mathrm {e}}^x-\frac {200\,x^3}{9}+\frac {125\,x^4}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((580*x)/3 - exp(x) - (200*x^2)/3 + (500*x^3)/81 - (exp(1)*(900*x - 3240))/81 - (exp(5)*(900*x - 3240))/81
- 120,x)

[Out]

x*(40*exp(1) + 40*exp(5) - 120) - x^2*((50*exp(1))/9 + (50*exp(5))/9 - 290/3) - exp(x) - (200*x^3)/9 + (125*x^
4)/81

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sympy [A]  time = 0.10, size = 49, normalized size = 1.40 \begin {gather*} \frac {125 x^{4}}{81} - \frac {200 x^{3}}{9} + x^{2} \left (- \frac {50 e^{5}}{9} - \frac {50 e}{9} + \frac {290}{3}\right ) + x \left (-120 + 40 e + 40 e^{5}\right ) - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)+1/81*(-900*x+3240)*exp(5)+1/81*(-900*x+3240)*exp(1)+500/81*x**3-200/3*x**2+580/3*x-120,x)

[Out]

125*x**4/81 - 200*x**3/9 + x**2*(-50*exp(5)/9 - 50*E/9 + 290/3) + x*(-120 + 40*E + 40*exp(5)) - exp(x)

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