3.29.57 \(\int \frac {1}{5} (36-e^4-26 x+3 x^2+e^{4 x} (4+14 x-4 x^2)+e^{2 x} (-24-36 x+12 x^2)) \, dx\)

Optimal. Leaf size=33 \[ 2+\frac {1}{5} (4-x) \left (e^4-x \left (-\left (3-e^{2 x}\right )^2+x\right )\right ) \]

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Rubi [B]  time = 0.12, antiderivative size = 71, normalized size of antiderivative = 2.15, number of steps used = 18, number of rules used = 4, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {12, 2196, 2194, 2176} \begin {gather*} \frac {x^3}{5}+\frac {6}{5} e^{2 x} x^2-\frac {1}{5} e^{4 x} x^2-\frac {13 x^2}{5}-\frac {24}{5} e^{2 x} x+\frac {4}{5} e^{4 x} x+\frac {1}{5} \left (36-e^4\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 - E^4 - 26*x + 3*x^2 + E^(4*x)*(4 + 14*x - 4*x^2) + E^(2*x)*(-24 - 36*x + 12*x^2))/5,x]

[Out]

(-24*E^(2*x)*x)/5 + (4*E^(4*x)*x)/5 + ((36 - E^4)*x)/5 - (13*x^2)/5 + (6*E^(2*x)*x^2)/5 - (E^(4*x)*x^2)/5 + x^
3/5

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]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \left (36-e^4-26 x+3 x^2+e^{4 x} \left (4+14 x-4 x^2\right )+e^{2 x} \left (-24-36 x+12 x^2\right )\right ) \, dx\\ &=\frac {1}{5} \left (36-e^4\right ) x-\frac {13 x^2}{5}+\frac {x^3}{5}+\frac {1}{5} \int e^{4 x} \left (4+14 x-4 x^2\right ) \, dx+\frac {1}{5} \int e^{2 x} \left (-24-36 x+12 x^2\right ) \, dx\\ &=\frac {1}{5} \left (36-e^4\right ) x-\frac {13 x^2}{5}+\frac {x^3}{5}+\frac {1}{5} \int \left (-24 e^{2 x}-36 e^{2 x} x+12 e^{2 x} x^2\right ) \, dx+\frac {1}{5} \int \left (4 e^{4 x}+14 e^{4 x} x-4 e^{4 x} x^2\right ) \, dx\\ &=\frac {1}{5} \left (36-e^4\right ) x-\frac {13 x^2}{5}+\frac {x^3}{5}+\frac {4}{5} \int e^{4 x} \, dx-\frac {4}{5} \int e^{4 x} x^2 \, dx+\frac {12}{5} \int e^{2 x} x^2 \, dx+\frac {14}{5} \int e^{4 x} x \, dx-\frac {24}{5} \int e^{2 x} \, dx-\frac {36}{5} \int e^{2 x} x \, dx\\ &=-\frac {12 e^{2 x}}{5}+\frac {e^{4 x}}{5}-\frac {18}{5} e^{2 x} x+\frac {7}{10} e^{4 x} x+\frac {1}{5} \left (36-e^4\right ) x-\frac {13 x^2}{5}+\frac {6}{5} e^{2 x} x^2-\frac {1}{5} e^{4 x} x^2+\frac {x^3}{5}+\frac {2}{5} \int e^{4 x} x \, dx-\frac {7}{10} \int e^{4 x} \, dx-\frac {12}{5} \int e^{2 x} x \, dx+\frac {18}{5} \int e^{2 x} \, dx\\ &=-\frac {3 e^{2 x}}{5}+\frac {e^{4 x}}{40}-\frac {24}{5} e^{2 x} x+\frac {4}{5} e^{4 x} x+\frac {1}{5} \left (36-e^4\right ) x-\frac {13 x^2}{5}+\frac {6}{5} e^{2 x} x^2-\frac {1}{5} e^{4 x} x^2+\frac {x^3}{5}-\frac {1}{10} \int e^{4 x} \, dx+\frac {6}{5} \int e^{2 x} \, dx\\ &=-\frac {24}{5} e^{2 x} x+\frac {4}{5} e^{4 x} x+\frac {1}{5} \left (36-e^4\right ) x-\frac {13 x^2}{5}+\frac {6}{5} e^{2 x} x^2-\frac {1}{5} e^{4 x} x^2+\frac {x^3}{5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 38, normalized size = 1.15 \begin {gather*} \frac {1}{5} x \left (36-e^4+6 e^{2 x} (-4+x)-e^{4 x} (-4+x)-13 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 - E^4 - 26*x + 3*x^2 + E^(4*x)*(4 + 14*x - 4*x^2) + E^(2*x)*(-24 - 36*x + 12*x^2))/5,x]

[Out]

(x*(36 - E^4 + 6*E^(2*x)*(-4 + x) - E^(4*x)*(-4 + x) - 13*x + x^2))/5

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fricas [A]  time = 0.83, size = 45, normalized size = 1.36 \begin {gather*} \frac {1}{5} \, x^{3} - \frac {13}{5} \, x^{2} - \frac {1}{5} \, x e^{4} - \frac {1}{5} \, {\left (x^{2} - 4 \, x\right )} e^{\left (4 \, x\right )} + \frac {6}{5} \, {\left (x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )} + \frac {36}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-4*x^2+14*x+4)*exp(2*x)^2+1/5*(12*x^2-36*x-24)*exp(2*x)-1/5*exp(4)+3/5*x^2-26/5*x+36/5,x, algor
ithm="fricas")

[Out]

1/5*x^3 - 13/5*x^2 - 1/5*x*e^4 - 1/5*(x^2 - 4*x)*e^(4*x) + 6/5*(x^2 - 4*x)*e^(2*x) + 36/5*x

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giac [A]  time = 0.24, size = 45, normalized size = 1.36 \begin {gather*} \frac {1}{5} \, x^{3} - \frac {13}{5} \, x^{2} - \frac {1}{5} \, x e^{4} - \frac {1}{5} \, {\left (x^{2} - 4 \, x\right )} e^{\left (4 \, x\right )} + \frac {6}{5} \, {\left (x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )} + \frac {36}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-4*x^2+14*x+4)*exp(2*x)^2+1/5*(12*x^2-36*x-24)*exp(2*x)-1/5*exp(4)+3/5*x^2-26/5*x+36/5,x, algor
ithm="giac")

[Out]

1/5*x^3 - 13/5*x^2 - 1/5*x*e^4 - 1/5*(x^2 - 4*x)*e^(4*x) + 6/5*(x^2 - 4*x)*e^(2*x) + 36/5*x

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maple [A]  time = 0.08, size = 50, normalized size = 1.52

method	result	size
risch	\(\frac {\left (-x^{2}+4 x \right ) {\mathrm e}^{4 x}}{5}+\frac {\left (6 x^{2}-24 x \right ) {\mathrm e}^{2 x}}{5}-\frac {x \,{\mathrm e}^{4}}{5}+\frac {x^{3}}{5}-\frac {13 x^{2}}{5}+\frac {36 x}{5}\)	\(50\)
derivativedivides	\(\frac {36 x}{5}-\frac {13 x^{2}}{5}+\frac {x^{3}}{5}+\frac {4 x \,{\mathrm e}^{4 x}}{5}-\frac {x^{2} {\mathrm e}^{4 x}}{5}+\frac {6 \,{\mathrm e}^{2 x} x^{2}}{5}-\frac {24 x \,{\mathrm e}^{2 x}}{5}-\frac {x \,{\mathrm e}^{4}}{5}\)	\(56\)
default	\(\frac {36 x}{5}-\frac {13 x^{2}}{5}+\frac {x^{3}}{5}+\frac {4 x \,{\mathrm e}^{4 x}}{5}-\frac {x^{2} {\mathrm e}^{4 x}}{5}+\frac {6 \,{\mathrm e}^{2 x} x^{2}}{5}-\frac {24 x \,{\mathrm e}^{2 x}}{5}-\frac {x \,{\mathrm e}^{4}}{5}\)	\(56\)
norman	\(\left (-\frac {{\mathrm e}^{4}}{5}+\frac {36}{5}\right ) x -\frac {13 x^{2}}{5}+\frac {x^{3}}{5}-\frac {24 x \,{\mathrm e}^{2 x}}{5}+\frac {4 x \,{\mathrm e}^{4 x}}{5}-\frac {x^{2} {\mathrm e}^{4 x}}{5}+\frac {6 \,{\mathrm e}^{2 x} x^{2}}{5}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(-4*x^2+14*x+4)*exp(2*x)^2+1/5*(12*x^2-36*x-24)*exp(2*x)-1/5*exp(4)+3/5*x^2-26/5*x+36/5,x,method=_RETU
RNVERBOSE)

[Out]

1/5*(-x^2+4*x)*exp(4*x)+1/5*(6*x^2-24*x)*exp(2*x)-1/5*x*exp(4)+1/5*x^3-13/5*x^2+36/5*x

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maxima [A]  time = 0.44, size = 45, normalized size = 1.36 \begin {gather*} \frac {1}{5} \, x^{3} - \frac {13}{5} \, x^{2} - \frac {1}{5} \, x e^{4} - \frac {1}{5} \, {\left (x^{2} - 4 \, x\right )} e^{\left (4 \, x\right )} + \frac {6}{5} \, {\left (x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )} + \frac {36}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-4*x^2+14*x+4)*exp(2*x)^2+1/5*(12*x^2-36*x-24)*exp(2*x)-1/5*exp(4)+3/5*x^2-26/5*x+36/5,x, algor
ithm="maxima")

[Out]

1/5*x^3 - 13/5*x^2 - 1/5*x*e^4 - 1/5*(x^2 - 4*x)*e^(4*x) + 6/5*(x^2 - 4*x)*e^(2*x) + 36/5*x

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mupad [B]  time = 1.71, size = 40, normalized size = 1.21 \begin {gather*} -\frac {x\,\left (13\,x+24\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^4-6\,x\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{4\,x}-x^2-36\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x)*(14*x - 4*x^2 + 4))/5 - exp(4)/5 - (26*x)/5 - (exp(2*x)*(36*x - 12*x^2 + 24))/5 + (3*x^2)/5 + 36
/5,x)

[Out]

-(x*(13*x + 24*exp(2*x) - 4*exp(4*x) + exp(4) - 6*x*exp(2*x) + x*exp(4*x) - x^2 - 36))/5

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sympy [B]  time = 0.14, size = 51, normalized size = 1.55 \begin {gather*} \frac {x^{3}}{5} - \frac {13 x^{2}}{5} + x \left (\frac {36}{5} - \frac {e^{4}}{5}\right ) + \frac {\left (- 5 x^{2} + 20 x\right ) e^{4 x}}{25} + \frac {\left (30 x^{2} - 120 x\right ) e^{2 x}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-4*x**2+14*x+4)*exp(2*x)**2+1/5*(12*x**2-36*x-24)*exp(2*x)-1/5*exp(4)+3/5*x**2-26/5*x+36/5,x)

[Out]

x**3/5 - 13*x**2/5 + x*(36/5 - exp(4)/5) + (-5*x**2 + 20*x)*exp(4*x)/25 + (30*x**2 - 120*x)*exp(2*x)/25

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