3.77.2 \(\int \frac {1}{15} (-243 x^2+5 x^4+e^x (162 x+81 x^2-4 x^3-x^4)) \, dx\)

Optimal. Leaf size=23 \[ 5+\frac {27}{5} \left (e^x-x\right ) x \left (x-\frac {x^3}{81}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 35, normalized size of antiderivative = 1.52, number of steps used = 18, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 2196, 2176, 2194} \begin {gather*} \frac {x^5}{15}-\frac {e^x x^4}{15}-\frac {27 x^3}{5}+\frac {27 e^x x^2}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-243*x^2 + 5*x^4 + E^x*(162*x + 81*x^2 - 4*x^3 - x^4))/15,x]

[Out]

(27*E^x*x^2)/5 - (27*x^3)/5 - (E^x*x^4)/15 + x^5/15

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}{15} \int \left (-243 x^2+5 x^4+e^x \left (162 x+81 x^2-4 x^3-x^4\right )\right ) \, dx\\ &=-\frac {27 x^3}{5}+\frac {x^5}{15}+\frac {1}{15} \int e^x \left (162 x+81 x^2-4 x^3-x^4\right ) \, dx\\ &=-\frac {27 x^3}{5}+\frac {x^5}{15}+\frac {1}{15} \int \left (162 e^x x+81 e^x x^2-4 e^x x^3-e^x x^4\right ) \, dx\\ &=-\frac {27 x^3}{5}+\frac {x^5}{15}-\frac {1}{15} \int e^x x^4 \, dx-\frac {4}{15} \int e^x x^3 \, dx+\frac {27}{5} \int e^x x^2 \, dx+\frac {54}{5} \int e^x x \, dx\\ &=\frac {54 e^x x}{5}+\frac {27 e^x x^2}{5}-\frac {27 x^3}{5}-\frac {4 e^x x^3}{15}-\frac {e^x x^4}{15}+\frac {x^5}{15}+\frac {4}{15} \int e^x x^3 \, dx+\frac {4}{5} \int e^x x^2 \, dx-\frac {54 \int e^x \, dx}{5}-\frac {54}{5} \int e^x x \, dx\\ &=-\frac {54 e^x}{5}+\frac {31 e^x x^2}{5}-\frac {27 x^3}{5}-\frac {e^x x^4}{15}+\frac {x^5}{15}-\frac {4}{5} \int e^x x^2 \, dx-\frac {8}{5} \int e^x x \, dx+\frac {54 \int e^x \, dx}{5}\\ &=-\frac {8 e^x x}{5}+\frac {27 e^x x^2}{5}-\frac {27 x^3}{5}-\frac {e^x x^4}{15}+\frac {x^5}{15}+\frac {8 \int e^x \, dx}{5}+\frac {8}{5} \int e^x x \, dx\\ &=\frac {8 e^x}{5}+\frac {27 e^x x^2}{5}-\frac {27 x^3}{5}-\frac {e^x x^4}{15}+\frac {x^5}{15}-\frac {8 \int e^x \, dx}{5}\\ &=\frac {27 e^x x^2}{5}-\frac {27 x^3}{5}-\frac {e^x x^4}{15}+\frac {x^5}{15}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 19, normalized size = 0.83 \begin {gather*} -\frac {1}{15} \left (e^x-x\right ) x^2 \left (-81+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-243*x^2 + 5*x^4 + E^x*(162*x + 81*x^2 - 4*x^3 - x^4))/15,x]

[Out]

-1/15*((E^x - x)*x^2*(-81 + x^2))

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{15} \, x^{5} - \frac {27}{5} \, x^{3} - \frac {1}{15} \, {\left (x^{4} - 81 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(-x^4-4*x^3+81*x^2+162*x)*exp(x)+1/3*x^4-81/5*x^2,x, algorithm="fricas")

[Out]

1/15*x^5 - 27/5*x^3 - 1/15*(x^4 - 81*x^2)*e^x

________________________________________________________________________________________

giac [A]  time = 0.15, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{15} \, x^{5} - \frac {27}{5} \, x^{3} - \frac {1}{15} \, {\left (x^{4} - 81 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(-x^4-4*x^3+81*x^2+162*x)*exp(x)+1/3*x^4-81/5*x^2,x, algorithm="giac")

[Out]

1/15*x^5 - 27/5*x^3 - 1/15*(x^4 - 81*x^2)*e^x

________________________________________________________________________________________

maple [A]  time = 0.02, size = 26, normalized size = 1.13

method	result	size
default	\(-\frac {27 x^{3}}{5}+\frac {x^{5}}{15}+\frac {27 \,{\mathrm e}^{x} x^{2}}{5}-\frac {{\mathrm e}^{x} x^{4}}{15}\)	\(26\)
norman	\(-\frac {27 x^{3}}{5}+\frac {x^{5}}{15}+\frac {27 \,{\mathrm e}^{x} x^{2}}{5}-\frac {{\mathrm e}^{x} x^{4}}{15}\)	\(26\)
risch	\(\frac {\left (-x^{4}+81 x^{2}\right ) {\mathrm e}^{x}}{15}+\frac {x^{5}}{15}-\frac {27 x^{3}}{5}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/15*(-x^4-4*x^3+81*x^2+162*x)*exp(x)+1/3*x^4-81/5*x^2,x,method=_RETURNVERBOSE)

[Out]

-27/5*x^3+1/15*x^5+27/5*exp(x)*x^2-1/15*exp(x)*x^4

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{15} \, x^{5} - \frac {27}{5} \, x^{3} - \frac {1}{15} \, {\left (x^{4} - 81 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(-x^4-4*x^3+81*x^2+162*x)*exp(x)+1/3*x^4-81/5*x^2,x, algorithm="maxima")

[Out]

1/15*x^5 - 27/5*x^3 - 1/15*(x^4 - 81*x^2)*e^x

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 16, normalized size = 0.70 \begin {gather*} \frac {x^2\,\left (x-{\mathrm {e}}^x\right )\,\left (x^2-81\right )}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(162*x + 81*x^2 - 4*x^3 - x^4))/15 - (81*x^2)/5 + x^4/3,x)

[Out]

(x^2*(x - exp(x))*(x^2 - 81))/15

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{5}}{15} - \frac {27 x^{3}}{5} + \frac {\left (- x^{4} + 81 x^{2}\right ) e^{x}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(-x**4-4*x**3+81*x**2+162*x)*exp(x)+1/3*x**4-81/5*x**2,x)

[Out]

x**5/15 - 27*x**3/5 + (-x**4 + 81*x**2)*exp(x)/15

________________________________________________________________________________________