3.29.15 \(\int \frac {1}{4} e^{\frac {1}{4} (24+6 e^x+5 x+x^4)} (5+6 e^x+4 x^3) \, dx\)

Optimal. Leaf size=20 \[ e^{5+x+\frac {1}{4} \left (4+6 e^x+x+x^4\right )} \]

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Rubi [A]  time = 0.10, antiderivative size = 19, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {12, 6706} \begin {gather*} e^{\frac {1}{4} \left (x^4+5 x+6 e^x+24\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((24 + 6*E^x + 5*x + x^4)/4)*(5 + 6*E^x + 4*x^3))/4,x]

[Out]

E^((24 + 6*E^x + 5*x + x^4)/4)

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}{4} \int e^{\frac {1}{4} \left (24+6 e^x+5 x+x^4\right )} \left (5+6 e^x+4 x^3\right ) \, dx\\ &=e^{\frac {1}{4} \left (24+6 e^x+5 x+x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 19, normalized size = 0.95 \begin {gather*} e^{\frac {1}{4} \left (24+6 e^x+5 x+x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((24 + 6*E^x + 5*x + x^4)/4)*(5 + 6*E^x + 4*x^3))/4,x]

[Out]

E^((24 + 6*E^x + 5*x + x^4)/4)

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fricas [A]  time = 1.60, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (\frac {1}{4} \, x^{4} + \frac {5}{4} \, x + \frac {3}{2} \, e^{x} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(6*exp(x)+4*x^3+5)*exp(3/2*exp(x)+1/4*x^4+5/4*x+6),x, algorithm="fricas")

[Out]

e^(1/4*x^4 + 5/4*x + 3/2*e^x + 6)

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giac [A]  time = 0.30, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (\frac {1}{4} \, x^{4} + \frac {5}{4} \, x + \frac {3}{2} \, e^{x} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(6*exp(x)+4*x^3+5)*exp(3/2*exp(x)+1/4*x^4+5/4*x+6),x, algorithm="giac")

[Out]

e^(1/4*x^4 + 5/4*x + 3/2*e^x + 6)

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maple [A]  time = 0.05, size = 16, normalized size = 0.80

method	result	size
norman	\({\mathrm e}^{\frac {3 \,{\mathrm e}^{x}}{2}+\frac {x^{4}}{4}+\frac {5 x}{4}+6}\)	\(16\)
risch	\({\mathrm e}^{\frac {3 \,{\mathrm e}^{x}}{2}+\frac {x^{4}}{4}+\frac {5 x}{4}+6}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(6*exp(x)+4*x^3+5)*exp(3/2*exp(x)+1/4*x^4+5/4*x+6),x,method=_RETURNVERBOSE)

[Out]

exp(3/2*exp(x)+1/4*x^4+5/4*x+6)

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maxima [A]  time = 0.35, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (\frac {1}{4} \, x^{4} + \frac {5}{4} \, x + \frac {3}{2} \, e^{x} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(6*exp(x)+4*x^3+5)*exp(3/2*exp(x)+1/4*x^4+5/4*x+6),x, algorithm="maxima")

[Out]

e^(1/4*x^4 + 5/4*x + 3/2*e^x + 6)

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mupad [B]  time = 1.73, size = 18, normalized size = 0.90 \begin {gather*} {\mathrm {e}}^{\frac {5\,x}{4}}\,{\mathrm {e}}^6\,{\mathrm {e}}^{\frac {x^4}{4}}\,{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^x}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((5*x)/4 + (3*exp(x))/2 + x^4/4 + 6)*(6*exp(x) + 4*x^3 + 5))/4,x)

[Out]

exp((5*x)/4)*exp(6)*exp(x^4/4)*exp((3*exp(x))/2)

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sympy [A]  time = 0.15, size = 19, normalized size = 0.95 \begin {gather*} e^{\frac {x^{4}}{4} + \frac {5 x}{4} + \frac {3 e^{x}}{2} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(6*exp(x)+4*x**3+5)*exp(3/2*exp(x)+1/4*x**4+5/4*x+6),x)

[Out]

exp(x**4/4 + 5*x/4 + 3*exp(x)/2 + 6)

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