3.55.56 \(\int e^{10 x+2 x^2+10 x^4+2 x^5} (10+4 x+40 x^3+10 x^4) \, dx\)

Optimal. Leaf size=21 \[ e^{2 (5+x) \left (x+x^4\right )}-\frac {4 \log ^2(3)}{5} \]

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Rubi [A]  time = 0.07, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6706} \begin {gather*} e^{2 x^5+10 x^4+2 x^2+10 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(10*x + 2*x^2 + 10*x^4 + 2*x^5)*(10 + 4*x + 40*x^3 + 10*x^4),x]

[Out]

E^(10*x + 2*x^2 + 10*x^4 + 2*x^5)

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} &=e^{10 x+2 x^2+10 x^4+2 x^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 16, normalized size = 0.76 \begin {gather*} e^{2 x \left (5+x+5 x^3+x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(10*x + 2*x^2 + 10*x^4 + 2*x^5)*(10 + 4*x + 40*x^3 + 10*x^4),x]

[Out]

E^(2*x*(5 + x + 5*x^3 + x^4))

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fricas [A]  time = 1.13, size = 20, normalized size = 0.95 \begin {gather*} e^{\left (2 \, x^{5} + 10 \, x^{4} + 2 \, x^{2} + 10 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^4+40*x^3+4*x+10)*exp(x^5+5*x^4+x^2+5*x)^2,x, algorithm="fricas")

[Out]

e^(2*x^5 + 10*x^4 + 2*x^2 + 10*x)

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giac [A]  time = 0.15, size = 20, normalized size = 0.95 \begin {gather*} e^{\left (2 \, x^{5} + 10 \, x^{4} + 2 \, x^{2} + 10 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^4+40*x^3+4*x+10)*exp(x^5+5*x^4+x^2+5*x)^2,x, algorithm="giac")

[Out]

e^(2*x^5 + 10*x^4 + 2*x^2 + 10*x)

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maple [A]  time = 0.03, size = 19, normalized size = 0.90

method	result	size
gosper	\({\mathrm e}^{2 x^{5}+10 x^{4}+2 x^{2}+10 x}\)	\(19\)
default	\({\mathrm e}^{2 x^{5}+10 x^{4}+2 x^{2}+10 x}\)	\(19\)
norman	\({\mathrm e}^{2 x^{5}+10 x^{4}+2 x^{2}+10 x}\)	\(19\)
risch	\({\mathrm e}^{2 x \left (5+x \right ) \left (x +1\right ) \left (x^{2}-x +1\right )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x^4+40*x^3+4*x+10)*exp(x^5+5*x^4+x^2+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(x^5+5*x^4+x^2+5*x)^2

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maxima [A]  time = 0.37, size = 20, normalized size = 0.95 \begin {gather*} e^{\left (2 \, x^{5} + 10 \, x^{4} + 2 \, x^{2} + 10 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^4+40*x^3+4*x+10)*exp(x^5+5*x^4+x^2+5*x)^2,x, algorithm="maxima")

[Out]

e^(2*x^5 + 10*x^4 + 2*x^2 + 10*x)

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mupad [B]  time = 3.40, size = 23, normalized size = 1.10 \begin {gather*} {\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{2\,x^2}\,{\mathrm {e}}^{2\,x^5}\,{\mathrm {e}}^{10\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(10*x + 2*x^2 + 10*x^4 + 2*x^5)*(4*x + 40*x^3 + 10*x^4 + 10),x)

[Out]

exp(10*x)*exp(2*x^2)*exp(2*x^5)*exp(10*x^4)

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sympy [A]  time = 0.10, size = 19, normalized size = 0.90 \begin {gather*} e^{2 x^{5} + 10 x^{4} + 2 x^{2} + 10 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x**4+40*x**3+4*x+10)*exp(x**5+5*x**4+x**2+5*x)**2,x)

[Out]

exp(2*x**5 + 10*x**4 + 2*x**2 + 10*x)

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