3.79.25 \(\int (2 x+e^{2 x+5 x^2} (2+10 x)) \, dx\)

Optimal. Leaf size=16 \[ -2+e^{2 x+5 x^2}+x^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2236} \begin {gather*} x^2+e^{5 x^2+2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*x + 5*x^2) + x^2

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x^2+\int e^{2 x+5 x^2} (2+10 x) \, dx\\ &=e^{2 x+5 x^2}+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 25, normalized size = 1.56 \begin {gather*} 2 \left (\frac {1}{2} e^{2 x+5 x^2}+\frac {x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*(E^(2*x + 5*x^2)/2 + x^2/2)

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fricas [A]  time = 1.07, size = 14, normalized size = 0.88 \begin {gather*} x^{2} + e^{\left (5 \, x^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + e^(5*x^2 + 2*x)

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giac [A]  time = 0.14, size = 14, normalized size = 0.88 \begin {gather*} x^{2} + e^{\left (5 \, x^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + e^(5*x^2 + 2*x)

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maple [A]  time = 0.02, size = 13, normalized size = 0.81

method	result	size
risch	\({\mathrm e}^{x \left (5 x +2\right )}+x^{2}\)	\(13\)
default	\({\mathrm e}^{5 x^{2}+2 x}+x^{2}\)	\(15\)
norman	\({\mathrm e}^{5 x^{2}+2 x}+x^{2}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 14, normalized size = 0.88 \begin {gather*} x^{2} + e^{\left (5 \, x^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + e^(5*x^2 + 2*x)

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mupad [B]  time = 0.10, size = 14, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^{5\,x^2+2\,x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2*x + 5*x^2) + x^2

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sympy [A]  time = 0.09, size = 12, normalized size = 0.75 \begin {gather*} x^{2} + e^{5 x^{2} + 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x+2)*exp(5*x**2+2*x)+2*x,x)

[Out]

x**2 + exp(5*x**2 + 2*x)

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