3.5.51 \(\int (2-18 e^{9 x^2} x+3 x^2) \, dx\)

Optimal. Leaf size=24 \[ -e^{9 x^2}+2 x+\frac {-x+x^4}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2209} \begin {gather*} x^3-e^{9 x^2}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 - 18*E^(9*x^2)*x + 3*x^2,x]

[Out]

-E^(9*x^2) + 2*x + x^3

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 x+x^3-18 \int e^{9 x^2} x \, dx\\ &=-e^{9 x^2}+2 x+x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.67 \begin {gather*} -e^{9 x^2}+2 x+x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 - 18*E^(9*x^2)*x + 3*x^2,x]

[Out]

-E^(9*x^2) + 2*x + x^3

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fricas [A]  time = 0.67, size = 15, normalized size = 0.62 \begin {gather*} x^{3} + 2 \, x - e^{\left (9 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-18*x*exp(9*x^2)+3*x^2+2,x, algorithm="fricas")

[Out]

x^3 + 2*x - e^(9*x^2)

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giac [A]  time = 0.68, size = 15, normalized size = 0.62 \begin {gather*} x^{3} + 2 \, x - e^{\left (9 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-18*x*exp(9*x^2)+3*x^2+2,x, algorithm="giac")

[Out]

x^3 + 2*x - e^(9*x^2)

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maple [A]  time = 0.02, size = 16, normalized size = 0.67

method	result	size
default	\(x^{3}+2 x -{\mathrm e}^{9 x^{2}}\)	\(16\)
norman	\(x^{3}+2 x -{\mathrm e}^{9 x^{2}}\)	\(16\)
risch	\(x^{3}+2 x -{\mathrm e}^{9 x^{2}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-18*x*exp(9*x^2)+3*x^2+2,x,method=_RETURNVERBOSE)

[Out]

x^3+2*x-exp(9*x^2)

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maxima [A]  time = 0.49, size = 15, normalized size = 0.62 \begin {gather*} x^{3} + 2 \, x - e^{\left (9 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-18*x*exp(9*x^2)+3*x^2+2,x, algorithm="maxima")

[Out]

x^3 + 2*x - e^(9*x^2)

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mupad [B]  time = 0.05, size = 15, normalized size = 0.62 \begin {gather*} 2\,x-{\mathrm {e}}^{9\,x^2}+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*x^2 - 18*x*exp(9*x^2) + 2,x)

[Out]

2*x - exp(9*x^2) + x^3

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sympy [A]  time = 0.08, size = 12, normalized size = 0.50 \begin {gather*} x^{3} + 2 x - e^{9 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-18*x*exp(9*x**2)+3*x**2+2,x)

[Out]

x**3 + 2*x - exp(9*x**2)

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