3.100.85 \(\int (98-196 x+e^{4 x^2} (98+784 x^2)) \, dx\)

Optimal. Leaf size=17 \[ 98 \left (x+\left (e^{4 x^2}-x\right ) x\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2226, 2204, 2212} \begin {gather*} -98 x^2+98 e^{4 x^2} x+98 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[98 - 196*x + E^(4*x^2)*(98 + 784*x^2),x]

[Out]

98*x + 98*E^(4*x^2)*x - 98*x^2

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=98 x-98 x^2+\int e^{4 x^2} \left (98+784 x^2\right ) \, dx\\ &=98 x-98 x^2+\int \left (98 e^{4 x^2}+784 e^{4 x^2} x^2\right ) \, dx\\ &=98 x-98 x^2+98 \int e^{4 x^2} \, dx+784 \int e^{4 x^2} x^2 \, dx\\ &=98 x+98 e^{4 x^2} x-98 x^2+\frac {49}{2} \sqrt {\pi } \text {erfi}(2 x)-98 \int e^{4 x^2} \, dx\\ &=98 x+98 e^{4 x^2} x-98 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.06 \begin {gather*} 98 \left (x+e^{4 x^2} x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[98 - 196*x + E^(4*x^2)*(98 + 784*x^2),x]

[Out]

98*(x + E^(4*x^2)*x - x^2)

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fricas [A]  time = 0.62, size = 18, normalized size = 1.06 \begin {gather*} -98 \, x^{2} + 98 \, x e^{\left (4 \, x^{2}\right )} + 98 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((784*x^2+98)*exp(4*x^2)-196*x+98,x, algorithm="fricas")

[Out]

-98*x^2 + 98*x*e^(4*x^2) + 98*x

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giac [A]  time = 0.17, size = 18, normalized size = 1.06 \begin {gather*} -98 \, x^{2} + 98 \, x e^{\left (4 \, x^{2}\right )} + 98 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((784*x^2+98)*exp(4*x^2)-196*x+98,x, algorithm="giac")

[Out]

-98*x^2 + 98*x*e^(4*x^2) + 98*x

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maple [A]  time = 0.02, size = 19, normalized size = 1.12

method	result	size
default	\(98 x -98 x^{2}+98 x \,{\mathrm e}^{4 x^{2}}\)	\(19\)
norman	\(98 x -98 x^{2}+98 x \,{\mathrm e}^{4 x^{2}}\)	\(19\)
risch	\(98 x -98 x^{2}+98 x \,{\mathrm e}^{4 x^{2}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((784*x^2+98)*exp(4*x^2)-196*x+98,x,method=_RETURNVERBOSE)

[Out]

98*x+98*x*exp(x^2)^4-98*x^2

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maxima [A]  time = 0.36, size = 18, normalized size = 1.06 \begin {gather*} -98 \, x^{2} + 98 \, x e^{\left (4 \, x^{2}\right )} + 98 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((784*x^2+98)*exp(4*x^2)-196*x+98,x, algorithm="maxima")

[Out]

-98*x^2 + 98*x*e^(4*x^2) + 98*x

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mupad [B]  time = 0.11, size = 14, normalized size = 0.82 \begin {gather*} 98\,x\,\left ({\mathrm {e}}^{4\,x^2}-x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4*x^2)*(784*x^2 + 98) - 196*x + 98,x)

[Out]

98*x*(exp(4*x^2) - x + 1)

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sympy [A]  time = 0.09, size = 17, normalized size = 1.00 \begin {gather*} - 98 x^{2} + 98 x e^{4 x^{2}} + 98 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((784*x**2+98)*exp(4*x**2)-196*x+98,x)

[Out]

-98*x**2 + 98*x*exp(4*x**2) + 98*x

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