3.102.69 \(\int \frac {2 x^2+e^{3968-1512 x+144 x^2} x (1-1512 x+288 x^2)}{x} \, dx\)

Optimal. Leaf size=22 \[ e^{-1+9 (-9+3 (-4+x)+x)^2} x+x^2 \]

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Rubi [A]  time = 0.10, antiderivative size = 33, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14, 2288} \begin {gather*} x^2+\frac {e^{144 x^2-1512 x+3968} \left (21 x-4 x^2\right )}{21-4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^2 + E^(3968 - 1512*x + 144*x^2)*x*(1 - 1512*x + 288*x^2))/x,x]

[Out]

x^2 + (E^(3968 - 1512*x + 144*x^2)*(21*x - 4*x^2))/(21 - 4*x)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 x+e^{3968-1512 x+144 x^2} \left (1-1512 x+288 x^2\right )\right ) \, dx\\ &=x^2+\int e^{3968-1512 x+144 x^2} \left (1-1512 x+288 x^2\right ) \, dx\\ &=x^2+\frac {e^{3968-1512 x+144 x^2} \left (21 x-4 x^2\right )}{21-4 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 18, normalized size = 0.82 \begin {gather*} x \left (e^{8 \left (496-189 x+18 x^2\right )}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2 + E^(3968 - 1512*x + 144*x^2)*x*(1 - 1512*x + 288*x^2))/x,x]

[Out]

x*(E^(8*(496 - 189*x + 18*x^2)) + x)

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fricas [A]  time = 0.51, size = 17, normalized size = 0.77 \begin {gather*} x^{2} + e^{\left (144 \, x^{2} - 1512 \, x + \log \relax (x) + 3968\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((288*x^2-1512*x+1)*exp(log(x)+144*x^2-1512*x+3968)+2*x^2)/x,x, algorithm="fricas")

[Out]

x^2 + e^(144*x^2 - 1512*x + log(x) + 3968)

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giac [A]  time = 0.17, size = 17, normalized size = 0.77 \begin {gather*} x^{2} + x e^{\left (144 \, x^{2} - 1512 \, x + 3968\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((288*x^2-1512*x+1)*exp(log(x)+144*x^2-1512*x+3968)+2*x^2)/x,x, algorithm="giac")

[Out]

x^2 + x*e^(144*x^2 - 1512*x + 3968)

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maple [A]  time = 0.04, size = 18, normalized size = 0.82

method	result	size
default	\(x \,{\mathrm e}^{144 x^{2}-1512 x +3968}+x^{2}\)	\(18\)
norman	\(x^{2}+{\mathrm e}^{\ln \relax (x )+144 x^{2}-1512 x +3968}\)	\(18\)
risch	\(x^{2}+x \,{\mathrm e}^{8 \left (6 x -31\right ) \left (3 x -16\right )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((288*x^2-1512*x+1)*exp(ln(x)+144*x^2-1512*x+3968)+2*x^2)/x,x,method=_RETURNVERBOSE)

[Out]

x*exp(144*x^2-1512*x+3968)+x^2

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maxima [C]  time = 0.49, size = 153, normalized size = 6.95 \begin {gather*} -\frac {1}{24} i \, \sqrt {\pi } \operatorname {erf}\left (12 i \, x - 63 i\right ) e^{\left (-1\right )} + x^{2} - \frac {1}{12} \, {\left (\frac {{\left (4 \, x - 21\right )}^{3} \Gamma \left (\frac {3}{2}, -9 \, {\left (4 \, x - 21\right )}^{2}\right )}{\left (-{\left (4 \, x - 21\right )}^{2}\right )^{\frac {3}{2}}} - \frac {3969 \, \sqrt {\pi } {\left (4 \, x - 21\right )} {\left (\operatorname {erf}\left (3 \, \sqrt {-{\left (4 \, x - 21\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (4 \, x - 21\right )}^{2}}} - 126 \, e^{\left (9 \, {\left (4 \, x - 21\right )}^{2}\right )}\right )} e^{\left (-1\right )} - \frac {21}{4} \, {\left (\frac {63 \, \sqrt {\pi } {\left (4 \, x - 21\right )} {\left (\operatorname {erf}\left (3 \, \sqrt {-{\left (4 \, x - 21\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (4 \, x - 21\right )}^{2}}} + e^{\left (9 \, {\left (4 \, x - 21\right )}^{2}\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((288*x^2-1512*x+1)*exp(log(x)+144*x^2-1512*x+3968)+2*x^2)/x,x, algorithm="maxima")

[Out]

-1/24*I*sqrt(pi)*erf(12*I*x - 63*I)*e^(-1) + x^2 - 1/12*((4*x - 21)^3*gamma(3/2, -9*(4*x - 21)^2)/(-(4*x - 21)
^2)^(3/2) - 3969*sqrt(pi)*(4*x - 21)*(erf(3*sqrt(-(4*x - 21)^2)) - 1)/sqrt(-(4*x - 21)^2) - 126*e^(9*(4*x - 21
)^2))*e^(-1) - 21/4*(63*sqrt(pi)*(4*x - 21)*(erf(3*sqrt(-(4*x - 21)^2)) - 1)/sqrt(-(4*x - 21)^2) + e^(9*(4*x -
21)^2))*e^(-1)

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mupad [B]  time = 6.90, size = 18, normalized size = 0.82 \begin {gather*} x^2+x\,{\mathrm {e}}^{-1512\,x}\,{\mathrm {e}}^{3968}\,{\mathrm {e}}^{144\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x) - 1512*x + 144*x^2 + 3968)*(288*x^2 - 1512*x + 1) + 2*x^2)/x,x)

[Out]

x^2 + x*exp(-1512*x)*exp(3968)*exp(144*x^2)

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sympy [A]  time = 0.11, size = 15, normalized size = 0.68 \begin {gather*} x^{2} + x e^{144 x^{2} - 1512 x + 3968} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((288*x**2-1512*x+1)*exp(ln(x)+144*x**2-1512*x+3968)+2*x**2)/x,x)

[Out]

x**2 + x*exp(144*x**2 - 1512*x + 3968)

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