3.35.58 \(\int \frac {1}{9} (432-1056 x+564 x^2-32 x^3-20 x^4+e^{x^2} (144-360 x+288 x^2-72 x^3)) \, dx\)

Optimal. Leaf size=29 \[ (-4+2 x)^2 \left (-e^{x^2}+x \left (4-\frac {1}{9} (3+x)^2\right )\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.97, number of steps used = 10, number of rules used = 5, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2226, 2204, 2209, 2212} \begin {gather*} -\frac {4 x^5}{9}-\frac {8 x^4}{9}+\frac {188 x^3}{9}-4 e^{x^2} x^2-\frac {176 x^2}{3}+16 e^{x^2} x-16 e^{x^2}+48 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(432 - 1056*x + 564*x^2 - 32*x^3 - 20*x^4 + E^x^2*(144 - 360*x + 288*x^2 - 72*x^3))/9,x]

[Out]

-16*E^x^2 + 48*x + 16*E^x^2*x - (176*x^2)/3 - 4*E^x^2*x^2 + (188*x^3)/9 - (8*x^4)/9 - (4*x^5)/9

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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]

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} &=\frac {1}{9} \int \left (432-1056 x+564 x^2-32 x^3-20 x^4+e^{x^2} \left (144-360 x+288 x^2-72 x^3\right )\right ) \, dx\\ &=48 x-\frac {176 x^2}{3}+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}+\frac {1}{9} \int e^{x^2} \left (144-360 x+288 x^2-72 x^3\right ) \, dx\\ &=48 x-\frac {176 x^2}{3}+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}+\frac {1}{9} \int \left (144 e^{x^2}-360 e^{x^2} x+288 e^{x^2} x^2-72 e^{x^2} x^3\right ) \, dx\\ &=48 x-\frac {176 x^2}{3}+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}-8 \int e^{x^2} x^3 \, dx+16 \int e^{x^2} \, dx+32 \int e^{x^2} x^2 \, dx-40 \int e^{x^2} x \, dx\\ &=-20 e^{x^2}+48 x+16 e^{x^2} x-\frac {176 x^2}{3}-4 e^{x^2} x^2+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}+8 \sqrt {\pi } \text {erfi}(x)+8 \int e^{x^2} x \, dx-16 \int e^{x^2} \, dx\\ &=-16 e^{x^2}+48 x+16 e^{x^2} x-\frac {176 x^2}{3}-4 e^{x^2} x^2+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 27, normalized size = 0.93 \begin {gather*} -\frac {4}{9} (-2+x)^2 \left (9 e^{x^2}+x \left (-27+6 x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(432 - 1056*x + 564*x^2 - 32*x^3 - 20*x^4 + E^x^2*(144 - 360*x + 288*x^2 - 72*x^3))/9,x]

[Out]

(-4*(-2 + x)^2*(9*E^x^2 + x*(-27 + 6*x + x^2)))/9

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fricas [A]  time = 0.71, size = 38, normalized size = 1.31 \begin {gather*} -\frac {4}{9} \, x^{5} - \frac {8}{9} \, x^{4} + \frac {188}{9} \, x^{3} - \frac {176}{3} \, x^{2} - 4 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 48 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-72*x^3+288*x^2-360*x+144)*exp(x^2)-20/9*x^4-32/9*x^3+188/3*x^2-352/3*x+48,x, algorithm="fricas
")

[Out]

-4/9*x^5 - 8/9*x^4 + 188/9*x^3 - 176/3*x^2 - 4*(x^2 - 4*x + 4)*e^(x^2) + 48*x

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giac [A]  time = 0.19, size = 38, normalized size = 1.31 \begin {gather*} -\frac {4}{9} \, x^{5} - \frac {8}{9} \, x^{4} + \frac {188}{9} \, x^{3} - \frac {176}{3} \, x^{2} - 4 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 48 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-72*x^3+288*x^2-360*x+144)*exp(x^2)-20/9*x^4-32/9*x^3+188/3*x^2-352/3*x+48,x, algorithm="giac")

[Out]

-4/9*x^5 - 8/9*x^4 + 188/9*x^3 - 176/3*x^2 - 4*(x^2 - 4*x + 4)*e^(x^2) + 48*x

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maple [A]  time = 0.02, size = 41, normalized size = 1.41

method	result	size
risch	\(\frac {\left (-36 x^{2}+144 x -144\right ) {\mathrm e}^{x^{2}}}{9}-\frac {4 x^{5}}{9}-\frac {8 x^{4}}{9}+\frac {188 x^{3}}{9}-\frac {176 x^{2}}{3}+48 x\)	\(41\)
default	\(48 x -\frac {176 x^{2}}{3}+\frac {188 x^{3}}{9}-\frac {8 x^{4}}{9}-\frac {4 x^{5}}{9}-16 \,{\mathrm e}^{x^{2}}+16 \,{\mathrm e}^{x^{2}} x -4 x^{2} {\mathrm e}^{x^{2}}\)	\(47\)
norman	\(48 x -\frac {176 x^{2}}{3}+\frac {188 x^{3}}{9}-\frac {8 x^{4}}{9}-\frac {4 x^{5}}{9}-16 \,{\mathrm e}^{x^{2}}+16 \,{\mathrm e}^{x^{2}} x -4 x^{2} {\mathrm e}^{x^{2}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(-72*x^3+288*x^2-360*x+144)*exp(x^2)-20/9*x^4-32/9*x^3+188/3*x^2-352/3*x+48,x,method=_RETURNVERBOSE)

[Out]

1/9*(-36*x^2+144*x-144)*exp(x^2)-4/9*x^5-8/9*x^4+188/9*x^3-176/3*x^2+48*x

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maxima [A]  time = 0.43, size = 38, normalized size = 1.31 \begin {gather*} -\frac {4}{9} \, x^{5} - \frac {8}{9} \, x^{4} + \frac {188}{9} \, x^{3} - \frac {176}{3} \, x^{2} - 4 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 48 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-72*x^3+288*x^2-360*x+144)*exp(x^2)-20/9*x^4-32/9*x^3+188/3*x^2-352/3*x+48,x, algorithm="maxima
")

[Out]

-4/9*x^5 - 8/9*x^4 + 188/9*x^3 - 176/3*x^2 - 4*(x^2 - 4*x + 4)*e^(x^2) + 48*x

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mupad [B]  time = 0.07, size = 25, normalized size = 0.86 \begin {gather*} -\frac {4\,{\left (x-2\right )}^2\,\left (9\,{\mathrm {e}}^{x^2}-27\,x+6\,x^2+x^3\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((188*x^2)/3 - (exp(x^2)*(360*x - 288*x^2 + 72*x^3 - 144))/9 - (352*x)/3 - (32*x^3)/9 - (20*x^4)/9 + 48,x)

[Out]

-(4*(x - 2)^2*(9*exp(x^2) - 27*x + 6*x^2 + x^3))/9

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sympy [B]  time = 0.12, size = 44, normalized size = 1.52 \begin {gather*} - \frac {4 x^{5}}{9} - \frac {8 x^{4}}{9} + \frac {188 x^{3}}{9} - \frac {176 x^{2}}{3} + 48 x + \left (- 4 x^{2} + 16 x - 16\right ) e^{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-72*x**3+288*x**2-360*x+144)*exp(x**2)-20/9*x**4-32/9*x**3+188/3*x**2-352/3*x+48,x)

[Out]

-4*x**5/9 - 8*x**4/9 + 188*x**3/9 - 176*x**2/3 + 48*x + (-4*x**2 + 16*x - 16)*exp(x**2)

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