\(\int (9378906250 x^9+e^{6+2 x^2} (7503125000 x^7+3751562500 x^9)+e^{3+x^2} (-16882031250 x^8-3751562500 x^{10})) \, dx\) [644]

Optimal result

Integrand size = 46, antiderivative size = 18 \[ \int \left (9378906250 x^9+e^{6+2 x^2} \left (7503125000 x^7+3751562500 x^9\right )+e^{3+x^2} \left (-16882031250 x^8-3751562500 x^{10}\right )\right ) \, dx=937890625 x^8 \left (-e^{3+x^2}+x\right )^2 \]

[Out]

937890625*(x-exp(x^2)*exp(3))^2*x^8

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.19 (sec) , antiderivative size = 188, normalized size of antiderivative = 10.44, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1607, 2258, 2243, 2240, 2249, 2235, 2250} \[ \int \left (9378906250 x^9+e^{6+2 x^2} \left (7503125000 x^7+3751562500 x^9\right )+e^{3+x^2} \left (-16882031250 x^8-3751562500 x^{10}\right )\right ) \, dx=-\frac {886306640625}{16} e^3 \sqrt {\pi } \text {erfi}(x)[phone] x^{10}[phone] e^{2 x^2+6} x^2+\frac {886306640625}{8} e^{x^2+3} x-\frac {2813671875}{2} e^{2 x^2+6}+\frac {1875781250 e^3 x^{11} \Gamma \left (\frac {11}{2},-x^2\right )}{\left (-x^2\right )^{11/2}}-8441015625 e^{x^2+3} x^7+1875781250 e^{2 x^2+6} x^6+\frac {59087109375}{2} e^{x^2+3} x^5-2813671875 e^{2 x^2+6} x^4-\frac {295435546875}{4} e^{x^2+3} x^3+\frac {937890625}{2} e^{2 x^2+6} \left (2 x^8-4 x^6+6 x^4-6 x^2+3\right ) \]

[In]

Int[9378906250*x^9 + E^(6 + 2*x^2)*(7503125000*x^7 + 3751562500*x^9) + E^(3 + x^2)*(-16882031250*x^8 - 3751562
500*x^10),x]

[Out]

(-2813671875*E^(6 + 2*x^2))/2 + (886306640625*E^(3 + x^2)*x)/8 + 2813671875*E^(6 + 2*x^2)*x^2 - (295435546875*
E^(3 + x^2)*x^3)/4 - 2813671875*E^(6 + 2*x^2)*x^4 + (59087109375*E^(3 + x^2)*x^5)/2 + 1875781250*E^(6 + 2*x^2)
*x^6 - 8441015625*E^(3 + x^2)*x^7 + 937890625*x^10 + (937890625*E^(6 + 2*x^2)*(3 - 6*x^2 + 6*x^4 - 4*x^6 + 2*x
^8))/2 - (886306640625*E^3*Sqrt[Pi]*Erfi[x])/16 + (1875781250*E^3*x^11*Gamma[11/2, -x^2])/(-x^2)^(11/2)

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2235

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 2240

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 2243

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 2249

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{p = Simplify
[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n*((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Lo
g[F]]]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] &&  !TrueQ[$UseGamma]

Rule 2250

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

Rule 2258

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{align*} \text {integral}& = 937890625 x^{10}+\int e^{6+2 x^2} \left (7503125000 x^7+3751562500 x^9\right ) \, dx+\int e^{3+x^2} \left (-16882031250 x^8-3751562500 x^{10}\right ) \, dx \\ & = 937890625 x^{10}+\int e^{3+x^2} x^8 \left (-16882031250-3751562500 x^2\right ) \, dx+\int e^{6+2 x^2} x^7 \left (7503125000+3751562500 x^2\right ) \, dx \\ & = 937890625 x^{10}+\int \left (7503125000 e^{6+2 x^2} x^7+3751562500 e^{6+2 x^2} x^9\right ) \, dx+\int \left (-16882031250 e^{3+x^2} x^8-3751562500 e^{3+x^2} x^{10}\right ) \, dx \\ & = 937890625 x^{10}[phone] \int e^{6+2 x^2} x^9 \, dx-3751562500 \int e^{3+x^2} x^{10} \, dx+7503125000 \int e^{6+2 x^2} x^7 \, dx-16882031250 \int e^{3+x^2} x^8 \, dx \\ & = 1875781250 e^{6+2 x^2} x^6-8441015625 e^{3+x^2} x^7+937890625 x^{10}+\frac {937890625}{2} e^{6+2 x^2} \left (3-6 x^2+6 x^4-4 x^6+2 x^8\right )+\frac {1875781250 e^3 x^{11} \Gamma \left (\frac {11}{2},-x^2\right )}{\left (-x^2\right )^{11/2}}-11254687500 \int e^{6+2 x^2} x^5 \, dx+59087109375 \int e^{3+x^2} x^6 \, dx \\ & = -2813671875 e^{6+2 x^2} x^4+\frac {59087109375}{2} e^{3+x^2} x^5+1875781250 e^{6+2 x^2} x^6-8441015625 e^{3+x^2} x^7+937890625 x^{10}+\frac {937890625}{2} e^{6+2 x^2} \left (3-6 x^2+6 x^4-4 x^6+2 x^8\right )+\frac {1875781250 e^3 x^{11} \Gamma \left (\frac {11}{2},-x^2\right )}{\left (-x^2\right )^{11/2}}[phone] \int e^{6+2 x^2} x^3 \, dx-\frac {295435546875}{2} \int e^{3+x^2} x^4 \, dx \\ & = 2813671875 e^{6+2 x^2} x^2-\frac {295435546875}{4} e^{3+x^2} x^3-2813671875 e^{6+2 x^2} x^4+\frac {59087109375}{2} e^{3+x^2} x^5+1875781250 e^{6+2 x^2} x^6-8441015625 e^{3+x^2} x^7+937890625 x^{10}+\frac {937890625}{2} e^{6+2 x^2} \left (3-6 x^2+6 x^4-4 x^6+2 x^8\right )+\frac {1875781250 e^3 x^{11} \Gamma \left (\frac {11}{2},-x^2\right )}{\left (-x^2\right )^{11/2}}-5627343750 \int e^{6+2 x^2} x \, dx+\frac {886306640625}{4} \int e^{3+x^2} x^2 \, dx \\ & = -\frac {2813671875}{2} e^{6+2 x^2}+\frac {886306640625}{8} e^{3+x^2} x+2813671875 e^{6+2 x^2} x^2-\frac {295435546875}{4} e^{3+x^2} x^3-2813671875 e^{6+2 x^2} x^4+\frac {59087109375}{2} e^{3+x^2} x^5+1875781250 e^{6+2 x^2} x^6-8441015625 e^{3+x^2} x^7+937890625 x^{10}+\frac {937890625}{2} e^{6+2 x^2} \left (3-6 x^2+6 x^4-4 x^6+2 x^8\right )+\frac {1875781250 e^3 x^{11} \Gamma \left (\frac {11}{2},-x^2\right )}{\left (-x^2\right )^{11/2}}-\frac {886306640625}{8} \int e^{3+x^2} \, dx \\ & = -\frac {2813671875}{2} e^{6+2 x^2}+\frac {886306640625}{8} e^{3+x^2} x+2813671875 e^{6+2 x^2} x^2-\frac {295435546875}{4} e^{3+x^2} x^3-2813671875 e^{6+2 x^2} x^4+\frac {59087109375}{2} e^{3+x^2} x^5+1875781250 e^{6+2 x^2} x^6-8441015625 e^{3+x^2} x^7+937890625 x^{10}+\frac {937890625}{2} e^{6+2 x^2} \left (3-6 x^2+6 x^4-4 x^6+2 x^8\right )-\frac {886306640625}{16} e^3 \sqrt {\pi } \text {erfi}(x)+\frac {1875781250 e^3 x^{11} \Gamma \left (\frac {11}{2},-x^2\right )}{\left (-x^2\right )^{11/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.53 (sec) , antiderivative size = 120, normalized size of antiderivative = 6.67 \[ \int \left (9378906250 x^9+e^{6+2 x^2} \left (7503125000 x^7+3751562500 x^9\right )+e^{3+x^2} \left (-16882031250 x^8-3751562500 x^{10}\right )\right ) \, dx=\frac {937890625 \left (2 x \left (8 x^{10}+4 e^{6+2 x^2} \left (-3+6 x^2-6 x^4+4 x^6\right )-9 e^{3+x^2} x \left (-105+70 x^2-28 x^4+8 x^6\right )\right )-945 e^3 \sqrt {\pi } x \text {erfi}(x)+e^6 x \Gamma \left (5,-2 x^2\right )+32 e^3 \sqrt {-x^2} \Gamma \left (\frac {11}{2},-x^2\right )\right )}{16 x} \]

[In]

Integrate[9378906250*x^9 + E^(6 + 2*x^2)*(7503125000*x^7 + 3751562500*x^9) + E^(3 + x^2)*(-16882031250*x^8 - 3
751562500*x^10),x]

[Out]

(937890625*(2*x*(8*x^10 + 4*E^(6 + 2*x^2)*(-3 + 6*x^2 - 6*x^4 + 4*x^6) - 9*E^(3 + x^2)*x*(-105 + 70*x^2 - 28*x
^4 + 8*x^6)) - 945*E^3*Sqrt[Pi]*x*Erfi[x] + E^6*x*Gamma[5, -2*x^2] + 32*E^3*Sqrt[-x^2]*Gamma[11/2, -x^2]))/(16
*x)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72

method	result	size
risch	\(937890625 x^{8} {\mathrm e}^{2 x^{2}+6}-1875781250 x^{9} {\mathrm e}^{x^{2}+3}[phone] x^{10}\)	\(31\)
default	\(937890625 \,{\mathrm e}^{6} {\mathrm e}^{2 x^{2}} x^{8}-1875781250 \,{\mathrm e}^{3} {\mathrm e}^{x^{2}} x^{9}[phone] x^{10}\)	\(33\)
norman	\(937890625 \,{\mathrm e}^{6} {\mathrm e}^{2 x^{2}} x^{8}-1875781250 \,{\mathrm e}^{3} {\mathrm e}^{x^{2}} x^{9}[phone] x^{10}\)	\(33\)
parallelrisch	\(937890625 \,{\mathrm e}^{6} {\mathrm e}^{2 x^{2}} x^{8}-1875781250 \,{\mathrm e}^{3} {\mathrm e}^{x^{2}} x^{9}[phone] x^{10}\)	\(33\)
parts	\(937890625 \,{\mathrm e}^{6} {\mathrm e}^{2 x^{2}} x^{8}-1875781250 \,{\mathrm e}^{3} {\mathrm e}^{x^{2}} x^{9}[phone] x^{10}\)	\(33\)

[In]

int((3751562500*x^9+7503125000*x^7)*exp(3)^2*exp(x^2)^2+(-3751562500*x^10-16882031250*x^8)*exp(3)*exp(x^2)+937
8906250*x^9,x,method=_RETURNVERBOSE)

[Out]

937890625*x^8*exp(2*x^2+6)-1875781250*x^9*exp(x^2+3)[phone]*x^10

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \left (9378906250 x^9+e^{6+2 x^2} \left (7503125000 x^7+3751562500 x^9\right )+e^{3+x^2} \left (-16882031250 x^8-3751562500 x^{10}\right )\right ) \, dx=937890625 \, x^{10} - 1875781250 \, x^{9} e^{\left (x^{2} + 3\right )} + 937890625 \, x^{8} e^{\left (2 \, x^{2} + 6\right )} \]

[In]

integrate((3751562500*x^9+7503125000*x^7)*exp(3)^2*exp(x^2)^2+(-3751562500*x^10-16882031250*x^8)*exp(3)*exp(x^
2)[phone]*x^9,x, algorithm="fricas")

[Out]

937890625*x^10 - 1875781250*x^9*e^(x^2 + 3) + 937890625*x^8*e^(2*x^2 + 6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \left (9378906250 x^9+e^{6+2 x^2} \left (7503125000 x^7+3751562500 x^9\right )+e^{3+x^2} \left (-16882031250 x^8-3751562500 x^{10}\right )\right ) \, dx=937890625 x^{10} - 1875781250 x^{9} e^{3} e^{x^{2}} + 937890625 x^{8} e^{6} e^{2 x^{2}} \]

[In]

integrate((3751562500*x**9+7503125000*x**7)*exp(3)**2*exp(x**2)**2+(-3751562500*x**10-16882031250*x**8)*exp(3)
*exp(x**2)[phone]*x**9,x)

[Out]

937890625*x**10 - 1875781250*x**9*exp(3)*exp(x**2) + 937890625*x**8*exp(6)*exp(2*x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (17) = 34\).

Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 5.11 \[ \int \left (9378906250 x^9+e^{6+2 x^2} \left (7503125000 x^7+3751562500 x^9\right )+e^{3+x^2} \left (-16882031250 x^8-3751562500 x^{10}\right )\right ) \, dx=937890625 \, x^{10} + 937890625 \, x^{8} e^{\left (2 \, x^{2} + 6\right )} - \frac {937890625}{8} \, {\left (16 \, x^{9} e^{3} - 72 \, x^{7} e^{3} + 252 \, x^{5} e^{3} - 630 \, x^{3} e^{3} + 945 \, x e^{3}\right )} e^{\left (x^{2}\right )} - \frac {8441015625}{8} \, {\left (8 \, x^{7} e^{3} - 28 \, x^{5} e^{3} + 70 \, x^{3} e^{3} - 105 \, x e^{3}\right )} e^{\left (x^{2}\right )} \]

[In]

integrate((3751562500*x^9+7503125000*x^7)*exp(3)^2*exp(x^2)^2+(-3751562500*x^10-16882031250*x^8)*exp(3)*exp(x^
2)[phone]*x^9,x, algorithm="maxima")

[Out]

937890625*x^10 + 937890625*x^8*e^(2*x^2 + 6) - 937890625/8*(16*x^9*e^3 - 72*x^7*e^3 + 252*x^5*e^3 - 630*x^3*e^
3 + 945*x*e^3)*e^(x^2) - 8441015625/8*(8*x^7*e^3 - 28*x^5*e^3 + 70*x^3*e^3 - 105*x*e^3)*e^(x^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.83 \[ \int \left (9378906250 x^9+e^{6+2 x^2} \left (7503125000 x^7+3751562500 x^9\right )+e^{3+x^2} \left (-16882031250 x^8-3751562500 x^{10}\right )\right ) \, dx=937890625 \, x^{10} - 1875781250 \, x^{9} e^{\left (x^{2} + 3\right )} + 937890625 \, {\left ({\left (x^{2} + 3\right )}^{4} - 12 \, {\left (x^{2} + 3\right )}^{3} + 54 \, {\left (x^{2} + 3\right )}^{2} - 108 \, x^{2} - 270\right )} e^{\left (2 \, x^{2} + 6\right )} + 25323046875 \, e^{\left (2 \, x^{2} + 6\right )} \]

[In]

integrate((3751562500*x^9+7503125000*x^7)*exp(3)^2*exp(x^2)^2+(-3751562500*x^10-16882031250*x^8)*exp(3)*exp(x^
2)[phone]*x^9,x, algorithm="giac")

[Out]

937890625*x^10 - 1875781250*x^9*e^(x^2 + 3) + 937890625*((x^2 + 3)^4 - 12*(x^2 + 3)^3 + 54*(x^2 + 3)^2 - 108*x
^2 - 270)*e^(2*x^2 + 6) + 25323046875*e^(2*x^2 + 6)

Mupad [B] (verification not implemented)

Time = 9.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \left (9378906250 x^9+e^{6+2 x^2} \left (7503125000 x^7+3751562500 x^9\right )+e^{3+x^2} \left (-16882031250 x^8-3751562500 x^{10}\right )\right ) \, dx=937890625\,x^8\,{\left (x-{\mathrm {e}}^{x^2+3}\right )}^2 \]

[In]

int(9378906250*x^9 - exp(x^2)*exp(3)*(16882031250*x^8 + 3751562500*x^10) + exp(6)*exp(2*x^2)*(7503125000*x^7 +
3751562500*x^9),x)

[Out]

937890625*x^8*(x - exp(x^2 + 3))^2