3.36.58 \(\int (311700512 x+46515384 x^2-60477948 x^3-3731570 x^4+2321298 x^5-8694 x^6+8 x^7+e^x (32 x+40 x^2+12 x^3+x^4)+e^{x/2} (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6)) \, dx\)

Optimal. Leaf size=30 \[ x^2 (4+x)^2 \left (4+e^{x/2}+625 (-5+x)-x^2\right )^2 \]

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Rubi [B]  time = 0.40, antiderivative size = 117, normalized size of antiderivative = 3.90, number of steps used = 46, number of rules used = 3, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2196, 2176, 2194} \begin {gather*} x^8-1242 x^7-2 e^{x/2} x^6+386883 x^6+1234 e^{x/2} x^5-746314 x^5+3726 e^{x/2} x^4+e^x x^4-15119487 x^4-29936 e^{x/2} x^3+8 e^x x^3+15505128 x^3-99872 e^{x/2} x^2+16 e^x x^2+155850256 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[311700512*x + 46515384*x^2 - 60477948*x^3 - 3731570*x^4 + 2321298*x^5 - 8694*x^6 + 8*x^7 + E^x*(32*x + 40*
x^2 + 12*x^3 + x^4) + E^(x/2)*(-199744*x - 139744*x^2 - 64*x^3 + 8033*x^4 + 605*x^5 - x^6),x]

[Out]

155850256*x^2 - 99872*E^(x/2)*x^2 + 16*E^x*x^2 + 15505128*x^3 - 29936*E^(x/2)*x^3 + 8*E^x*x^3 - 15119487*x^4 +
3726*E^(x/2)*x^4 + E^x*x^4 - 746314*x^5 + 1234*E^(x/2)*x^5 + 386883*x^6 - 2*E^(x/2)*x^6 - 1242*x^7 + x^8

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=155850256 x^2+15505128 x^3-15119487 x^4-746314 x^5+386883 x^6-1242 x^7+x^8+\int e^x \left (32 x+40 x^2+12 x^3+x^4\right ) \, dx+\int e^{x/2} \left (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6\right ) \, dx\\ &=155850256 x^2+15505128 x^3-15119487 x^4-746314 x^5+386883 x^6-1242 x^7+x^8+\int \left (32 e^x x+40 e^x x^2+12 e^x x^3+e^x x^4\right ) \, dx+\int \left (-199744 e^{x/2} x-139744 e^{x/2} x^2-64 e^{x/2} x^3+8033 e^{x/2} x^4+605 e^{x/2} x^5-e^{x/2} x^6\right ) \, dx\\ &=155850256 x^2+15505128 x^3-15119487 x^4-746314 x^5+386883 x^6-1242 x^7+x^8+12 \int e^x x^3 \, dx+32 \int e^x x \, dx+40 \int e^x x^2 \, dx-64 \int e^{x/2} x^3 \, dx+605 \int e^{x/2} x^5 \, dx+8033 \int e^{x/2} x^4 \, dx-139744 \int e^{x/2} x^2 \, dx-199744 \int e^{x/2} x \, dx+\int e^x x^4 \, dx-\int e^{x/2} x^6 \, dx\\ &=-399488 e^{x/2} x+32 e^x x+155850256 x^2-279488 e^{x/2} x^2+40 e^x x^2+15505128 x^3-128 e^{x/2} x^3+12 e^x x^3-15119487 x^4+16066 e^{x/2} x^4+e^x x^4-746314 x^5+1210 e^{x/2} x^5+386883 x^6-2 e^{x/2} x^6-1242 x^7+x^8-4 \int e^x x^3 \, dx+12 \int e^{x/2} x^5 \, dx-32 \int e^x \, dx-36 \int e^x x^2 \, dx-80 \int e^x x \, dx+384 \int e^{x/2} x^2 \, dx-6050 \int e^{x/2} x^4 \, dx-64264 \int e^{x/2} x^3 \, dx+399488 \int e^{x/2} \, dx+558976 \int e^{x/2} x \, dx\\ &=798976 e^{x/2}-32 e^x+718464 e^{x/2} x-48 e^x x+155850256 x^2-278720 e^{x/2} x^2+4 e^x x^2+15505128 x^3-128656 e^{x/2} x^3+8 e^x x^3-15119487 x^4+3966 e^{x/2} x^4+e^x x^4-746314 x^5+1234 e^{x/2} x^5+386883 x^6-2 e^{x/2} x^6-1242 x^7+x^8+12 \int e^x x^2 \, dx+72 \int e^x x \, dx+80 \int e^x \, dx-120 \int e^{x/2} x^4 \, dx-1536 \int e^{x/2} x \, dx+48400 \int e^{x/2} x^3 \, dx+385584 \int e^{x/2} x^2 \, dx-1117952 \int e^{x/2} \, dx\\ &=-1436928 e^{x/2}+48 e^x+715392 e^{x/2} x+24 e^x x+155850256 x^2+492448 e^{x/2} x^2+16 e^x x^2+15505128 x^3-31856 e^{x/2} x^3+8 e^x x^3-15119487 x^4+3726 e^{x/2} x^4+e^x x^4-746314 x^5+1234 e^{x/2} x^5+386883 x^6-2 e^{x/2} x^6-1242 x^7+x^8-24 \int e^x x \, dx-72 \int e^x \, dx+960 \int e^{x/2} x^3 \, dx+3072 \int e^{x/2} \, dx-290400 \int e^{x/2} x^2 \, dx-1542336 \int e^{x/2} x \, dx\\ &=-1430784 e^{x/2}-24 e^x-2369280 e^{x/2} x+155850256 x^2-88352 e^{x/2} x^2+16 e^x x^2+15505128 x^3-29936 e^{x/2} x^3+8 e^x x^3-15119487 x^4+3726 e^{x/2} x^4+e^x x^4-746314 x^5+1234 e^{x/2} x^5+386883 x^6-2 e^{x/2} x^6-1242 x^7+x^8+24 \int e^x \, dx-5760 \int e^{x/2} x^2 \, dx+1161600 \int e^{x/2} x \, dx+3084672 \int e^{x/2} \, dx\\ &=4738560 e^{x/2}-46080 e^{x/2} x+155850256 x^2-99872 e^{x/2} x^2+16 e^x x^2+15505128 x^3-29936 e^{x/2} x^3+8 e^x x^3-15119487 x^4+3726 e^{x/2} x^4+e^x x^4-746314 x^5+1234 e^{x/2} x^5+386883 x^6-2 e^{x/2} x^6-1242 x^7+x^8+23040 \int e^{x/2} x \, dx-2323200 \int e^{x/2} \, dx\\ &=92160 e^{x/2}[phone] x^2-99872 e^{x/2} x^2+16 e^x x^2+15505128 x^3-29936 e^{x/2} x^3+8 e^x x^3-15119487 x^4+3726 e^{x/2} x^4+e^x x^4-746314 x^5+1234 e^{x/2} x^5+386883 x^6-2 e^{x/2} x^6-1242 x^7+x^8-46080 \int e^{x/2} \, dx\\ &=155850256 x^2-99872 e^{x/2} x^2+16 e^x x^2+15505128 x^3-29936 e^{x/2} x^3+8 e^x x^3-15119487 x^4+3726 e^{x/2} x^4+e^x x^4-746314 x^5+1234 e^{x/2} x^5+386883 x^6-2 e^{x/2} x^6-1242 x^7+x^8\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 28, normalized size = 0.93 \begin {gather*} x^2 (4+x)^2 \left (3121-e^{x/2}-625 x+x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[311700512*x + 46515384*x^2 - 60477948*x^3 - 3731570*x^4 + 2321298*x^5 - 8694*x^6 + 8*x^7 + E^x*(32*x
+ 40*x^2 + 12*x^3 + x^4) + E^(x/2)*(-199744*x - 139744*x^2 - 64*x^3 + 8033*x^4 + 605*x^5 - x^6),x]

[Out]

x^2*(4 + x)^2*(3121 - E^(x/2) - 625*x + x^2)^2

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fricas [B]  time = 0.54, size = 81, normalized size = 2.70 \begin {gather*} x^{8} - 1242 \, x^{7} + 386883 \, x^{6} - 746314 \, x^{5} - 15119487 \, x^{4} + 15505128 \, x^{3} + 155850256 \, x^{2} - 2 \, {\left (x^{6} - 617 \, x^{5} - 1863 \, x^{4} + 14968 \, x^{3} + 49936 \, x^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+12*x^3+40*x^2+32*x)*exp(1/2*x)^2+(-x^6+605*x^5+8033*x^4-64*x^3-139744*x^2-199744*x)*exp(1/2*x)+
8*x^7-8694*x^6+2321298*x^5-3731570*x^4-60477948*x^3+46515384*x^2+311700512*x,x, algorithm="fricas")

[Out]

x^8 - 1242*x^7 + 386883*x^6 - 746314*x^5 - 15119487*x^4 + 15505128*x^3 + 155850256*x^2 - 2*(x^6 - 617*x^5 - 18
63*x^4 + 14968*x^3 + 49936*x^2)*e^(1/2*x) + (x^4 + 8*x^3 + 16*x^2)*e^x

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giac [B]  time = 0.17, size = 81, normalized size = 2.70 \begin {gather*} x^{8} - 1242 \, x^{7} + 386883 \, x^{6} - 746314 \, x^{5} - 15119487 \, x^{4} + 15505128 \, x^{3} + 155850256 \, x^{2} - 2 \, {\left (x^{6} - 617 \, x^{5} - 1863 \, x^{4} + 14968 \, x^{3} + 49936 \, x^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+12*x^3+40*x^2+32*x)*exp(1/2*x)^2+(-x^6+605*x^5+8033*x^4-64*x^3-139744*x^2-199744*x)*exp(1/2*x)+
8*x^7-8694*x^6+2321298*x^5-3731570*x^4-60477948*x^3+46515384*x^2+311700512*x,x, algorithm="giac")

[Out]

x^8 - 1242*x^7 + 386883*x^6 - 746314*x^5 - 15119487*x^4 + 15505128*x^3 + 155850256*x^2 - 2*(x^6 - 617*x^5 - 18
63*x^4 + 14968*x^3 + 49936*x^2)*e^(1/2*x) + (x^4 + 8*x^3 + 16*x^2)*e^x

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maple [B]  time = 0.09, size = 83, normalized size = 2.77

method	result	size
risch	\(\left (x^{4}+8 x^{3}+16 x^{2}\right ) {\mathrm e}^{x}+\left (-2 x^{6}+1234 x^{5}+3726 x^{4}-29936 x^{3}-99872 x^{2}\right ) {\mathrm e}^{\frac {x}{2}}+x^{8}-1242 x^{7}+386883 x^{6}-746314 x^{5}-15119487 x^{4}[phone] x^{3}[phone] x^{2}\)	\(83\)
derivativedivides	\(x^{8}-1242 x^{7}+386883 x^{6}-746314 x^{5}-15119487 x^{4}[phone] x^{3}[phone] x^{2}+{\mathrm e}^{x} x^{4}+8 \,{\mathrm e}^{x} x^{3}+16 \,{\mathrm e}^{x} x^{2}-99872 x^{2} {\mathrm e}^{\frac {x}{2}}-29936 \,{\mathrm e}^{\frac {x}{2}} x^{3}+3726 \,{\mathrm e}^{\frac {x}{2}} x^{4}+1234 \,{\mathrm e}^{\frac {x}{2}} x^{5}-2 \,{\mathrm e}^{\frac {x}{2}} x^{6}\)	\(112\)
default	\(x^{8}-1242 x^{7}+386883 x^{6}-746314 x^{5}-15119487 x^{4}[phone] x^{3}[phone] x^{2}+{\mathrm e}^{x} x^{4}+8 \,{\mathrm e}^{x} x^{3}+16 \,{\mathrm e}^{x} x^{2}-99872 x^{2} {\mathrm e}^{\frac {x}{2}}-29936 \,{\mathrm e}^{\frac {x}{2}} x^{3}+3726 \,{\mathrm e}^{\frac {x}{2}} x^{4}+1234 \,{\mathrm e}^{\frac {x}{2}} x^{5}-2 \,{\mathrm e}^{\frac {x}{2}} x^{6}\)	\(112\)
norman	\(x^{8}-1242 x^{7}+386883 x^{6}-746314 x^{5}-15119487 x^{4}[phone] x^{3}[phone] x^{2}+{\mathrm e}^{x} x^{4}+8 \,{\mathrm e}^{x} x^{3}+16 \,{\mathrm e}^{x} x^{2}-99872 x^{2} {\mathrm e}^{\frac {x}{2}}-29936 \,{\mathrm e}^{\frac {x}{2}} x^{3}+3726 \,{\mathrm e}^{\frac {x}{2}} x^{4}+1234 \,{\mathrm e}^{\frac {x}{2}} x^{5}-2 \,{\mathrm e}^{\frac {x}{2}} x^{6}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+12*x^3+40*x^2+32*x)*exp(1/2*x)^2+(-x^6+605*x^5+8033*x^4-64*x^3-139744*x^2-199744*x)*exp(1/2*x)+8*x^7-
8694*x^6+2321298*x^5-3731570*x^4-60477948*x^3+46515384*x^2+311700512*x,x,method=_RETURNVERBOSE)

[Out]

(x^4+8*x^3+16*x^2)*exp(x)+(-2*x^6+1234*x^5+3726*x^4-29936*x^3-99872*x^2)*exp(1/2*x)+x^8-1242*x^7+386883*x^6-74
6314*x^5-15119487*x^4+15505128*x^3+155850256*x^2

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maxima [B]  time = 0.68, size = 81, normalized size = 2.70 \begin {gather*} x^{8} - 1242 \, x^{7} + 386883 \, x^{6} - 746314 \, x^{5} - 15119487 \, x^{4} + 15505128 \, x^{3} + 155850256 \, x^{2} - 2 \, {\left (x^{6} - 617 \, x^{5} - 1863 \, x^{4} + 14968 \, x^{3} + 49936 \, x^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+12*x^3+40*x^2+32*x)*exp(1/2*x)^2+(-x^6+605*x^5+8033*x^4-64*x^3-139744*x^2-199744*x)*exp(1/2*x)+
8*x^7-8694*x^6+2321298*x^5-3731570*x^4-60477948*x^3+46515384*x^2+311700512*x,x, algorithm="maxima")

[Out]

x^8 - 1242*x^7 + 386883*x^6 - 746314*x^5 - 15119487*x^4 + 15505128*x^3 + 155850256*x^2 - 2*(x^6 - 617*x^5 - 18
63*x^4 + 14968*x^3 + 49936*x^2)*e^(1/2*x) + (x^4 + 8*x^3 + 16*x^2)*e^x

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mupad [B]  time = 2.20, size = 25, normalized size = 0.83 \begin {gather*} x^2\,{\left (x+4\right )}^2\,{\left (625\,x+{\mathrm {e}}^{x/2}-x^2-3121\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(311700512*x - exp(x/2)*(199744*x + 139744*x^2 + 64*x^3 - 8033*x^4 - 605*x^5 + x^6) + exp(x)*(32*x + 40*x^2
+ 12*x^3 + x^4) + 46515384*x^2 - 60477948*x^3 - 3731570*x^4 + 2321298*x^5 - 8694*x^6 + 8*x^7,x)

[Out]

x^2*(x + 4)^2*(625*x + exp(x/2) - x^2 - 3121)^2

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sympy [B]  time = 0.15, size = 80, normalized size = 2.67 \begin {gather*} x^{8} - 1242 x^{7} + 386883 x^{6} - 746314 x^{5} - 15119487 x^{4} + 15505128 x^{3} + 155850256 x^{2} + \left (x^{4} + 8 x^{3} + 16 x^{2}\right ) e^{x} + \left (- 2 x^{6} + 1234 x^{5} + 3726 x^{4} - 29936 x^{3} - 99872 x^{2}\right ) e^{\frac {x}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+12*x**3+40*x**2+32*x)*exp(1/2*x)**2+(-x**6+605*x**5+8033*x**4-64*x**3-139744*x**2-199744*x)*ex
p(1/2*x)+8*x**7-8694*x**6+2321298*x**5-3731570*x**4-60477948*x**3+46515384*x**2+311700512*x,x)

[Out]

x**8 - 1242*x**7 + 386883*x**6 - 746314*x**5 - 15119487*x**4 + 15505128*x**3 + 155850256*x**2 + (x**4 + 8*x**3
+ 16*x**2)*exp(x) + (-2*x**6 + 1234*x**5 + 3726*x**4 - 29936*x**3 - 99872*x**2)*exp(x/2)

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