3.9.23 \(\int \frac {1}{25} e^{16 x^6-32 x^7+16 x^8+e^{10} (x^2-2 x^3+x^4)+e^{15/2} (-8 x^3+16 x^4-8 x^5)+e^5 (24 x^4-48 x^5+24 x^6)+e^{5/2} (-32 x^5+64 x^6-32 x^7)} (1+96 x^6-224 x^7+128 x^8+e^{10} (2 x^2-6 x^3+4 x^4)+e^{15/2} (-24 x^3+64 x^4-40 x^5)+e^5 (96 x^4-240 x^5+144 x^6)+e^{5/2} (-160 x^5+384 x^6-224 x^7)) \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{25} e^{\left (e^{5/2}-2 x\right )^4 \left (-x+x^2\right )^2} x \]

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Rubi [B]  time = 4.70, antiderivative size = 295, normalized size of antiderivative = 10.54, number of steps used = 2, number of rules used = 2, integrand size = 203, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {12, 2288} \begin {gather*} \frac {\left (64 x^8-112 x^7+48 x^6-16 e^{5/2} \left (7 x^7-12 x^6+5 x^5\right )+24 e^5 \left (3 x^6-5 x^5+2 x^4\right )-4 e^{15/2} \left (5 x^5-8 x^4+3 x^3\right )+e^{10} \left (2 x^4-3 x^3+x^2\right )\right ) \exp \left (16 x^8-32 x^7+16 x^6-32 e^{5/2} \left (x^7-2 x^6+x^5\right )+24 e^5 \left (x^6-2 x^5+x^4\right )-8 e^{15/2} \left (x^5-2 x^4+x^3\right )+e^{10} \left (x^4-2 x^3+x^2\right )\right )}{25 \left (64 x^7-112 x^6+48 x^5+e^{10} \left (2 x^3-3 x^2+x\right )-16 e^{5/2} \left (7 x^6-12 x^5+5 x^4\right )+24 e^5 \left (3 x^5-5 x^4+2 x^3\right )-4 e^{15/2} \left (5 x^4-8 x^3+3 x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(16*x^6 - 32*x^7 + 16*x^8 + E^10*(x^2 - 2*x^3 + x^4) + E^(15/2)*(-8*x^3 + 16*x^4 - 8*x^5) + E^5*(24*x^4
- 48*x^5 + 24*x^6) + E^(5/2)*(-32*x^5 + 64*x^6 - 32*x^7))*(1 + 96*x^6 - 224*x^7 + 128*x^8 + E^10*(2*x^2 - 6*x
^3 + 4*x^4) + E^(15/2)*(-24*x^3 + 64*x^4 - 40*x^5) + E^5*(96*x^4 - 240*x^5 + 144*x^6) + E^(5/2)*(-160*x^5 + 38
4*x^6 - 224*x^7)))/25,x]

[Out]

(E^(16*x^6 - 32*x^7 + 16*x^8 + E^10*(x^2 - 2*x^3 + x^4) - 8*E^(15/2)*(x^3 - 2*x^4 + x^5) + 24*E^5*(x^4 - 2*x^5
+ x^6) - 32*E^(5/2)*(x^5 - 2*x^6 + x^7))*(48*x^6 - 112*x^7 + 64*x^8 + E^10*(x^2 - 3*x^3 + 2*x^4) - 4*E^(15/2)
*(3*x^3 - 8*x^4 + 5*x^5) + 24*E^5*(2*x^4 - 5*x^5 + 3*x^6) - 16*E^(5/2)*(5*x^5 - 12*x^6 + 7*x^7)))/(25*(48*x^5
- 112*x^6 + 64*x^7 + E^10*(x - 3*x^2 + 2*x^3) - 4*E^(15/2)*(3*x^2 - 8*x^3 + 5*x^4) + 24*E^5*(2*x^3 - 5*x^4 + 3
*x^5) - 16*E^(5/2)*(5*x^4 - 12*x^5 + 7*x^6)))

Rule 12

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

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} &=\frac {1}{25} \int \exp \left (16 x^6-32 x^7+16 x^8+e^{10} \left (x^2-2 x^3+x^4\right )+e^{15/2} \left (-8 x^3+16 x^4-8 x^5\right )+e^5 \left (24 x^4-48 x^5+24 x^6\right )+e^{5/2} \left (-32 x^5+64 x^6-32 x^7\right )\right ) \left (1+96 x^6-224 x^7+128 x^8+e^{10} \left (2 x^2-6 x^3+4 x^4\right )+e^{15/2} \left (-24 x^3+64 x^4-40 x^5\right )+e^5 \left (96 x^4-240 x^5+144 x^6\right )+e^{5/2} \left (-160 x^5+384 x^6-224 x^7\right )\right ) \, dx\\ &=\frac {\exp \left (16 x^6-32 x^7+16 x^8+e^{10} \left (x^2-2 x^3+x^4\right )-8 e^{15/2} \left (x^3-2 x^4+x^5\right )+24 e^5 \left (x^4-2 x^5+x^6\right )-32 e^{5/2} \left (x^5-2 x^6+x^7\right )\right ) \left (48 x^6-112 x^7+64 x^8+e^{10} \left (x^2-3 x^3+2 x^4\right )-4 e^{15/2} \left (3 x^3-8 x^4+5 x^5\right )+24 e^5 \left (2 x^4-5 x^5+3 x^6\right )-16 e^{5/2} \left (5 x^5-12 x^6+7 x^7\right )\right )}{25 \left (48 x^5-112 x^6+64 x^7+e^{10} \left (x-3 x^2+2 x^3\right )-4 e^{15/2} \left (3 x^2-8 x^3+5 x^4\right )+24 e^5 \left (2 x^3-5 x^4+3 x^5\right )-16 e^{5/2} \left (5 x^4-12 x^5+7 x^6\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 27, normalized size = 0.96 \begin {gather*} \frac {1}{25} e^{\left (e^{5/2}-2 x\right )^4 (-1+x)^2 x^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(16*x^6 - 32*x^7 + 16*x^8 + E^10*(x^2 - 2*x^3 + x^4) + E^(15/2)*(-8*x^3 + 16*x^4 - 8*x^5) + E^5*(
24*x^4 - 48*x^5 + 24*x^6) + E^(5/2)*(-32*x^5 + 64*x^6 - 32*x^7))*(1 + 96*x^6 - 224*x^7 + 128*x^8 + E^10*(2*x^2
- 6*x^3 + 4*x^4) + E^(15/2)*(-24*x^3 + 64*x^4 - 40*x^5) + E^5*(96*x^4 - 240*x^5 + 144*x^6) + E^(5/2)*(-160*x^
5 + 384*x^6 - 224*x^7)))/25,x]

[Out]

(E^((E^(5/2) - 2*x)^4*(-1 + x)^2*x^2)*x)/25

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fricas [B]  time = 1.24, size = 83, normalized size = 2.96 \begin {gather*} \frac {1}{25} \, x e^{\left (16 \, x^{8} - 32 \, x^{7} + 16 \, x^{6} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{10} - 8 \, {\left (x^{5} - 2 \, x^{4} + x^{3}\right )} e^{\frac {15}{2}} + 24 \, {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} e^{5} - 32 \, {\left (x^{7} - 2 \, x^{6} + x^{5}\right )} e^{\frac {5}{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((4*x^4-6*x^3+2*x^2)*exp(5/2)^4+(-40*x^5+64*x^4-24*x^3)*exp(5/2)^3+(144*x^6-240*x^5+96*x^4)*exp
(5/2)^2+(-224*x^7+384*x^6-160*x^5)*exp(5/2)+128*x^8-224*x^7+96*x^6+1)*exp((x^4-2*x^3+x^2)*exp(5/2)^4+(-8*x^5+1
6*x^4-8*x^3)*exp(5/2)^3+(24*x^6-48*x^5+24*x^4)*exp(5/2)^2+(-32*x^7+64*x^6-32*x^5)*exp(5/2)+16*x^8-32*x^7+16*x^
6),x, algorithm="fricas")

[Out]

1/25*x*e^(16*x^8 - 32*x^7 + 16*x^6 + (x^4 - 2*x^3 + x^2)*e^10 - 8*(x^5 - 2*x^4 + x^3)*e^(15/2) + 24*(x^6 - 2*x
^5 + x^4)*e^5 - 32*(x^7 - 2*x^6 + x^5)*e^(5/2))

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giac [B]  time = 1.14, size = 102, normalized size = 3.64 \begin {gather*} \frac {1}{25} \, x e^{\left (16 \, x^{8} - 32 \, x^{7} e^{\frac {5}{2}} - 32 \, x^{7} + 24 \, x^{6} e^{5} + 64 \, x^{6} e^{\frac {5}{2}} + 16 \, x^{6} - 8 \, x^{5} e^{\frac {15}{2}} - 48 \, x^{5} e^{5} - 32 \, x^{5} e^{\frac {5}{2}} + x^{4} e^{10} + 16 \, x^{4} e^{\frac {15}{2}} + 24 \, x^{4} e^{5} - 2 \, x^{3} e^{10} - 8 \, x^{3} e^{\frac {15}{2}} + x^{2} e^{10}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((4*x^4-6*x^3+2*x^2)*exp(5/2)^4+(-40*x^5+64*x^4-24*x^3)*exp(5/2)^3+(144*x^6-240*x^5+96*x^4)*exp
(5/2)^2+(-224*x^7+384*x^6-160*x^5)*exp(5/2)+128*x^8-224*x^7+96*x^6+1)*exp((x^4-2*x^3+x^2)*exp(5/2)^4+(-8*x^5+1
6*x^4-8*x^3)*exp(5/2)^3+(24*x^6-48*x^5+24*x^4)*exp(5/2)^2+(-32*x^7+64*x^6-32*x^5)*exp(5/2)+16*x^8-32*x^7+16*x^
6),x, algorithm="giac")

[Out]

1/25*x*e^(16*x^8 - 32*x^7*e^(5/2) - 32*x^7 + 24*x^6*e^5 + 64*x^6*e^(5/2) + 16*x^6 - 8*x^5*e^(15/2) - 48*x^5*e^
5 - 32*x^5*e^(5/2) + x^4*e^10 + 16*x^4*e^(15/2) + 24*x^4*e^5 - 2*x^3*e^10 - 8*x^3*e^(15/2) + x^2*e^10)

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maple [B]  time = 0.02, size = 121, normalized size = 4.32 \[\frac {x \,{\mathrm e}^{x^{4} {\mathrm e}^{10}-8 \,{\mathrm e}^{\frac {15}{2}} x^{5}+24 x^{6} {\mathrm e}^{5}-32 \,{\mathrm e}^{\frac {5}{2}} x^{7}+16 x^{8}-2 x^{3} {\mathrm e}^{10}+16 \,{\mathrm e}^{\frac {15}{2}} x^{4}-48 x^{5} {\mathrm e}^{5}+64 \,{\mathrm e}^{\frac {5}{2}} x^{6}-32 x^{7}+x^{2} {\mathrm e}^{10}-8 \,{\mathrm e}^{\frac {15}{2}} x^{3}+24 x^{4} {\mathrm e}^{5}-32 \,{\mathrm e}^{\frac {5}{2}} x^{5}+16 x^{6}}}{25}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*((4*x^4-6*x^3+2*x^2)*exp(5/2)^4+(-40*x^5+64*x^4-24*x^3)*exp(5/2)^3+(144*x^6-240*x^5+96*x^4)*exp(5/2)^
2+(-224*x^7+384*x^6-160*x^5)*exp(5/2)+128*x^8-224*x^7+96*x^6+1)*exp((x^4-2*x^3+x^2)*exp(5/2)^4+(-8*x^5+16*x^4-
8*x^3)*exp(5/2)^3+(24*x^6-48*x^5+24*x^4)*exp(5/2)^2+(-32*x^7+64*x^6-32*x^5)*exp(5/2)+16*x^8-32*x^7+16*x^6),x)

[Out]

1/25*x*exp(exp(5/2)^4*x^4-8*exp(5/2)^3*x^5+24*exp(5/2)^2*x^6-32*exp(5/2)*x^7+16*x^8-2*exp(5/2)^4*x^3+16*exp(5/
2)^3*x^4-48*exp(5/2)^2*x^5+64*exp(5/2)*x^6-32*x^7+exp(5/2)^4*x^2-8*exp(5/2)^3*x^3+24*exp(5/2)^2*x^4-32*exp(5/2
)*x^5+16*x^6)

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maxima [B]  time = 1.21, size = 102, normalized size = 3.64 \begin {gather*} \frac {1}{25} \, x e^{\left (16 \, x^{8} - 32 \, x^{7} e^{\frac {5}{2}} - 32 \, x^{7} + 24 \, x^{6} e^{5} + 64 \, x^{6} e^{\frac {5}{2}} + 16 \, x^{6} - 8 \, x^{5} e^{\frac {15}{2}} - 48 \, x^{5} e^{5} - 32 \, x^{5} e^{\frac {5}{2}} + x^{4} e^{10} + 16 \, x^{4} e^{\frac {15}{2}} + 24 \, x^{4} e^{5} - 2 \, x^{3} e^{10} - 8 \, x^{3} e^{\frac {15}{2}} + x^{2} e^{10}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((4*x^4-6*x^3+2*x^2)*exp(5/2)^4+(-40*x^5+64*x^4-24*x^3)*exp(5/2)^3+(144*x^6-240*x^5+96*x^4)*exp
(5/2)^2+(-224*x^7+384*x^6-160*x^5)*exp(5/2)+128*x^8-224*x^7+96*x^6+1)*exp((x^4-2*x^3+x^2)*exp(5/2)^4+(-8*x^5+1
6*x^4-8*x^3)*exp(5/2)^3+(24*x^6-48*x^5+24*x^4)*exp(5/2)^2+(-32*x^7+64*x^6-32*x^5)*exp(5/2)+16*x^8-32*x^7+16*x^
6),x, algorithm="maxima")

[Out]

1/25*x*e^(16*x^8 - 32*x^7*e^(5/2) - 32*x^7 + 24*x^6*e^5 + 64*x^6*e^(5/2) + 16*x^6 - 8*x^5*e^(15/2) - 48*x^5*e^
5 - 32*x^5*e^(5/2) + x^4*e^10 + 16*x^4*e^(15/2) + 24*x^4*e^5 - 2*x^3*e^10 - 8*x^3*e^(15/2) + x^2*e^10)

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mupad [B]  time = 1.17, size = 115, normalized size = 4.11 \begin {gather*} \frac {x\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{x^4\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{-2\,x^3\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{-8\,x^3\,{\mathrm {e}}^{15/2}}\,{\mathrm {e}}^{-8\,x^5\,{\mathrm {e}}^{15/2}}\,{\mathrm {e}}^{24\,x^4\,{\mathrm {e}}^5}\,{\mathrm {e}}^{24\,x^6\,{\mathrm {e}}^5}\,{\mathrm {e}}^{16\,x^4\,{\mathrm {e}}^{15/2}}\,{\mathrm {e}}^{-32\,x^5\,{\mathrm {e}}^{5/2}}\,{\mathrm {e}}^{-32\,x^7\,{\mathrm {e}}^{5/2}}\,{\mathrm {e}}^{-48\,x^5\,{\mathrm {e}}^5}\,{\mathrm {e}}^{64\,x^6\,{\mathrm {e}}^{5/2}}\,{\mathrm {e}}^{16\,x^6}\,{\mathrm {e}}^{16\,x^8}\,{\mathrm {e}}^{-32\,x^7}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(10)*(x^2 - 2*x^3 + x^4) - exp(15/2)*(8*x^3 - 16*x^4 + 8*x^5) + exp(5)*(24*x^4 - 48*x^5 + 24*x^6)
- exp(5/2)*(32*x^5 - 64*x^6 + 32*x^7) + 16*x^6 - 32*x^7 + 16*x^8)*(exp(10)*(2*x^2 - 6*x^3 + 4*x^4) - exp(15/2)
*(24*x^3 - 64*x^4 + 40*x^5) + exp(5)*(96*x^4 - 240*x^5 + 144*x^6) - exp(5/2)*(160*x^5 - 384*x^6 + 224*x^7) + 9
6*x^6 - 224*x^7 + 128*x^8 + 1))/25,x)

[Out]

(x*exp(x^2*exp(10))*exp(x^4*exp(10))*exp(-2*x^3*exp(10))*exp(-8*x^3*exp(15/2))*exp(-8*x^5*exp(15/2))*exp(24*x^
4*exp(5))*exp(24*x^6*exp(5))*exp(16*x^4*exp(15/2))*exp(-32*x^5*exp(5/2))*exp(-32*x^7*exp(5/2))*exp(-48*x^5*exp
(5))*exp(64*x^6*exp(5/2))*exp(16*x^6)*exp(16*x^8)*exp(-32*x^7))/25

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sympy [B]  time = 0.45, size = 94, normalized size = 3.36 \begin {gather*} \frac {x e^{16 x^{8} - 32 x^{7} + 16 x^{6} + \left (x^{4} - 2 x^{3} + x^{2}\right ) e^{10} + \left (- 8 x^{5} + 16 x^{4} - 8 x^{3}\right ) e^{\frac {15}{2}} + \left (24 x^{6} - 48 x^{5} + 24 x^{4}\right ) e^{5} + \left (- 32 x^{7} + 64 x^{6} - 32 x^{5}\right ) e^{\frac {5}{2}}}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((4*x**4-6*x**3+2*x**2)*exp(5/2)**4+(-40*x**5+64*x**4-24*x**3)*exp(5/2)**3+(144*x**6-240*x**5+9
6*x**4)*exp(5/2)**2+(-224*x**7+384*x**6-160*x**5)*exp(5/2)+128*x**8-224*x**7+96*x**6+1)*exp((x**4-2*x**3+x**2)
*exp(5/2)**4+(-8*x**5+16*x**4-8*x**3)*exp(5/2)**3+(24*x**6-48*x**5+24*x**4)*exp(5/2)**2+(-32*x**7+64*x**6-32*x
**5)*exp(5/2)+16*x**8-32*x**7+16*x**6),x)

[Out]

x*exp(16*x**8 - 32*x**7 + 16*x**6 + (x**4 - 2*x**3 + x**2)*exp(10) + (-8*x**5 + 16*x**4 - 8*x**3)*exp(15/2) +
(24*x**6 - 48*x**5 + 24*x**4)*exp(5) + (-32*x**7 + 64*x**6 - 32*x**5)*exp(5/2))/25

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