3.36.51 \(\int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8)}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx\)

Optimal. Leaf size=20 \[ e^{\left (2+\left (x+\frac {45 (5+x)}{1+x}\right )^2\right )^2} \]

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Rubi [F]  time = 7.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}\right ) \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2563093129 + 2096362816*x + 688675888*x^2 + 115651112*x^3 + 10505414*x^4 + 513920*x^5 + 13600*x^6 + 1
84*x^7 + x^8)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4))*(-8156009700 - 4911736672*x - 1030398440*x^2 - 73629456*x^3 + 2
569600*x^4 + 595520*x^5 + 28488*x^6 + 560*x^7 + 4*x^8))/(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5),x]

[Out]

461340*Defer[Int][E^((50627 + 20704*x + 2568*x^2 + 92*x^3 + x^4)^2/(1 + x)^4), x] + 25748*Defer[Int][E^((50627
+ 20704*x + 2568*x^2 + 92*x^3 + x^4)^2/(1 + x)^4)*x, x] + 540*Defer[Int][E^((50627 + 20704*x + 2568*x^2 + 92*
x^3 + x^4)^2/(1 + x)^4)*x^2, x] + 4*Defer[Int][E^((50627 + 20704*x + 2568*x^2 + 92*x^3 + x^4)^2/(1 + x)^4)*x^3
, x] - 4199040000*Defer[Int][E^((50627 + 20704*x + 2568*x^2 + 92*x^3 + x^4)^2/(1 + x)^4)/(1 + x)^5, x] - 30792
96000*Defer[Int][E^((50627 + 20704*x + 2568*x^2 + 92*x^3 + x^4)^2/(1 + x)^4)/(1 + x)^4, x] - 799632000*Defer[I
nt][E^((50627 + 20704*x + 2568*x^2 + 92*x^3 + x^4)^2/(1 + x)^4)/(1 + x)^3, x] - 78503040*Defer[Int][E^((50627
+ 20704*x + 2568*x^2 + 92*x^3 + x^4)^2/(1 + x)^4)/(1 + x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}} \left (-2039002425-1227934168 x-257599610 x^2-18407364 x^3+642400 x^4+148880 x^5+7122 x^6+140 x^7+x^8\right )}{(1+x)^5} \, dx\\ &=4 \int \frac {e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}} \left (-2039002425-1227934168 x-257599610 x^2-18407364 x^3+642400 x^4+148880 x^5+7122 x^6+140 x^7+x^8\right )}{(1+x)^5} \, dx\\ &=4 \int \left (115335 e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}}+6437 e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}} x+135 e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}} x^2+e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}} x^3-\frac {1049760000 e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}}}{(1+x)^5}-\frac {769824000 e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}}}{(1+x)^4}-\frac {199908000 e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}}}{(1+x)^3}-\frac {19625760 e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}}}{(1+x)^2}\right ) \, dx\\ &=4 \int e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}} x^3 \, dx+540 \int e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}} x^2 \, dx+25748 \int e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}} x \, dx+461340 \int e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}} \, dx-78503040 \int \frac {e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}}}{(1+x)^2} \, dx-799632000 \int \frac {e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}}}{(1+x)^3} \, dx-3079296000 \int \frac {e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}}}{(1+x)^4} \, dx-4199040000 \int \frac {e^{\frac {\left (50627+20704 x+2568 x^2+92 x^3+x^4\right )^2}{(1+x)^4}}}{(1+x)^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 4.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^((2563093129 + 2096362816*x + 688675888*x^2 + 115651112*x^3 + 10505414*x^4 + 513920*x^5 + 13600*x
^6 + 184*x^7 + x^8)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4))*(-8156009700 - 4911736672*x - 1030398440*x^2 - 73629456*x
^3 + 2569600*x^4 + 595520*x^5 + 28488*x^6 + 560*x^7 + 4*x^8))/(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5),x]

[Out]

Integrate[(E^((2563093129 + 2096362816*x + 688675888*x^2 + 115651112*x^3 + 10505414*x^4 + 513920*x^5 + 13600*x
^6 + 184*x^7 + x^8)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4))*(-8156009700 - 4911736672*x - 1030398440*x^2 - 73629456*x
^3 + 2569600*x^4 + 595520*x^5 + 28488*x^6 + 560*x^7 + 4*x^8))/(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5), x]

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fricas [B]  time = 0.69, size = 60, normalized size = 3.00 \begin {gather*} e^{\left (\frac {x^{8} + 184 \, x^{7} + 13600 \, x^{6} + 513920 \, x^{5} + 10505414 \, x^{4} + 115651112 \, x^{3} + 688675888 \, x^{2} + 2096362816 \, x + 2563093129}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-1030398440*x^2-4911736672*x-8156009700)
*exp((x^8+184*x^7+13600*x^6+513920*x^5+10505414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+
4*x^3+6*x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x, algorithm="fricas")

[Out]

e^((x^8 + 184*x^7 + 13600*x^6 + 513920*x^5 + 10505414*x^4 + 115651112*x^3 + 688675888*x^2 + 2096362816*x + 256
3093129)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1))

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giac [B]  time = 0.18, size = 221, normalized size = 11.05 \begin {gather*} e^{\left (\frac {x^{8}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {184 \, x^{7}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {13600 \, x^{6}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {513920 \, x^{5}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {10505414 \, x^{4}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {115651112 \, x^{3}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {688675888 \, x^{2}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {2096362816 \, x}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {2563093129}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-1030398440*x^2-4911736672*x-8156009700)
*exp((x^8+184*x^7+13600*x^6+513920*x^5+10505414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+
4*x^3+6*x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x, algorithm="giac")

[Out]

e^(x^8/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 184*x^7/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 13600*x^6/(x^4 + 4*x^3 + 6*
x^2 + 4*x + 1) + 513920*x^5/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 10505414*x^4/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 1
15651112*x^3/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 688675888*x^2/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 2096362816*x/(x
^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 2563093129/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1))

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 hanged

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-1030398440*x^2-4911736672*x-8156009700)*exp((
x^8+184*x^7+13600*x^6+513920*x^5+10505414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+4*x^3+
6*x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x,method=_RETURNVERBOSE)

[Out]

int((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-1030398440*x^2-4911736672*x-8156009700)*exp((
x^8+184*x^7+13600*x^6+513920*x^5+10505414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+4*x^3+
6*x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x,method=_RETURNVERBOSE)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-1030398440*x^2-4911736672*x-8156009700)
*exp((x^8+184*x^7+13600*x^6+513920*x^5+10505414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+
4*x^3+6*x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 2.46, size = 229, normalized size = 11.45 \begin {gather*} {\mathrm {e}}^{\frac {2563093129}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {x^8}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {184\,x^7}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {13600\,x^6}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {513920\,x^5}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {10505414\,x^4}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {115651112\,x^3}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {2096362816\,x}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {688675888\,x^2}{x^4+4\,x^3+6\,x^2+4\,x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2096362816*x + 688675888*x^2 + 115651112*x^3 + 10505414*x^4 + 513920*x^5 + 13600*x^6 + 184*x^7 + x^8
+ 2563093129)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*(2569600*x^4 - 1030398440*x^2 - 73629456*x^3 - 4911736672*x +
595520*x^5 + 28488*x^6 + 560*x^7 + 4*x^8 - 8156009700))/(5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5 + 1),x)

[Out]

exp(2563093129/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp(x^8/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp((184*x^7)/(4*x +
6*x^2 + 4*x^3 + x^4 + 1))*exp((13600*x^6)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp((513920*x^5)/(4*x + 6*x^2 + 4*x
^3 + x^4 + 1))*exp((10505414*x^4)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp((115651112*x^3)/(4*x + 6*x^2 + 4*x^3 +
x^4 + 1))*exp((2096362816*x)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp((688675888*x^2)/(4*x + 6*x^2 + 4*x^3 + x^4 +
1))

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sympy [B]  time = 0.32, size = 58, normalized size = 2.90 \begin {gather*} e^{\frac {x^{8} + 184 x^{7} + 13600 x^{6} + 513920 x^{5} + 10505414 x^{4} + 115651112 x^{3} + 688675888 x^{2} + 2096362816 x + 2563093129}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**8+560*x**7+28488*x**6+595520*x**5+2569600*x**4-73629456*x**3-1030398440*x**2-4911736672*x-8156
009700)*exp((x**8+184*x**7+13600*x**6+513920*x**5+10505414*x**4+115651112*x**3+688675888*x**2+2096362816*x+256
3093129)/(x**4+4*x**3+6*x**2+4*x+1))/(x**5+5*x**4+10*x**3+10*x**2+5*x+1),x)

[Out]

exp((x**8 + 184*x**7 + 13600*x**6 + 513920*x**5 + 10505414*x**4 + 115651112*x**3 + 688675888*x**2 + 2096362816
*x + 2563093129)/(x**4 + 4*x**3 + 6*x**2 + 4*x + 1))

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