∫ 3750000000x16+3250000000x17+1050000000x18+150000000x19+8000000x20+ex(2500000000x15+1500000000x16+175000000x17−55000000x18−15000000x19−1000000x20)_________________________________________________________________________________________________________________________ 81e5x+810e4xx+3240e3xx2+6480e2xx3+6480exx4+2592x5 dx

3.57.62 \(\int \frac {3750000000 x^{16}+3250000000 x^{17}+1050000000 x^{18}+150000000 x^{19}+8000000 x^{20}+e^x (2500000000 x^{15}+1500000000 x^{16}+175000000 x^{17}-55000000 x^{18}-15000000 x^{19}-1000000 x^{20})}{81 e^{5 x}+810 e^{4 x} x+3240 e^{3 x} x^2+6480 e^{2 x} x^3+6480 e^x x^4+2592 x^5} \, dx\)

Optimal. Leaf size=23 \[ \frac {250000 x^{12} (5+x)^4}{81 \left (2+\frac {e^x}{x}\right )^4} \]

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Rubi [F] time = 2.18, 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 {3750000000 x^{16}+3250000000 x^{17}+1050000000 x^{18}+150000000 x^{19}+8000000 x^{20}+e^x \left (2500000000 x^{15}+1500000000 x^{16}+175000000 x^{17}-55000000 x^{18}-15000000 x^{19}-1000000 x^{20}\right )}{81 e^{5 x}+810 e^{4 x} x+3240 e^{3 x} x^2+6480 e^{2 x} x^3+6480 e^x x^4+2592 x^5} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(3750000000*x^16 + 3250000000*x^17 + 1050000000*x^18 + 150000000*x^19 + 8000000*x^20 + E^x*(2500000000*x^1

5 + 1500000000*x^16 + 175000000*x^17 - 55000000*x^18 - 15000000*x^19 - 1000000*x^20))/(81*E^(5*x) + 810*E^(4*x

)*x + 3240*E^(3*x)*x^2 + 6480*E^(2*x)*x^3 + 6480*E^x*x^4 + 2592*x^5),x]

[Out]

(-1250000000*Defer[Int][x^16/(E^x + 2*x)^5, x])/81 + (250000000*Defer[Int][x^17/(E^x + 2*x)^5, x])/81 + (70000

0000*Defer[Int][x^18/(E^x + 2*x)^5, x])/81 + (260000000*Defer[Int][x^19/(E^x + 2*x)^5, x])/81 + (38000000*Defe

r[Int][x^20/(E^x + 2*x)^5, x])/81 + (2000000*Defer[Int][x^21/(E^x + 2*x)^5, x])/81 + (2500000000*Defer[Int][x^

15/(E^x + 2*x)^4, x])/81 + (500000000*Defer[Int][x^16/(E^x + 2*x)^4, x])/27 + (175000000*Defer[Int][x^17/(E^x

+ 2*x)^4, x])/81 - (55000000*Defer[Int][x^18/(E^x + 2*x)^4, x])/81 - (5000000*Defer[Int][x^19/(E^x + 2*x)^4, x

])/27 - (1000000*Defer[Int][x^20/(E^x + 2*x)^4, x])/81

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1000000 x^{15} (5+x)^3 \left (2 x (15+4 x)-e^x \left (-20+x^2\right )\right )}{81 \left (e^x+2 x\right )^5} \, dx\\ &=\frac {1000000}{81} \int \frac {x^{15} (5+x)^3 \left (2 x (15+4 x)-e^x \left (-20+x^2\right )\right )}{\left (e^x+2 x\right )^5} \, dx\\ &=\frac {1000000}{81} \int \left (\frac {2 (-1+x) x^{16} (5+x)^4}{\left (e^x+2 x\right )^5}-\frac {x^{15} (5+x)^3 \left (-20+x^2\right )}{\left (e^x+2 x\right )^4}\right ) \, dx\\ &=-\left (\frac {1000000}{81} \int \frac {x^{15} (5+x)^3 \left (-20+x^2\right )}{\left (e^x+2 x\right )^4} \, dx\right )+\frac {2000000}{81} \int \frac {(-1+x) x^{16} (5+x)^4}{\left (e^x+2 x\right )^5} \, dx\\ &=-\left (\frac {1000000}{81} \int \left (-\frac {2500 x^{15}}{\left (e^x+2 x\right )^4}-\frac {1500 x^{16}}{\left (e^x+2 x\right )^4}-\frac {175 x^{17}}{\left (e^x+2 x\right )^4}+\frac {55 x^{18}}{\left (e^x+2 x\right )^4}+\frac {15 x^{19}}{\left (e^x+2 x\right )^4}+\frac {x^{20}}{\left (e^x+2 x\right )^4}\right ) \, dx\right )+\frac {2000000}{81} \int \left (-\frac {625 x^{16}}{\left (e^x+2 x\right )^5}+\frac {125 x^{17}}{\left (e^x+2 x\right )^5}+\frac {350 x^{18}}{\left (e^x+2 x\right )^5}+\frac {130 x^{19}}{\left (e^x+2 x\right )^5}+\frac {19 x^{20}}{\left (e^x+2 x\right )^5}+\frac {x^{21}}{\left (e^x+2 x\right )^5}\right ) \, dx\\ &=-\left (\frac {1000000}{81} \int \frac {x^{20}}{\left (e^x+2 x\right )^4} \, dx\right )+\frac {2000000}{81} \int \frac {x^{21}}{\left (e^x+2 x\right )^5} \, dx-\frac {5000000}{27} \int \frac {x^{19}}{\left (e^x+2 x\right )^4} \, dx+\frac {38000000}{81} \int \frac {x^{20}}{\left (e^x+2 x\right )^5} \, dx-\frac {55000000}{81} \int \frac {x^{18}}{\left (e^x+2 x\right )^4} \, dx+\frac {175000000}{81} \int \frac {x^{17}}{\left (e^x+2 x\right )^4} \, dx+\frac {250000000}{81} \int \frac {x^{17}}{\left (e^x+2 x\right )^5} \, dx+\frac {260000000}{81} \int \frac {x^{19}}{\left (e^x+2 x\right )^5} \, dx+\frac {700000000}{81} \int \frac {x^{18}}{\left (e^x+2 x\right )^5} \, dx-\frac {1250000000}{81} \int \frac {x^{16}}{\left (e^x+2 x\right )^5} \, dx+\frac {500000000}{27} \int \frac {x^{16}}{\left (e^x+2 x\right )^4} \, dx+\frac {2500000000}{81} \int \frac {x^{15}}{\left (e^x+2 x\right )^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.31, size = 21, normalized size = 0.91 \begin {gather*} \frac {250000 x^{16} (5+x)^4}{81 \left (e^x+2 x\right )^4} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3750000000*x^16 + 3250000000*x^17 + 1050000000*x^18 + 150000000*x^19 + 8000000*x^20 + E^x*(25000000

00*x^15 + 1500000000*x^16 + 175000000*x^17 - 55000000*x^18 - 15000000*x^19 - 1000000*x^20))/(81*E^(5*x) + 810*

E^(4*x)*x + 3240*E^(3*x)*x^2 + 6480*E^(2*x)*x^3 + 6480*E^x*x^4 + 2592*x^5),x]

[Out]

(250000*x^16*(5 + x)^4)/(81*(E^x + 2*x)^4)

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fricas [B] time = 0.68, size = 61, normalized size = 2.65 \begin {gather*} \frac {250000 \, {\left (x^{20} + 20 \, x^{19} + 150 \, x^{18} + 500 \, x^{17} + 625 \, x^{16}\right )}}{81 \, {\left (16 \, x^{4} + 32 \, x^{3} e^{x} + 24 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1000000*x^20-15000000*x^19-55000000*x^18+175000000*x^17+1500000000*x^16+2500000000*x^15)*exp(x)+8

000000*x^20+150000000*x^19+1050000000*x^18+3250000000*x^17+3750000000*x^16)/(81*exp(x)^5+810*x*exp(x)^4+3240*x

^2*exp(x)^3+6480*exp(x)^2*x^3+6480*exp(x)*x^4+2592*x^5),x, algorithm="fricas")

[Out]

250000/81*(x^20 + 20*x^19 + 150*x^18 + 500*x^17 + 625*x^16)/(16*x^4 + 32*x^3*e^x + 24*x^2*e^(2*x) + 8*x*e^(3*x

) + e^(4*x))

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giac [B] time = 0.15, size = 61, normalized size = 2.65 \begin {gather*} \frac {250000 \, {\left (x^{20} + 20 \, x^{19} + 150 \, x^{18} + 500 \, x^{17} + 625 \, x^{16}\right )}}{81 \, {\left (16 \, x^{4} + 32 \, x^{3} e^{x} + 24 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1000000*x^20-15000000*x^19-55000000*x^18+175000000*x^17+1500000000*x^16+2500000000*x^15)*exp(x)+8

000000*x^20+150000000*x^19+1050000000*x^18+3250000000*x^17+3750000000*x^16)/(81*exp(x)^5+810*x*exp(x)^4+3240*x

^2*exp(x)^3+6480*exp(x)^2*x^3+6480*exp(x)*x^4+2592*x^5),x, algorithm="giac")

[Out]

250000/81*(x^20 + 20*x^19 + 150*x^18 + 500*x^17 + 625*x^16)/(16*x^4 + 32*x^3*e^x + 24*x^2*e^(2*x) + 8*x*e^(3*x

) + e^(4*x))

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maple [A] time = 0.05, size = 32, normalized size = 1.39


method	result	size

risch	\(\frac {250000 \left (x^{4}+20 x^{3}+150 x^{2}+500 x +625\right ) x^{16}}{81 \left ({\mathrm e}^{x}+2 x \right )^{4}}\)	\(32\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-1000000*x^20-15000000*x^19-55000000*x^18+175000000*x^17+1500000000*x^16+2500000000*x^15)*exp(x)+8000000

*x^20+150000000*x^19+1050000000*x^18+3250000000*x^17+3750000000*x^16)/(81*exp(x)^5+810*x*exp(x)^4+3240*x^2*exp

(x)^3+6480*exp(x)^2*x^3+6480*exp(x)*x^4+2592*x^5),x,method=_RETURNVERBOSE)

[Out]

250000/81*(x^4+20*x^3+150*x^2+500*x+625)*x^16/(exp(x)+2*x)^4

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maxima [B] time = 0.41, size = 61, normalized size = 2.65 \begin {gather*} \frac {250000 \, {\left (x^{20} + 20 \, x^{19} + 150 \, x^{18} + 500 \, x^{17} + 625 \, x^{16}\right )}}{81 \, {\left (16 \, x^{4} + 32 \, x^{3} e^{x} + 24 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1000000*x^20-15000000*x^19-55000000*x^18+175000000*x^17+1500000000*x^16+2500000000*x^15)*exp(x)+8

000000*x^20+150000000*x^19+1050000000*x^18+3250000000*x^17+3750000000*x^16)/(81*exp(x)^5+810*x*exp(x)^4+3240*x

^2*exp(x)^3+6480*exp(x)^2*x^3+6480*exp(x)*x^4+2592*x^5),x, algorithm="maxima")

[Out]

250000/81*(x^20 + 20*x^19 + 150*x^18 + 500*x^17 + 625*x^16)/(16*x^4 + 32*x^3*e^x + 24*x^2*e^(2*x) + 8*x*e^(3*x

) + e^(4*x))

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mupad [B] time = 6.84, size = 71, normalized size = 3.09 \begin {gather*} \frac {250000\,\left (x^{21}+19\,x^{20}+130\,x^{19}+350\,x^{18}+125\,x^{17}-625\,x^{16}\right )}{81\,\left (x-1\right )\,\left ({\mathrm {e}}^{4\,x}+8\,x\,{\mathrm {e}}^{3\,x}+32\,x^3\,{\mathrm {e}}^x+24\,x^2\,{\mathrm {e}}^{2\,x}+16\,x^4\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((3750000000*x^16 + 3250000000*x^17 + 1050000000*x^18 + 150000000*x^19 + 8000000*x^20 + exp(x)*(2500000000*

x^15 + 1500000000*x^16 + 175000000*x^17 - 55000000*x^18 - 15000000*x^19 - 1000000*x^20))/(81*exp(5*x) + 810*x*

exp(4*x) + 6480*x^4*exp(x) + 3240*x^2*exp(3*x) + 6480*x^3*exp(2*x) + 2592*x^5),x)

[Out]

(250000*(125*x^17 - 625*x^16 + 350*x^18 + 130*x^19 + 19*x^20 + x^21))/(81*(x - 1)*(exp(4*x) + 8*x*exp(3*x) + 3

2*x^3*exp(x) + 24*x^2*exp(2*x) + 16*x^4))

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sympy [B] time = 0.22, size = 63, normalized size = 2.74 \begin {gather*} \frac {250000 x^{20} + 5000000 x^{19} + 37500000 x^{18} + 125000000 x^{17} + 156250000 x^{16}}{1296 x^{4} + 2592 x^{3} e^{x} + 1944 x^{2} e^{2 x} + 648 x e^{3 x} + 81 e^{4 x}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1000000*x**20-15000000*x**19-55000000*x**18+175000000*x**17+1500000000*x**16+2500000000*x**15)*ex

p(x)+8000000*x**20+150000000*x**19+1050000000*x**18+3250000000*x**17+3750000000*x**16)/(81*exp(x)**5+810*x*exp

(x)**4+3240*x**2*exp(x)**3+6480*exp(x)**2*x**3+6480*exp(x)*x**4+2592*x**5),x)

[Out]

(250000*x**20 + 5000000*x**19 + 37500000*x**18 + 125000000*x**17 + 156250000*x**16)/(1296*x**4 + 2592*x**3*exp

(x) + 1944*x**2*exp(2*x) + 648*x*exp(3*x) + 81*exp(4*x))

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