3.5.16 $\int \frac{-2500-3000x+18800x^2+11840x^3+47360x^5-300800x^6-192000x^7+640000x^8}{81x^5+432e^5{x}^5+864e^{10}{x}^5+768e^{15}{x}^5+256e^{20}{x}^5}dx$

Optimal. Leaf size=22 $$\frac{{(2+\frac{5}{x}-20x)}^4}{{(3+4e^5)}^4}$$

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Rubi [B]  time = 0.06, antiderivative size = 111, normalized size of antiderivative = 5.05, number of steps used = 7, number of rules used = 3, integrand size = 76, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.039, Rules used = {6, 12, 14} $$\begin{array}{c}\frac{160000x^4}{{(3+4e^5)}^4}+\frac{625}{{(3+4e^5)}^4{x}^4}-\frac{64000x^3}{{(3+4e^5)}^4}+\frac{1000}{{(3+4e^5)}^4{x}^3}-\frac{150400x^2}{{(3+4e^5)}^4}-\frac{9400}{{(3+4e^5)}^4{x}^2}+\frac{47360x}{{(3+4e^5)}^4}-\frac{11840}{{(3+4e^5)}^{4}x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-2500 - 3000*x + 18800*x^2 + 11840*x^3 + 47360*x^5 - 300800*x^6 - 192000*x^7 + 640000*x^8)/(81*x^5 + 432*
E^5*x^5 + 864*E^10*x^5 + 768*E^15*x^5 + 256*E^20*x^5),x]

[Out]

625/((3 + 4*E^5)^4*x^4) + 1000/((3 + 4*E^5)^4*x^3) - 9400/((3 + 4*E^5)^4*x^2) - 11840/((3 + 4*E^5)^4*x) + (473
60*x)/(3 + 4*E^5)^4 - (150400*x^2)/(3 + 4*E^5)^4 - (64000*x^3)/(3 + 4*E^5)^4 + (160000*x^4)/(3 + 4*E^5)^4

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-2500-3000x+18800x^2+11840x^3+47360x^5-300800x^6-192000x^7+640000x^8}{864e^{10}{x}^5+768e^{15}{x}^5+256e^{20}{x}^5+(81+432e^5)x^5}dx\\ & =\int \frac{-2500-3000x+18800x^2+11840x^3+47360x^5-300800x^6-192000x^7+640000x^8}{256e^{20}{x}^5+(81+432e^5)x^5+(864e^{10}+768e^{15})x^5}dx\\ & =\int \frac{-2500-3000x+18800x^2+11840x^3+47360x^5-300800x^6-192000x^7+640000x^8}{(864e^{10}+768e^{15})x^5+(81+432e^5+256e^{20})x^5}dx\\ & =\int \frac{-2500-3000x+18800x^2+11840x^3+47360x^5-300800x^6-192000x^7+640000x^8}{(81+432e^5+864e^{10}+768e^{15}+256e^{20})x^5}dx\\ & =\frac{\int \frac{-2500-3000x+18800x^2+11840x^3+47360x^5-300800x^6-192000x^7+640000x^8}{x^5}dx}{{(3+4e^5)}^4}\\ & =\frac{\int (47360-\frac{2500}{x^5}-\frac{3000}{x^4}+\frac{18800}{x^3}+\frac{11840}{x^2}-300800x-192000x^2+640000x^3)dx}{{(3+4e^5)}^4}\\ & =\frac{625}{{(3+4e^5)}^4{x}^4}+\frac{1000}{{(3+4e^5)}^4{x}^3}-\frac{9400}{{(3+4e^5)}^4{x}^2}-\frac{11840}{{(3+4e^5)}^{4}x}+\frac{47360x}{{(3+4e^5)}^4}-\frac{150400x^2}{{(3+4e^5)}^4}-\frac{64000x^3}{{(3+4e^5)}^4}+\frac{160000x^4}{{(3+4e^5)}^4}\end{array}\end{array}$$

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Mathematica [B]  time = 0.01, size = 52, normalized size = 2.36 $$\begin{array}{c}\frac{20(\frac{125}{4x^4}+\frac{50}{x^3}-\frac{470}{x^2}-\frac{592}{x}+2368x-7520x^2-3200x^3+8000x^4)}{{(3+4e^5)}^4}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-2500 - 3000*x + 18800*x^2 + 11840*x^3 + 47360*x^5 - 300800*x^6 - 192000*x^7 + 640000*x^8)/(81*x^5
+ 432*E^5*x^5 + 864*E^10*x^5 + 768*E^15*x^5 + 256*E^20*x^5),x]

[Out]

(20*(125/(4*x^4) + 50/x^3 - 470/x^2 - 592/x + 2368*x - 7520*x^2 - 3200*x^3 + 8000*x^4))/(3 + 4*E^5)^4

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fricas [B]  time = 0.53, size = 73, normalized size = 3.32 $$\begin{array}{c}\frac{5(32000x^8-12800x^7-30080x^6+9472x^5-2368x^3-1880x^2+200x+125)}{256x^4{e}^{20}+768x^4{e}^{15}+864x^4{e}^{10}+432x^4{e}^5+81x^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((640000*x^8-192000*x^7-300800*x^6+47360*x^5+11840*x^3+18800*x^2-3000*x-2500)/(256*x^5*exp(5)^4+768*x
^5*exp(5)^3+864*x^5*exp(5)^2+432*x^5*exp(5)+81*x^5),x, algorithm="fricas")

[Out]

5*(32000*x^8 - 12800*x^7 - 30080*x^6 + 9472*x^5 - 2368*x^3 - 1880*x^2 + 200*x + 125)/(256*x^4*e^20 + 768*x^4*e
^15 + 864*x^4*e^10 + 432*x^4*e^5 + 81*x^4)

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giac [B]  time = 1.08, size = 442, normalized size = 20.09 $$\begin{array}{c}\frac{640(4194304000x^4{e}^{60}+37748736000x^4{e}^{55}+155713536000x^4{e}^{50}+389283840000x^4{e}^{45}+656916480000x^4{e}^{40}+788299776000x^4{e}^{35}+689762304000x^4{e}^{30}+443418624000x^4{e}^{25}+207852480000x^4{e}^{20}+69284160000x^4{e}^{15}+15588936000x^4{e}^{10}+2125764000x^4{e}^5+132860250x^4-1677721600x^3{e}^{60}-15099494400x^3{e}^{55}-62285414400x^3{e}^{50}-155713536000x^3{e}^{45}-262766592000x^3{e}^{40}-315319910400x^3{e}^{35}-275904921600x^3{e}^{30}-177367449600x^3{e}^{25}-83140992000x^3{e}^{20}-27713664000x^3{e}^{15}-6235574400x^3{e}^{10}-850305600x^3{e}^5-53144100x^3-3942645760x^2{e}^{60}-35483811840x^2{e}^{55}-146370723840x^2{e}^{50}-365926809600x^2{e}^{45}-617501491200x^2{e}^{40}-741001789440x^2{e}^{35}-648376565760x^2{e}^{30}-416813506560x^2{e}^{25}-195381331200x^2{e}^{20}-65127110400x^2{e}^{15}-14653599840x^2{e}^{10}-1998218160x^2{e}^5-124888635x^2+1241513984x{e}^{60}+11173625856x{e}^{55}+46091206656x{e}^{50}+115228016640x{e}^{45}+194447278080x{e}^{40}+233336733696x{e}^{35}+204169641984x{e}^{30}+131251912704x{e}^{25}+61524334080x{e}^{20}+20508111360x{e}^{15}+4614325056x{e}^{10}+629226144x{e}^5+39326634x)}{4294967296e^{80}+51539607552e^{75}+289910292480e^{70}+1014686023680e^{65}+2473297182720e^{60}+4451934928896e^{55}+6121410527232e^{50}+6558654136320e^{45}+5533864427520e^{40}+3689242951680e^{35}+1936852549632e^{30}+792348770304e^{25}+247608990720e^{20}+57140536320e^{15}+9183300480e^{10}+918330048e^5+43046721}-\frac{5(2368x^3+1880x^2-200x-125)}{x^4(256e^{20}+768e^{15}+864e^{10}+432e^5+81)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((640000*x^8-192000*x^7-300800*x^6+47360*x^5+11840*x^3+18800*x^2-3000*x-2500)/(256*x^5*exp(5)^4+768*x
^5*exp(5)^3+864*x^5*exp(5)^2+432*x^5*exp(5)+81*x^5),x, algorithm="giac")

[Out]

640*(4194304000*x^4*e^60 + 37748736000*x^4*e^55 + 155713536000*x^4*e^50 + 389283840000*x^4*e^45 + 656916480000
*x^4*e^40 + 788299776000*x^4*e^35 + 689762304000*x^4*e^30 + 443418624000*x^4*e^25 + 207852480000*x^4*e^20 + 69
284160000*x^4*e^15 + 15588936000*x^4*e^10 + 2125764000*x^4*e^5 + 132860250*x^4 - 1677721600*x^3*e^60 - 1509949
4400*x^3*e^55 - 62285414400*x^3*e^50 - 155713536000*x^3*e^45 - 262766592000*x^3*e^40 - 315319910400*x^3*e^35 -
275904921600*x^3*e^30 - 177367449600*x^3*e^25 - 83140992000*x^3*e^20 - 27713664000*x^3*e^15 - 6235574400*x^3*
e^10 - 850305600*x^3*e^5 - 53144100*x^3 - 3942645760*x^2*e^60 - 35483811840*x^2*e^55 - 146370723840*x^2*e^50 -
365926809600*x^2*e^45 - 617501491200*x^2*e^40 - 741001789440*x^2*e^35 - 648376565760*x^2*e^30 - 416813506560*
x^2*e^25 - 195381331200*x^2*e^20 - 65127110400*x^2*e^15 - 14653599840*x^2*e^10 - 1998218160*x^2*e^5 - 12488863
5*x^2 + 1241513984*x*e^60 + 11173625856*x*e^55 + 46091206656*x*e^50 + 115228016640*x*e^45 + 194447278080*x*e^4
0 + 233336733696*x*e^35 + 204169641984*x*e^30 + 131251912704*x*e^25 + 61524334080*x*e^20 + 20508111360*x*e^15
+ 4614325056*x*e^10 + 629226144*x*e^5 + 39326634*x)/(4294967296*e^80 + 51539607552*e^75 + 289910292480*e^70 +
1014686023680*e^65 + 2473297182720*e^60 + 4451934928896*e^55 + 6121410527232*e^50 + 6558654136320*e^45 + 55338
64427520*e^40 + 3689242951680*e^35 + 1936852549632*e^30 + 792348770304*e^25 + 247608990720*e^20 + 57140536320*
e^15 + 9183300480*e^10 + 918330048*e^5 + 43046721) - 5*(2368*x^3 + 1880*x^2 - 200*x - 125)/(x^4*(256*e^20 + 76
8*e^15 + 864*e^10 + 432*e^5 + 81))

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maple [B]  time = 0.22, size = 67, normalized size = 3.05

method	result	size
gosper	$\frac{160000x^8-64000x^7-150400x^6+47360x^5-11840x^3-9400x^2+1000x+625}{x^4(256{\mathrm{e}}^{20}+768{\mathrm{e}}^{15}+864{\mathrm{e}}^{10}+432{\mathrm{e}}^5+81)}$	$67$
default	$\frac{160000x^4-64000x^3-150400x^2+47360x-\frac{9400}{x^2}+\frac{1000}{x^3}-\frac{11840}{x}+\frac{625}{x^4}}{256{\mathrm{e}}^{20}+768{\mathrm{e}}^{15}+864{\mathrm{e}}^{10}+432{\mathrm{e}}^5+81}$	$68$
norman	$\frac{\frac{625}{4{\mathrm{e}}^5+3}+\frac{1000x}{4{\mathrm{e}}^5+3}-\frac{9400x^2}{4{\mathrm{e}}^5+3}-\frac{11840x^3}{4{\mathrm{e}}^5+3}+\frac{47360x^5}{4{\mathrm{e}}^5+3}-\frac{150400x^6}{4{\mathrm{e}}^5+3}-\frac{64000x^7}{4{\mathrm{e}}^5+3}+\frac{160000x^8}{4{\mathrm{e}}^5+3}}{x^4{(4{\mathrm{e}}^5+3)}^3}$	$113$
risch	$\frac{160000x^4}{256{\mathrm{e}}^{20}+768{\mathrm{e}}^{15}+864{\mathrm{e}}^{10}+432{\mathrm{e}}^5+81}-\frac{64000x^3}{256{\mathrm{e}}^{20}+768{\mathrm{e}}^{15}+864{\mathrm{e}}^{10}+432{\mathrm{e}}^5+81}-\frac{150400x^2}{256{\mathrm{e}}^{20}+768{\mathrm{e}}^{15}+864{\mathrm{e}}^{10}+432{\mathrm{e}}^5+81}+\frac{47360x}{256{\mathrm{e}}^{20}+768{\mathrm{e}}^{15}+864{\mathrm{e}}^{10}+432{\mathrm{e}}^5+81}+\frac{256(-11840{\mathrm{e}}^{20}-35520{\mathrm{e}}^{15}-39960{\mathrm{e}}^{10}-19980{\mathrm{e}}^5-\frac{14985}{4})x^3+256(-9400{\mathrm{e}}^{20}-28200{\mathrm{e}}^{15}-31725{\mathrm{e}}^{10}-\frac{31725{\mathrm{e}}^5}{2}-\frac{95175}{32})x^2+256(1000{\mathrm{e}}^{20}+3000{\mathrm{e}}^{15}+3375{\mathrm{e}}^{10}+\frac{3375{\mathrm{e}}^5}{2}+\frac{10125}{32})x+160000{\mathrm{e}}^{20}+480000{\mathrm{e}}^{15}+540000{\mathrm{e}}^{10}+270000{\mathrm{e}}^5+50625}{{(256{\mathrm{e}}^{20}+768{\mathrm{e}}^{15}+864{\mathrm{e}}^{10}+432{\mathrm{e}}^5+81)}^2{x}^4}$	$207$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((640000*x^8-192000*x^7-300800*x^6+47360*x^5+11840*x^3+18800*x^2-3000*x-2500)/(256*x^5*exp(5)^4+768*x^5*exp
(5)^3+864*x^5*exp(5)^2+432*x^5*exp(5)+81*x^5),x,method=_RETURNVERBOSE)

[Out]

5*(32000*x^8-12800*x^7-30080*x^6+9472*x^5-2368*x^3-1880*x^2+200*x+125)/x^4/(256*exp(5)^4+768*exp(5)^3+864*exp(
5)^2+432*exp(5)+81)

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maxima [B]  time = 0.39, size = 82, normalized size = 3.73 $$\begin{array}{c}\frac{640(250x^4-100x^3-235x^2+74x)}{256e^{20}+768e^{15}+864e^{10}+432e^5+81}-\frac{5(2368x^3+1880x^2-200x-125)}{x^4(256e^{20}+768e^{15}+864e^{10}+432e^5+81)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((640000*x^8-192000*x^7-300800*x^6+47360*x^5+11840*x^3+18800*x^2-3000*x-2500)/(256*x^5*exp(5)^4+768*x
^5*exp(5)^3+864*x^5*exp(5)^2+432*x^5*exp(5)+81*x^5),x, algorithm="maxima")

[Out]

640*(250*x^4 - 100*x^3 - 235*x^2 + 74*x)/(256*e^20 + 768*e^15 + 864*e^10 + 432*e^5 + 81) - 5*(2368*x^3 + 1880*
x^2 - 200*x - 125)/(x^4*(256*e^20 + 768*e^15 + 864*e^10 + 432*e^5 + 81))

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mupad [B]  time = 0.46, size = 90, normalized size = 4.09 $$\begin{array}{c}\frac{47360x}{{(4{\mathrm{e}}^5+3)}^4}-\frac{150400x^2}{{(4{\mathrm{e}}^5+3)}^4}-\frac{64000x^3}{{(4{\mathrm{e}}^5+3)}^4}+\frac{160000x^4}{{(4{\mathrm{e}}^5+3)}^4}+\frac{-11840x^3-9400x^2+1000x+625}{x^4(432{\mathrm{e}}^5+864{\mathrm{e}}^{10}+768{\mathrm{e}}^{15}+256{\mathrm{e}}^{20}+81)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3000*x - 18800*x^2 - 11840*x^3 - 47360*x^5 + 300800*x^6 + 192000*x^7 - 640000*x^8 + 2500)/(432*x^5*exp(5
) + 864*x^5*exp(10) + 768*x^5*exp(15) + 256*x^5*exp(20) + 81*x^5),x)

[Out]

(47360*x)/(4*exp(5) + 3)^4 - (150400*x^2)/(4*exp(5) + 3)^4 - (64000*x^3)/(4*exp(5) + 3)^4 + (160000*x^4)/(4*ex
p(5) + 3)^4 + (1000*x - 9400*x^2 - 11840*x^3 + 625)/(x^4*(432*exp(5) + 864*exp(10) + 768*exp(15) + 256*exp(20)
+ 81))

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sympy [B]  time = 0.12, size = 58, normalized size = 2.64 $$\begin{array}{c}\frac{160000x^4-64000x^3-150400x^2+47360x+\frac{-11840x^3-9400x^2+1000x+625}{x^4}}{81+432e^5+864e^{10}+768e^{15}+256e^{20}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((640000*x**8-192000*x**7-300800*x**6+47360*x**5+11840*x**3+18800*x**2-3000*x-2500)/(256*x**5*exp(5)*
*4+768*x**5*exp(5)**3+864*x**5*exp(5)**2+432*x**5*exp(5)+81*x**5),x)

[Out]

(160000*x**4 - 64000*x**3 - 150400*x**2 + 47360*x + (-11840*x**3 - 9400*x**2 + 1000*x + 625)/x**4)/(81 + 432*e
xp(5) + 864*exp(10) + 768*exp(15) + 256*exp(20))

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