3.44.4 \(\int \frac {-32000 x+9600 x^2-960 x^3-3968 x^4+720 x^5-32 x^6-80 x^7+6 x^8+e^5 (-8000+2400 x-240 x^2-792 x^3+140 x^4-6 x^5)}{-1000+300 x-30 x^2+x^3} \, dx\)

Optimal. Leaf size=21 \[ -1+\left (e^5+4 x-\frac {x^4}{-10+x}\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.14, antiderivative size = 73, normalized size of antiderivative = 3.48, number of steps used = 2, number of rules used = 1, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2074} \begin {gather*} x^6+20 x^5+292 x^4+2 \left (1960-e^5\right ) x^3+4 \left (12304-5 e^5\right ) x^2+64 \left (9250-3 e^5\right ) x-\frac {20000 \left (3960-e^5\right )}{10-x}+\frac {100000000}{(10-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32000*x + 9600*x^2 - 960*x^3 - 3968*x^4 + 720*x^5 - 32*x^6 - 80*x^7 + 6*x^8 + E^5*(-8000 + 2400*x - 240*
x^2 - 792*x^3 + 140*x^4 - 6*x^5))/(-1000 + 300*x - 30*x^2 + x^3),x]

[Out]

100000000/(10 - x)^2 - (20000*(3960 - E^5))/(10 - x) + 64*(9250 - 3*E^5)*x + 4*(12304 - 5*E^5)*x^2 + 2*(1960 -
E^5)*x^3 + 292*x^4 + 20*x^5 + x^6

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-64 \left (-9250+3 e^5\right )-\frac {200000000}{(-10+x)^3}+\frac {20000 \left (-3960+e^5\right )}{(-10+x)^2}-8 \left (-12304+5 e^5\right ) x-6 \left (-1960+e^5\right ) x^2+1168 x^3+100 x^4+6 x^5\right ) \, dx\\ &=\frac {100000000}{(10-x)^2}-\frac {20000 \left (3960-e^5\right )}{10-x}+64 \left (9250-3 e^5\right ) x+4 \left (12304-5 e^5\right ) x^2+2 \left (1960-e^5\right ) x^3+292 x^4+20 x^5+x^6\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.04, size = 68, normalized size = 3.24 \begin {gather*} \frac {-2760160000+552032000 x-27600000 x^2-320 x^3+16 x^4+80 x^5-8 x^6+x^8+e^5 \left (792000-157600 x+7760 x^2+8 x^3+20 x^4-2 x^5\right )}{(-10+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32000*x + 9600*x^2 - 960*x^3 - 3968*x^4 + 720*x^5 - 32*x^6 - 80*x^7 + 6*x^8 + E^5*(-8000 + 2400*x
- 240*x^2 - 792*x^3 + 140*x^4 - 6*x^5))/(-1000 + 300*x - 30*x^2 + x^3),x]

[Out]

(-2760160000 + 552032000*x - 27600000*x^2 - 320*x^3 + 16*x^4 + 80*x^5 - 8*x^6 + x^8 + E^5*(792000 - 157600*x +
7760*x^2 + 8*x^3 + 20*x^4 - 2*x^5))/(-10 + x)^2

________________________________________________________________________________________

fricas [B]  time = 0.50, size = 71, normalized size = 3.38 \begin {gather*} \frac {x^{8} - 8 \, x^{6} + 80 \, x^{5} + 16 \, x^{4} - 320 \, x^{3} - 6918400 \, x^{2} - 2 \, {\left (x^{5} - 10 \, x^{4} - 4 \, x^{3} - 920 \, x^{2} + 19600 \, x - 100000\right )} e^{5} + 138400000 \, x - 692000000}{x^{2} - 20 \, x + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^5+140*x^4-792*x^3-240*x^2+2400*x-8000)*exp(5)+6*x^8-80*x^7-32*x^6+720*x^5-3968*x^4-960*x^3+96
00*x^2-32000*x)/(x^3-30*x^2+300*x-1000),x, algorithm="fricas")

[Out]

(x^8 - 8*x^6 + 80*x^5 + 16*x^4 - 320*x^3 - 6918400*x^2 - 2*(x^5 - 10*x^4 - 4*x^3 - 920*x^2 + 19600*x - 100000)
*e^5 + 138400000*x - 692000000)/(x^2 - 20*x + 100)

________________________________________________________________________________________

giac [B]  time = 0.13, size = 66, normalized size = 3.14 \begin {gather*} x^{6} + 20 \, x^{5} + 292 \, x^{4} - 2 \, x^{3} e^{5} + 3920 \, x^{3} - 20 \, x^{2} e^{5} + 49216 \, x^{2} - 192 \, x e^{5} + 592000 \, x - \frac {20000 \, {\left (x e^{5} - 3960 \, x - 10 \, e^{5} + 34600\right )}}{{\left (x - 10\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^5+140*x^4-792*x^3-240*x^2+2400*x-8000)*exp(5)+6*x^8-80*x^7-32*x^6+720*x^5-3968*x^4-960*x^3+96
00*x^2-32000*x)/(x^3-30*x^2+300*x-1000),x, algorithm="giac")

[Out]

x^6 + 20*x^5 + 292*x^4 - 2*x^3*e^5 + 3920*x^3 - 20*x^2*e^5 + 49216*x^2 - 192*x*e^5 + 592000*x - 20000*(x*e^5 -
3960*x - 10*e^5 + 34600)/(x - 10)^2

________________________________________________________________________________________

maple [B]  time = 0.06, size = 59, normalized size = 2.81

method	result	size
norman	\(\frac {x^{8}+\left (-320+8 \,{\mathrm e}^{5}\right ) x^{3}+\left (80-2 \,{\mathrm e}^{5}\right ) x^{5}+\left (20 \,{\mathrm e}^{5}+16\right ) x^{4}+\left (32000-2400 \,{\mathrm e}^{5}\right ) x -8 x^{6}-160000+16000 \,{\mathrm e}^{5}}{\left (x -10\right )^{2}}\)	\(59\)
default	\(x^{6}+20 x^{5}-2 x^{3} {\mathrm e}^{5}+292 x^{4}-20 x^{2} {\mathrm e}^{5}+3920 x^{3}-192 x \,{\mathrm e}^{5}+49216 x^{2}+592000 x -\frac {2 \left (-39600000+10000 \,{\mathrm e}^{5}\right )}{x -10}+\frac {100000000}{\left (x -10\right )^{2}}\)	\(67\)
risch	\(x^{6}+20 x^{5}-2 x^{3} {\mathrm e}^{5}+292 x^{4}-20 x^{2} {\mathrm e}^{5}+3920 x^{3}-192 x \,{\mathrm e}^{5}+49216 x^{2}+592000 x +\frac {\left (79200000-20000 \,{\mathrm e}^{5}\right ) x -692000000+200000 \,{\mathrm e}^{5}}{x^{2}-20 x +100}\)	\(72\)
gosper	\(-\frac {-x^{8}+2 x^{5} {\mathrm e}^{5}+8 x^{6}-20 x^{4} {\mathrm e}^{5}-80 x^{5}-8 x^{3} {\mathrm e}^{5}-16 x^{4}+320 x^{3}+2400 x \,{\mathrm e}^{5}-16000 \,{\mathrm e}^{5}-32000 x +160000}{x^{2}-20 x +100}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^5+140*x^4-792*x^3-240*x^2+2400*x-8000)*exp(5)+6*x^8-80*x^7-32*x^6+720*x^5-3968*x^4-960*x^3+9600*x^2
-32000*x)/(x^3-30*x^2+300*x-1000),x,method=_RETURNVERBOSE)

[Out]

(x^8+(-320+8*exp(5))*x^3+(80-2*exp(5))*x^5+(20*exp(5)+16)*x^4+(32000-2400*exp(5))*x-8*x^6-160000+16000*exp(5))
/(x-10)^2

________________________________________________________________________________________

maxima [B]  time = 0.37, size = 67, normalized size = 3.19 \begin {gather*} x^{6} + 20 \, x^{5} + 292 \, x^{4} - 2 \, x^{3} {\left (e^{5} - 1960\right )} - 4 \, x^{2} {\left (5 \, e^{5} - 12304\right )} - 64 \, x {\left (3 \, e^{5} - 9250\right )} - \frac {20000 \, {\left (x {\left (e^{5} - 3960\right )} - 10 \, e^{5} + 34600\right )}}{x^{2} - 20 \, x + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^5+140*x^4-792*x^3-240*x^2+2400*x-8000)*exp(5)+6*x^8-80*x^7-32*x^6+720*x^5-3968*x^4-960*x^3+96
00*x^2-32000*x)/(x^3-30*x^2+300*x-1000),x, algorithm="maxima")

[Out]

x^6 + 20*x^5 + 292*x^4 - 2*x^3*(e^5 - 1960) - 4*x^2*(5*e^5 - 12304) - 64*x*(3*e^5 - 9250) - 20000*(x*(e^5 - 39
60) - 10*e^5 + 34600)/(x^2 - 20*x + 100)

________________________________________________________________________________________

mupad [B]  time = 3.09, size = 71, normalized size = 3.38 \begin {gather*} 292\,x^4-x^3\,\left (2\,{\mathrm {e}}^5-3920\right )-x^2\,\left (20\,{\mathrm {e}}^5-49216\right )-\frac {x\,\left (20000\,{\mathrm {e}}^5-79200000\right )-200000\,{\mathrm {e}}^5+692000000}{x^2-20\,x+100}+20\,x^5+x^6-x\,\left (192\,{\mathrm {e}}^5-592000\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32000*x + exp(5)*(240*x^2 - 2400*x + 792*x^3 - 140*x^4 + 6*x^5 + 8000) - 9600*x^2 + 960*x^3 + 3968*x^4 -
720*x^5 + 32*x^6 + 80*x^7 - 6*x^8)/(300*x - 30*x^2 + x^3 - 1000),x)

[Out]

292*x^4 - x^3*(2*exp(5) - 3920) - x^2*(20*exp(5) - 49216) - (x*(20000*exp(5) - 79200000) - 200000*exp(5) + 692
000000)/(x^2 - 20*x + 100) + 20*x^5 + x^6 - x*(192*exp(5) - 592000)

________________________________________________________________________________________

sympy [B]  time = 0.28, size = 65, normalized size = 3.10 \begin {gather*} x^{6} + 20 x^{5} + 292 x^{4} + x^{3} \left (3920 - 2 e^{5}\right ) + x^{2} \left (49216 - 20 e^{5}\right ) + x \left (592000 - 192 e^{5}\right ) + \frac {x \left (79200000 - 20000 e^{5}\right ) - 692000000 + 200000 e^{5}}{x^{2} - 20 x + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**5+140*x**4-792*x**3-240*x**2+2400*x-8000)*exp(5)+6*x**8-80*x**7-32*x**6+720*x**5-3968*x**4-9
60*x**3+9600*x**2-32000*x)/(x**3-30*x**2+300*x-1000),x)

[Out]

x**6 + 20*x**5 + 292*x**4 + x**3*(3920 - 2*exp(5)) + x**2*(49216 - 20*exp(5)) + x*(592000 - 192*exp(5)) + (x*(
79200000 - 20000*exp(5)) - 692000000 + 200000*exp(5))/(x**2 - 20*x + 100)

________________________________________________________________________________________