3.103.49 \(\int \frac {618-164770 x-24360 x^2-33206 x^3-4820 x^4-240 x^5-4 x^6+e^{15} (20 x+4 x^3)+e^{10} (-1212 x-60 x^2-240 x^3-12 x^4)+e^5 (-30+24480 x+2418 x^2+4860 x^3+480 x^4+12 x^5)}{-8000+e^{15}+e^{10} (-60-3 x)-1200 x-60 x^2-x^3+e^5 (1200+120 x+3 x^2)} \, dx\)

Optimal. Leaf size=21 \[ 3+\left (5+x^2+\frac {3}{20-e^5+x}\right )^2 \]

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Rubi [B]  time = 0.16, antiderivative size = 46, normalized size of antiderivative = 2.19, number of steps used = 2, number of rules used = 1, integrand size = 139, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {2074} \begin {gather*} x^4+10 x^2+6 x+\frac {6 \left (405-40 e^5+e^{10}\right )}{x-e^5+20}+\frac {9}{\left (x-e^5+20\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(618 - 164770*x - 24360*x^2 - 33206*x^3 - 4820*x^4 - 240*x^5 - 4*x^6 + E^15*(20*x + 4*x^3) + E^10*(-1212*x
- 60*x^2 - 240*x^3 - 12*x^4) + E^5*(-30 + 24480*x + 2418*x^2 + 4860*x^3 + 480*x^4 + 12*x^5))/(-8000 + E^15 +
E^10*(-60 - 3*x) - 1200*x - 60*x^2 - x^3 + E^5*(1200 + 120*x + 3*x^2)),x]

[Out]

6*x + 10*x^2 + x^4 + 9/(20 - E^5 + x)^2 + (6*(405 - 40*E^5 + E^10))/(20 - E^5 + x)

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 (6+\frac {18}{\left (-20+e^5-x\right )^3}-\frac {6 \left (405-40 e^5+e^{10}\right )}{\left (-20+e^5-x\right )^2}+20 x+4 x^3\right ) \, dx\\ &=6 x+10 x^2+x^4+\frac {9}{\left (20-e^5+x\right )^2}+\frac {6 \left (405-40 e^5+e^{10}\right )}{20-e^5+x}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 83, normalized size = 3.95 \begin {gather*} -163880+80 e^{15}-e^{20}+e^5 \left (32394+\frac {240}{-20+e^5-x}\right )+6 x+10 x^2+x^4+\frac {9}{\left (20-e^5+x\right )^2}+\frac {2430}{20-e^5+x}+e^{10} \left (-2410+\frac {6}{20-e^5+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(618 - 164770*x - 24360*x^2 - 33206*x^3 - 4820*x^4 - 240*x^5 - 4*x^6 + E^15*(20*x + 4*x^3) + E^10*(-
1212*x - 60*x^2 - 240*x^3 - 12*x^4) + E^5*(-30 + 24480*x + 2418*x^2 + 4860*x^3 + 480*x^4 + 12*x^5))/(-8000 + E
^15 + E^10*(-60 - 3*x) - 1200*x - 60*x^2 - x^3 + E^5*(1200 + 120*x + 3*x^2)),x]

[Out]

-163880 + 80*E^15 - E^20 + E^5*(32394 + 240/(-20 + E^5 - x)) + 6*x + 10*x^2 + x^4 + 9/(20 - E^5 + x)^2 + 2430/
(20 - E^5 + x) + E^10*(-2410 + 6/(20 - E^5 + x))

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fricas [B]  time = 0.54, size = 95, normalized size = 4.52 \begin {gather*} \frac {x^{6} + 40 \, x^{5} + 410 \, x^{4} + 406 \, x^{3} + 4240 \, x^{2} + {\left (x^{4} + 10 \, x^{2} + 12 \, x + 360\right )} e^{10} - 2 \, {\left (x^{5} + 20 \, x^{4} + 10 \, x^{3} + 206 \, x^{2} + 240 \, x + 3615\right )} e^{5} + 4830 \, x - 6 \, e^{15} + 48609}{x^{2} - 2 \, {\left (x + 20\right )} e^{5} + 40 \, x + e^{10} + 400} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+20*x)*exp(5)^3+(-12*x^4-240*x^3-60*x^2-1212*x)*exp(5)^2+(12*x^5+480*x^4+4860*x^3+2418*x^2+24
480*x-30)*exp(5)-4*x^6-240*x^5-4820*x^4-33206*x^3-24360*x^2-164770*x+618)/(exp(5)^3+(-3*x-60)*exp(5)^2+(3*x^2+
120*x+1200)*exp(5)-x^3-60*x^2-1200*x-8000),x, algorithm="fricas")

[Out]

(x^6 + 40*x^5 + 410*x^4 + 406*x^3 + 4240*x^2 + (x^4 + 10*x^2 + 12*x + 360)*e^10 - 2*(x^5 + 20*x^4 + 10*x^3 + 2
06*x^2 + 240*x + 3615)*e^5 + 4830*x - 6*e^15 + 48609)/(x^2 - 2*(x + 20)*e^5 + 40*x + e^10 + 400)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (2 \, x^{6} + 120 \, x^{5} + 2410 \, x^{4} + 16603 \, x^{3} + 12180 \, x^{2} - 2 \, {\left (x^{3} + 5 \, x\right )} e^{15} + 6 \, {\left (x^{4} + 20 \, x^{3} + 5 \, x^{2} + 101 \, x\right )} e^{10} - 3 \, {\left (2 \, x^{5} + 80 \, x^{4} + 810 \, x^{3} + 403 \, x^{2} + 4080 \, x - 5\right )} e^{5} + 82385 \, x - 309\right )}}{x^{3} + 60 \, x^{2} + 3 \, {\left (x + 20\right )} e^{10} - 3 \, {\left (x^{2} + 40 \, x + 400\right )} e^{5} + 1200 \, x - e^{15} + 8000}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+20*x)*exp(5)^3+(-12*x^4-240*x^3-60*x^2-1212*x)*exp(5)^2+(12*x^5+480*x^4+4860*x^3+2418*x^2+24
480*x-30)*exp(5)-4*x^6-240*x^5-4820*x^4-33206*x^3-24360*x^2-164770*x+618)/(exp(5)^3+(-3*x-60)*exp(5)^2+(3*x^2+
120*x+1200)*exp(5)-x^3-60*x^2-1200*x-8000),x, algorithm="giac")

[Out]

integrate(2*(2*x^6 + 120*x^5 + 2410*x^4 + 16603*x^3 + 12180*x^2 - 2*(x^3 + 5*x)*e^15 + 6*(x^4 + 20*x^3 + 5*x^2
+ 101*x)*e^10 - 3*(2*x^5 + 80*x^4 + 810*x^3 + 403*x^2 + 4080*x - 5)*e^5 + 82385*x - 309)/(x^3 + 60*x^2 + 3*(x
+ 20)*e^10 - 3*(x^2 + 40*x + 400)*e^5 + 1200*x - e^15 + 8000), x)

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maple [B]  time = 0.12, size = 61, normalized size = 2.90

method	result	size
risch	\(x^{4}+10 x^{2}+6 x +\frac {\left (6 \,{\mathrm e}^{10}-240 \,{\mathrm e}^{5}+2430\right ) x -6 \,{\mathrm e}^{15}+360 \,{\mathrm e}^{10}-7230 \,{\mathrm e}^{5}+48609}{{\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}-40 \,{\mathrm e}^{5}+40 x +400}\)	\(61\)
norman	\(\frac {x^{6}+\left (-20 \,{\mathrm e}^{5}+406\right ) x^{3}+\left (-2 \,{\mathrm e}^{5}+40\right ) x^{5}+\left ({\mathrm e}^{10}-40 \,{\mathrm e}^{5}+410\right ) x^{4}+\left (20 \,{\mathrm e}^{15}-1212 \,{\mathrm e}^{10}+24480 \,{\mathrm e}^{5}-164770\right ) x -10 \,{\mathrm e}^{20}+806 \,{\mathrm e}^{15}-24360 \,{\mathrm e}^{10}+327170 \,{\mathrm e}^{5}-1647391}{\left ({\mathrm e}^{5}-x -20\right )^{2}}\)	\(92\)
default	\(x^{4}+10 x^{2}+6 x -2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+\left (-3 \,{\mathrm e}^{5}+60\right ) \textit {\_Z}^{2}+\left (-120 \,{\mathrm e}^{5}+3 \,{\mathrm e}^{10}+1200\right ) \textit {\_Z} -1200 \,{\mathrm e}^{5}+60 \,{\mathrm e}^{10}-{\mathrm e}^{15}+8000\right )}{\sum }\frac {\left (8103-40 \textit {\_R} \,{\mathrm e}^{5}+\textit {\_R} \,{\mathrm e}^{10}-1205 \,{\mathrm e}^{5}+60 \,{\mathrm e}^{10}-{\mathrm e}^{15}+405 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{400+{\mathrm e}^{10}-2 \textit {\_R} \,{\mathrm e}^{5}+\textit {\_R}^{2}-40 \,{\mathrm e}^{5}+40 \textit {\_R}}\right )\)	\(112\)
gosper	\(-\frac {-x^{4} {\mathrm e}^{10}+2 x^{5} {\mathrm e}^{5}-x^{6}+40 x^{4} {\mathrm e}^{5}-40 x^{5}+10 \,{\mathrm e}^{20}-20 x \,{\mathrm e}^{15}+20 x^{3} {\mathrm e}^{5}-410 x^{4}-806 \,{\mathrm e}^{15}+1212 x \,{\mathrm e}^{10}-406 x^{3}+24360 \,{\mathrm e}^{10}-24480 x \,{\mathrm e}^{5}-327170 \,{\mathrm e}^{5}+164770 x +1647391}{{\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}-40 \,{\mathrm e}^{5}+40 x +400}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3+20*x)*exp(5)^3+(-12*x^4-240*x^3-60*x^2-1212*x)*exp(5)^2+(12*x^5+480*x^4+4860*x^3+2418*x^2+24480*x-
30)*exp(5)-4*x^6-240*x^5-4820*x^4-33206*x^3-24360*x^2-164770*x+618)/(exp(5)^3+(-3*x-60)*exp(5)^2+(3*x^2+120*x+
1200)*exp(5)-x^3-60*x^2-1200*x-8000),x,method=_RETURNVERBOSE)

[Out]

x^4+10*x^2+6*x+((6*exp(10)-240*exp(5)+2430)*x-6*exp(15)+360*exp(10)-7230*exp(5)+48609)/(exp(10)-2*x*exp(5)+x^2
-40*exp(5)+40*x+400)

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maxima [B]  time = 0.37, size = 59, normalized size = 2.81 \begin {gather*} x^{4} + 10 \, x^{2} + 6 \, x + \frac {3 \, {\left (2 \, x {\left (e^{10} - 40 \, e^{5} + 405\right )} - 2 \, e^{15} + 120 \, e^{10} - 2410 \, e^{5} + 16203\right )}}{x^{2} - 2 \, x {\left (e^{5} - 20\right )} + e^{10} - 40 \, e^{5} + 400} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+20*x)*exp(5)^3+(-12*x^4-240*x^3-60*x^2-1212*x)*exp(5)^2+(12*x^5+480*x^4+4860*x^3+2418*x^2+24
480*x-30)*exp(5)-4*x^6-240*x^5-4820*x^4-33206*x^3-24360*x^2-164770*x+618)/(exp(5)^3+(-3*x-60)*exp(5)^2+(3*x^2+
120*x+1200)*exp(5)-x^3-60*x^2-1200*x-8000),x, algorithm="maxima")

[Out]

x^4 + 10*x^2 + 6*x + 3*(2*x*(e^10 - 40*e^5 + 405) - 2*e^15 + 120*e^10 - 2410*e^5 + 16203)/(x^2 - 2*x*(e^5 - 20
) + e^10 - 40*e^5 + 400)

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mupad [B]  time = 6.04, size = 126, normalized size = 6.00 \begin {gather*} \frac {360\,{\mathrm {e}}^{10}-7230\,{\mathrm {e}}^5-6\,{\mathrm {e}}^{15}+x\,\left (6\,{\mathrm {e}}^{10}-240\,{\mathrm {e}}^5+2430\right )+48609}{x^2+\left (40-2\,{\mathrm {e}}^5\right )\,x-40\,{\mathrm {e}}^5+{\mathrm {e}}^{10}+400}-x^2\,\left (240\,{\mathrm {e}}^5-6\,{\mathrm {e}}^{10}+6\,{\left ({\mathrm {e}}^5-20\right )}^2-2410\right )+x^4-x\,\left (4860\,{\mathrm {e}}^5-240\,{\mathrm {e}}^{10}+4\,{\mathrm {e}}^{15}-4\,{\left ({\mathrm {e}}^5-20\right )}^3+\left (3\,{\mathrm {e}}^5-60\right )\,\left (480\,{\mathrm {e}}^5-12\,{\mathrm {e}}^{10}+12\,{\left ({\mathrm {e}}^5-20\right )}^2-4820\right )-33206\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((164770*x - exp(15)*(20*x + 4*x^3) + exp(10)*(1212*x + 60*x^2 + 240*x^3 + 12*x^4) - exp(5)*(24480*x + 2418
*x^2 + 4860*x^3 + 480*x^4 + 12*x^5 - 30) + 24360*x^2 + 33206*x^3 + 4820*x^4 + 240*x^5 + 4*x^6 - 618)/(1200*x -
exp(15) - exp(5)*(120*x + 3*x^2 + 1200) + 60*x^2 + x^3 + exp(10)*(3*x + 60) + 8000),x)

[Out]

(360*exp(10) - 7230*exp(5) - 6*exp(15) + x*(6*exp(10) - 240*exp(5) + 2430) + 48609)/(exp(10) - 40*exp(5) + x^2
- x*(2*exp(5) - 40) + 400) - x^2*(240*exp(5) - 6*exp(10) + 6*(exp(5) - 20)^2 - 2410) + x^4 - x*(4860*exp(5) -
240*exp(10) + 4*exp(15) - 4*(exp(5) - 20)^3 + (3*exp(5) - 60)*(480*exp(5) - 12*exp(10) + 12*(exp(5) - 20)^2 -
4820) - 33206)

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sympy [B]  time = 0.54, size = 63, normalized size = 3.00 \begin {gather*} x^{4} + 10 x^{2} + 6 x + \frac {x \left (- 240 e^{5} + 2430 + 6 e^{10}\right ) - 6 e^{15} - 7230 e^{5} + 48609 + 360 e^{10}}{x^{2} + x \left (40 - 2 e^{5}\right ) - 40 e^{5} + 400 + e^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3+20*x)*exp(5)**3+(-12*x**4-240*x**3-60*x**2-1212*x)*exp(5)**2+(12*x**5+480*x**4+4860*x**3+24
18*x**2+24480*x-30)*exp(5)-4*x**6-240*x**5-4820*x**4-33206*x**3-24360*x**2-164770*x+618)/(exp(5)**3+(-3*x-60)*
exp(5)**2+(3*x**2+120*x+1200)*exp(5)-x**3-60*x**2-1200*x-8000),x)

[Out]

x**4 + 10*x**2 + 6*x + (x*(-240*exp(5) + 2430 + 6*exp(10)) - 6*exp(15) - 7230*exp(5) + 48609 + 360*exp(10))/(x
**2 + x*(40 - 2*exp(5)) - 40*exp(5) + 400 + exp(10))

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