3.9.2 \(\int \frac {-20 x^5+120 x^9-160 x^{11}+60 x^{13}+e^{25} (-320+480 x^4-320 x^6+60 x^8)+e^{20} (-1280 x+480 x^3+1680 x^5-1400 x^7+300 x^9)+e^{15} (-1600 x^2+960 x^4+2400 x^6-2440 x^8+600 x^{10})+e^{10} (-880 x^3+600 x^5+1800 x^7-2120 x^9+600 x^{11})+e^5 (-220 x^4+120 x^6+720 x^8-920 x^{10}+300 x^{12})}{e^{25} x^3+5 e^{20} x^4+10 e^{15} x^5+10 e^{10} x^6+5 e^5 x^7+x^8} \, dx\) [802]

Optimal. Leaf size=23 \[ 4+\frac {10 \left (-2+x^2+\frac {x}{e^5+x}\right )^4}{x^2} \]

[Out]

4+10*(x/(exp(5)+x)+x^2-2)^4/x^2

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(133\) vs. \(2(23)=46\).
time = 0.22, antiderivative size = 133, normalized size of antiderivative = 5.78, number of steps used = 2, number of rules used = 1, integrand size = 206, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2099} \begin {gather*} 10 x^6-40 x^4-40 e^5 x^3+20 \left (3+2 e^{10}\right ) x^2+\frac {160}{x^2}+40 e^5 \left (3-e^{10}\right ) x+\frac {40 \left (8-3 e^{10}-e^{30}\right )}{e^5 \left (x+e^5\right )}+\frac {10 \left (17-12 e^{10}+6 e^{20}\right )}{\left (x+e^5\right )^2}+\frac {20 e^5 \left (3-2 e^{10}\right )}{\left (x+e^5\right )^3}+\frac {10 e^{10}}{\left (x+e^5\right )^4}-\frac {320}{e^5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-20*x^5 + 120*x^9 - 160*x^11 + 60*x^13 + E^25*(-320 + 480*x^4 - 320*x^6 + 60*x^8) + E^20*(-1280*x + 480*x
^3 + 1680*x^5 - 1400*x^7 + 300*x^9) + E^15*(-1600*x^2 + 960*x^4 + 2400*x^6 - 2440*x^8 + 600*x^10) + E^10*(-880
*x^3 + 600*x^5 + 1800*x^7 - 2120*x^9 + 600*x^11) + E^5*(-220*x^4 + 120*x^6 + 720*x^8 - 920*x^10 + 300*x^12))/(
E^25*x^3 + 5*E^20*x^4 + 10*E^15*x^5 + 10*E^10*x^6 + 5*E^5*x^7 + x^8),x]

[Out]

160/x^2 - 320/(E^5*x) + 40*E^5*(3 - E^10)*x + 20*(3 + 2*E^10)*x^2 - 40*E^5*x^3 - 40*x^4 + 10*x^6 + (10*E^10)/(
E^5 + x)^4 + (20*E^5*(3 - 2*E^10))/(E^5 + x)^3 + (10*(17 - 12*E^10 + 6*E^20))/(E^5 + x)^2 + (40*(8 - 3*E^10 -
E^30))/(E^5*(E^5 + x))

Rule 2099

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 (-40 e^5 \left (-3+e^{10}\right )-\frac {320}{x^3}+\frac {320}{e^5 x^2}+40 \left (3+2 e^{10}\right ) x-120 e^5 x^2-160 x^3+60 x^5-\frac {40 e^{10}}{\left (e^5+x\right )^5}+\frac {60 e^5 \left (-3+2 e^{10}\right )}{\left (e^5+x\right )^4}-\frac {20 \left (17-12 e^{10}+6 e^{20}\right )}{\left (e^5+x\right )^3}+\frac {40 \left (-8+3 e^{10}+e^{30}\right )}{e^5 \left (e^5+x\right )^2}\right ) \, dx\\ &=\frac {160}{x^2}-\frac {320}{e^5 x}+40 e^5 \left (3-e^{10}\right ) x+20 \left (3+2 e^{10}\right ) x^2-40 e^5 x^3-40 x^4+10 x^6+\frac {10 e^{10}}{\left (e^5+x\right )^4}+\frac {20 e^5 \left (3-2 e^{10}\right )}{\left (e^5+x\right )^3}+\frac {10 \left (17-12 e^{10}+6 e^{20}\right )}{\left (e^5+x\right )^2}+\frac {40 \left (8-3 e^{10}-e^{30}\right )}{e^5 \left (e^5+x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(23)=46\).
time = 0.07, size = 127, normalized size = 5.52 \begin {gather*} 10 \left (\frac {16}{x^2}-\frac {32}{e^5 x}-4 e^5 \left (-3+e^{10}\right ) x+2 \left (3+2 e^{10}\right ) x^2-4 e^5 x^3-4 x^4+x^6+\frac {e^{10}}{\left (e^5+x\right )^4}+\frac {6 e^5-4 e^{15}}{\left (e^5+x\right )^3}+\frac {17-12 e^{10}+6 e^{20}}{\left (e^5+x\right )^2}-\frac {4 \left (-8+3 e^{10}+e^{30}\right )}{e^5 \left (e^5+x\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20*x^5 + 120*x^9 - 160*x^11 + 60*x^13 + E^25*(-320 + 480*x^4 - 320*x^6 + 60*x^8) + E^20*(-1280*x +
480*x^3 + 1680*x^5 - 1400*x^7 + 300*x^9) + E^15*(-1600*x^2 + 960*x^4 + 2400*x^6 - 2440*x^8 + 600*x^10) + E^10
*(-880*x^3 + 600*x^5 + 1800*x^7 - 2120*x^9 + 600*x^11) + E^5*(-220*x^4 + 120*x^6 + 720*x^8 - 920*x^10 + 300*x^
12))/(E^25*x^3 + 5*E^20*x^4 + 10*E^15*x^5 + 10*E^10*x^6 + 5*E^5*x^7 + x^8),x]

[Out]

10*(16/x^2 - 32/(E^5*x) - 4*E^5*(-3 + E^10)*x + 2*(3 + 2*E^10)*x^2 - 4*E^5*x^3 - 4*x^4 + x^6 + E^10/(E^5 + x)^
4 + (6*E^5 - 4*E^15)/(E^5 + x)^3 + (17 - 12*E^10 + 6*E^20)/(E^5 + x)^2 - (4*(-8 + 3*E^10 + E^30))/(E^5*(E^5 +
x)))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.48, size = 231, normalized size = 10.04

method	result	size
risch	\(10 x^{6}-40 x \,{\mathrm e}^{15}+40 \,{\mathrm e}^{10} x^{2}-40 x^{3} {\mathrm e}^{5}-40 x^{4}+120 x \,{\mathrm e}^{5}+60 x^{2}+\frac {\left (-40 \,{\mathrm e}^{25}-120 \,{\mathrm e}^{5}\right ) x^{5}+\left (-120 \,{\mathrm e}^{30}+60 \,{\mathrm e}^{20}-480 \,{\mathrm e}^{10}+10\right ) x^{4}+\left (-120 \,{\mathrm e}^{35}+120 \,{\mathrm e}^{25}-640 \,{\mathrm e}^{15}+80 \,{\mathrm e}^{5}\right ) x^{3}+\left (-40 \,{\mathrm e}^{40}+60 \,{\mathrm e}^{30}-280 \,{\mathrm e}^{20}+240 \,{\mathrm e}^{10}\right ) x^{2}+320 x \,{\mathrm e}^{15}+160 \,{\mathrm e}^{20}}{x^{2} \left ({\mathrm e}^{20}+4 x \,{\mathrm e}^{15}+6 \,{\mathrm e}^{10} x^{2}+4 x^{3} {\mathrm e}^{5}+x^{4}\right )}\)	\(155\)
norman	\(\frac {\left (-40+60 \,{\mathrm e}^{10}\right ) x^{10}+\left (-280 \,{\mathrm e}^{15}+360 \,{\mathrm e}^{5}\right ) x^{7}+\left (40 \,{\mathrm e}^{15}-200 \,{\mathrm e}^{5}\right ) x^{9}+80 \,{\mathrm e}^{5} \left (4 \,{\mathrm e}^{30}-39 \,{\mathrm e}^{20}-8 \,{\mathrm e}^{10}+1\right ) x^{3}+\left (10 \,{\mathrm e}^{20}-360 \,{\mathrm e}^{10}+60\right ) x^{8}+\left (320 \,{\mathrm e}^{25}-2400 \,{\mathrm e}^{15}-120 \,{\mathrm e}^{5}\right ) x^{5}+\left (480 \,{\mathrm e}^{30}-4440 \,{\mathrm e}^{20}-480 \,{\mathrm e}^{10}+10\right ) x^{4}+\left (80 \,{\mathrm e}^{40}-780 \,{\mathrm e}^{30}-280 \,{\mathrm e}^{20}+240 \,{\mathrm e}^{10}\right ) x^{2}+10 x^{12}+160 \,{\mathrm e}^{20}+320 x \,{\mathrm e}^{15}+40 \,{\mathrm e}^{5} x^{11}}{x^{2} \left ({\mathrm e}^{5}+x \right )^{4}}\)	\(198\)
default	\(10 x^{6}-3840 x \,{\mathrm e}^{15}+40 \,{\mathrm e}^{10} x^{2}-40 x^{3} {\mathrm e}^{5}-40 x^{4}+3800 \,{\mathrm e}^{5} {\mathrm e}^{10} x +120 x \,{\mathrm e}^{5}+60 x^{2}-4 \,{\mathrm e}^{-50} \left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{25}+5 \textit {\_Z} \,{\mathrm e}^{20}+10 \textit {\_Z}^{2} {\mathrm e}^{15}+10 \textit {\_Z}^{3} {\mathrm e}^{10}+5 \textit {\_Z}^{4} {\mathrm e}^{5}+\textit {\_Z}^{5}\right )}{\sum }\frac {\left (2 \left (8 \,{\mathrm e}^{45}-3 \,{\mathrm e}^{55}-{\mathrm e}^{75}\right ) \textit {\_R}^{3}+\left (-30 \,{\mathrm e}^{60}+6 \,{\mathrm e}^{70}+65 \,{\mathrm e}^{50}-6 \,{\mathrm e}^{80}\right ) \textit {\_R}^{2}+\left (91 \,{\mathrm e}^{55}+12 \,{\mathrm e}^{75}-48 \,{\mathrm e}^{65}-6 \,{\mathrm e}^{85}\right ) \textit {\_R} +44 \,{\mathrm e}^{60}-24 \,{\mathrm e}^{70}+6 \,{\mathrm e}^{80}-2 \,{\mathrm e}^{90}\right ) \ln \left (x -\textit {\_R} \right )}{{\mathrm e}^{20}+4 \textit {\_R} \,{\mathrm e}^{15}+6 \,{\mathrm e}^{10} \textit {\_R}^{2}+4 \textit {\_R}^{3} {\mathrm e}^{5}+\textit {\_R}^{4}}\right )+\frac {160 \,{\mathrm e}^{-50} {\mathrm e}^{50}}{x^{2}}-\frac {320 \,{\mathrm e}^{45} {\mathrm e}^{-50}}{x}\)	\(231\)
gosper	\(\frac {-280 x^{2} {\mathrm e}^{20}+320 x \,{\mathrm e}^{15}-3120 x^{3} {\mathrm e}^{25}+360 x^{7} {\mathrm e}^{5}-120 x^{5} {\mathrm e}^{5}+80 x^{3} {\mathrm e}^{5}+160 \,{\mathrm e}^{20}+10 x^{12}+60 x^{8}-40 x^{10}+10 x^{4}+10 x^{8} {\mathrm e}^{20}+40 \,{\mathrm e}^{15} x^{9}+60 \,{\mathrm e}^{10} x^{10}+320 \,{\mathrm e}^{35} x^{3}+480 \,{\mathrm e}^{30} x^{4}+320 \,{\mathrm e}^{25} x^{5}-780 \,{\mathrm e}^{30} x^{2}-360 \,{\mathrm e}^{10} x^{8}-2400 \,{\mathrm e}^{15} x^{5}-4440 \,{\mathrm e}^{20} x^{4}-640 \,{\mathrm e}^{15} x^{3}-280 x^{7} {\mathrm e}^{15}+40 \,{\mathrm e}^{5} x^{11}-200 \,{\mathrm e}^{5} x^{9}-480 x^{4} {\mathrm e}^{10}+240 \,{\mathrm e}^{10} x^{2}+80 x^{2} {\mathrm e}^{40}}{x^{2} \left ({\mathrm e}^{20}+4 x \,{\mathrm e}^{15}+6 \,{\mathrm e}^{10} x^{2}+4 x^{3} {\mathrm e}^{5}+x^{4}\right )}\)	\(256\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((60*x^8-320*x^6+480*x^4-320)*exp(5)^5+(300*x^9-1400*x^7+1680*x^5+480*x^3-1280*x)*exp(5)^4+(600*x^10-2440*
x^8+2400*x^6+960*x^4-1600*x^2)*exp(5)^3+(600*x^11-2120*x^9+1800*x^7+600*x^5-880*x^3)*exp(5)^2+(300*x^12-920*x^
10+720*x^8+120*x^6-220*x^4)*exp(5)+60*x^13-160*x^11+120*x^9-20*x^5)/(x^3*exp(5)^5+5*x^4*exp(5)^4+10*x^5*exp(5)
^3+10*x^6*exp(5)^2+5*x^7*exp(5)+x^8),x,method=_RETURNVERBOSE)

[Out]

10*x^6-3000*x*exp(5)^3+300*x^2*exp(5)^2-40*x^3*exp(5)-40*x^4+3800*exp(5)*exp(10)*x-260*exp(10)*x^2+120*x*exp(5
)-840*x*exp(15)+60*x^2-4*exp(-50)*sum((2*(8*exp(45)-3*exp(55)-exp(75))*_R^3+(-30*exp(60)+6*exp(70)+65*exp(50)-
6*exp(80))*_R^2+(91*exp(55)+12*exp(75)-48*exp(65)-6*exp(85))*_R+44*exp(60)-24*exp(70)+6*exp(80)-2*exp(90))/(ex
p(20)+4*_R*exp(15)+6*exp(10)*_R^2+4*_R^3*exp(5)+_R^4)*ln(x-_R),_R=RootOf(exp(25)+5*_Z*exp(20)+10*_Z^2*exp(15)+
10*_Z^3*exp(10)+5*_Z^4*exp(5)+_Z^5))+160*exp(-50)*exp(50)/x^2-320*exp(45)*exp(-50)/x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (22) = 44\).
time = 0.27, size = 158, normalized size = 6.87 \begin {gather*} 10 \, x^{6} - 40 \, x^{4} - 40 \, x^{3} e^{5} + 20 \, x^{2} {\left (2 \, e^{10} + 3\right )} - 40 \, x {\left (e^{15} - 3 \, e^{5}\right )} - \frac {10 \, {\left (4 \, x^{5} {\left (e^{25} + 3 \, e^{5}\right )} + x^{4} {\left (12 \, e^{30} - 6 \, e^{20} + 48 \, e^{10} - 1\right )} + 4 \, x^{3} {\left (3 \, e^{35} - 3 \, e^{25} + 16 \, e^{15} - 2 \, e^{5}\right )} + 2 \, x^{2} {\left (2 \, e^{40} - 3 \, e^{30} + 14 \, e^{20} - 12 \, e^{10}\right )} - 32 \, x e^{15} - 16 \, e^{20}\right )}}{x^{6} + 4 \, x^{5} e^{5} + 6 \, x^{4} e^{10} + 4 \, x^{3} e^{15} + x^{2} e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^8-320*x^6+480*x^4-320)*exp(5)^5+(300*x^9-1400*x^7+1680*x^5+480*x^3-1280*x)*exp(5)^4+(600*x^10
-2440*x^8+2400*x^6+960*x^4-1600*x^2)*exp(5)^3+(600*x^11-2120*x^9+1800*x^7+600*x^5-880*x^3)*exp(5)^2+(300*x^12-
920*x^10+720*x^8+120*x^6-220*x^4)*exp(5)+60*x^13-160*x^11+120*x^9-20*x^5)/(x^3*exp(5)^5+5*x^4*exp(5)^4+10*x^5*
exp(5)^3+10*x^6*exp(5)^2+5*x^7*exp(5)+x^8),x, algorithm="maxima")

[Out]

10*x^6 - 40*x^4 - 40*x^3*e^5 + 20*x^2*(2*e^10 + 3) - 40*x*(e^15 - 3*e^5) - 10*(4*x^5*(e^25 + 3*e^5) + x^4*(12*
e^30 - 6*e^20 + 48*e^10 - 1) + 4*x^3*(3*e^35 - 3*e^25 + 16*e^15 - 2*e^5) + 2*x^2*(2*e^40 - 3*e^30 + 14*e^20 -
12*e^10) - 32*x*e^15 - 16*e^20)/(x^6 + 4*x^5*e^5 + 6*x^4*e^10 + 4*x^3*e^15 + x^2*e^20)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (22) = 44\).
time = 0.39, size = 201, normalized size = 8.74 \begin {gather*} \frac {10 \, {\left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + x^{4} - 16 \, x^{3} e^{35} - 4 \, x^{2} e^{40} - 6 \, {\left (4 \, x^{4} - x^{2}\right )} e^{30} - 8 \, {\left (2 \, x^{5} - 3 \, x^{3}\right )} e^{25} + {\left (x^{8} - 12 \, x^{6} + 60 \, x^{4} - 28 \, x^{2} + 16\right )} e^{20} + 4 \, {\left (x^{9} - 7 \, x^{7} + 24 \, x^{5} - 16 \, x^{3} + 8 \, x\right )} e^{15} + 6 \, {\left (x^{10} - 6 \, x^{8} + 14 \, x^{6} - 8 \, x^{4} + 4 \, x^{2}\right )} e^{10} + 4 \, {\left (x^{11} - 5 \, x^{9} + 9 \, x^{7} - 3 \, x^{5} + 2 \, x^{3}\right )} e^{5}\right )}}{x^{6} + 4 \, x^{5} e^{5} + 6 \, x^{4} e^{10} + 4 \, x^{3} e^{15} + x^{2} e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^8-320*x^6+480*x^4-320)*exp(5)^5+(300*x^9-1400*x^7+1680*x^5+480*x^3-1280*x)*exp(5)^4+(600*x^10
-2440*x^8+2400*x^6+960*x^4-1600*x^2)*exp(5)^3+(600*x^11-2120*x^9+1800*x^7+600*x^5-880*x^3)*exp(5)^2+(300*x^12-
920*x^10+720*x^8+120*x^6-220*x^4)*exp(5)+60*x^13-160*x^11+120*x^9-20*x^5)/(x^3*exp(5)^5+5*x^4*exp(5)^4+10*x^5*
exp(5)^3+10*x^6*exp(5)^2+5*x^7*exp(5)+x^8),x, algorithm="fricas")

[Out]

10*(x^12 - 4*x^10 + 6*x^8 + x^4 - 16*x^3*e^35 - 4*x^2*e^40 - 6*(4*x^4 - x^2)*e^30 - 8*(2*x^5 - 3*x^3)*e^25 + (
x^8 - 12*x^6 + 60*x^4 - 28*x^2 + 16)*e^20 + 4*(x^9 - 7*x^7 + 24*x^5 - 16*x^3 + 8*x)*e^15 + 6*(x^10 - 6*x^8 + 1
4*x^6 - 8*x^4 + 4*x^2)*e^10 + 4*(x^11 - 5*x^9 + 9*x^7 - 3*x^5 + 2*x^3)*e^5)/(x^6 + 4*x^5*e^5 + 6*x^4*e^10 + 4*
x^3*e^15 + x^2*e^20)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (19) = 38\).
time = 2.60, size = 170, normalized size = 7.39 \begin {gather*} 10 x^{6} - 40 x^{4} - 40 x^{3} e^{5} + x^{2} \cdot \left (60 + 40 e^{10}\right ) + x \left (- 40 e^{15} + 120 e^{5}\right ) + \frac {x^{5} \left (- 40 e^{25} - 120 e^{5}\right ) + x^{4} \left (- 120 e^{30} - 480 e^{10} + 10 + 60 e^{20}\right ) + x^{3} \left (- 120 e^{35} - 640 e^{15} + 80 e^{5} + 120 e^{25}\right ) + x^{2} \left (- 40 e^{40} - 280 e^{20} + 240 e^{10} + 60 e^{30}\right ) + 320 x e^{15} + 160 e^{20}}{x^{6} + 4 x^{5} e^{5} + 6 x^{4} e^{10} + 4 x^{3} e^{15} + x^{2} e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x**8-320*x**6+480*x**4-320)*exp(5)**5+(300*x**9-1400*x**7+1680*x**5+480*x**3-1280*x)*exp(5)**4+
(600*x**10-2440*x**8+2400*x**6+960*x**4-1600*x**2)*exp(5)**3+(600*x**11-2120*x**9+1800*x**7+600*x**5-880*x**3)
*exp(5)**2+(300*x**12-920*x**10+720*x**8+120*x**6-220*x**4)*exp(5)+60*x**13-160*x**11+120*x**9-20*x**5)/(x**3*
exp(5)**5+5*x**4*exp(5)**4+10*x**5*exp(5)**3+10*x**6*exp(5)**2+5*x**7*exp(5)+x**8),x)

[Out]

10*x**6 - 40*x**4 - 40*x**3*exp(5) + x**2*(60 + 40*exp(10)) + x*(-40*exp(15) + 120*exp(5)) + (x**5*(-40*exp(25
) - 120*exp(5)) + x**4*(-120*exp(30) - 480*exp(10) + 10 + 60*exp(20)) + x**3*(-120*exp(35) - 640*exp(15) + 80*
exp(5) + 120*exp(25)) + x**2*(-40*exp(40) - 280*exp(20) + 240*exp(10) + 60*exp(30)) + 320*x*exp(15) + 160*exp(
20))/(x**6 + 4*x**5*exp(5) + 6*x**4*exp(10) + 4*x**3*exp(15) + x**2*exp(20))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (22) = 44\).
time = 0.41, size = 149, normalized size = 6.48 \begin {gather*} 10 \, x^{6} - 40 \, x^{4} - 40 \, x^{3} e^{5} + 40 \, x^{2} e^{10} + 60 \, x^{2} - 40 \, x e^{15} + 120 \, x e^{5} - \frac {160 \, {\left (2 \, x - e^{5}\right )} e^{\left (-5\right )}}{x^{2}} - \frac {10 \, {\left (4 \, x^{3} e^{30} + 12 \, x^{3} e^{10} - 32 \, x^{3} + 12 \, x^{2} e^{35} - 6 \, x^{2} e^{25} + 48 \, x^{2} e^{15} - 113 \, x^{2} e^{5} + 12 \, x e^{40} - 12 \, x e^{30} + 64 \, x e^{20} - 136 \, x e^{10} + 4 \, e^{45} - 6 \, e^{35} + 28 \, e^{25} - 56 \, e^{15}\right )} e^{\left (-5\right )}}{{\left (x + e^{5}\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^8-320*x^6+480*x^4-320)*exp(5)^5+(300*x^9-1400*x^7+1680*x^5+480*x^3-1280*x)*exp(5)^4+(600*x^10
-2440*x^8+2400*x^6+960*x^4-1600*x^2)*exp(5)^3+(600*x^11-2120*x^9+1800*x^7+600*x^5-880*x^3)*exp(5)^2+(300*x^12-
920*x^10+720*x^8+120*x^6-220*x^4)*exp(5)+60*x^13-160*x^11+120*x^9-20*x^5)/(x^3*exp(5)^5+5*x^4*exp(5)^4+10*x^5*
exp(5)^3+10*x^6*exp(5)^2+5*x^7*exp(5)+x^8),x, algorithm="giac")

[Out]

10*x^6 - 40*x^4 - 40*x^3*e^5 + 40*x^2*e^10 + 60*x^2 - 40*x*e^15 + 120*x*e^5 - 160*(2*x - e^5)*e^(-5)/x^2 - 10*
(4*x^3*e^30 + 12*x^3*e^10 - 32*x^3 + 12*x^2*e^35 - 6*x^2*e^25 + 48*x^2*e^15 - 113*x^2*e^5 + 12*x*e^40 - 12*x*e
^30 + 64*x*e^20 - 136*x*e^10 + 4*e^45 - 6*e^35 + 28*e^25 - 56*e^15)*e^(-5)/(x + e^5)^4

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Mupad [B]
time = 0.91, size = 268, normalized size = 11.65 \begin {gather*} x^3\,\left (\frac {800\,{\mathrm {e}}^5}{3}-200\,{\mathrm {e}}^{15}+\frac {40\,{\mathrm {e}}^5\,\left (15\,{\mathrm {e}}^{10}-23\right )}{3}\right )+\frac {\left (-120\,{\mathrm {e}}^5-40\,{\mathrm {e}}^{25}\right )\,x^5+\left (60\,{\mathrm {e}}^{20}-480\,{\mathrm {e}}^{10}-120\,{\mathrm {e}}^{30}+10\right )\,x^4+\left (80\,{\mathrm {e}}^5-640\,{\mathrm {e}}^{15}+120\,{\mathrm {e}}^{25}-120\,{\mathrm {e}}^{35}\right )\,x^3+\left (240\,{\mathrm {e}}^{10}-280\,{\mathrm {e}}^{20}+60\,{\mathrm {e}}^{30}-40\,{\mathrm {e}}^{40}\right )\,x^2+320\,{\mathrm {e}}^{15}\,x+160\,{\mathrm {e}}^{20}}{x^6+4\,{\mathrm {e}}^5\,x^5+6\,{\mathrm {e}}^{10}\,x^4+4\,{\mathrm {e}}^{15}\,x^3+{\mathrm {e}}^{20}\,x^2}+x\,\left (1600\,{\mathrm {e}}^{15}-60\,{\mathrm {e}}^{25}+5\,{\mathrm {e}}^5\,\left (520\,{\mathrm {e}}^{10}+5\,{\mathrm {e}}^5\,\left (800\,{\mathrm {e}}^5-600\,{\mathrm {e}}^{15}+40\,{\mathrm {e}}^5\,\left (15\,{\mathrm {e}}^{10}-23\right )\right )-120\right )-10\,{\mathrm {e}}^{10}\,\left (800\,{\mathrm {e}}^5-600\,{\mathrm {e}}^{15}+40\,{\mathrm {e}}^5\,\left (15\,{\mathrm {e}}^{10}-23\right )\right )+20\,{\mathrm {e}}^5\,\left (3\,{\mathrm {e}}^{20}-122\,{\mathrm {e}}^{10}+36\right )\right )-40\,x^4+10\,x^6-x^2\,\left (260\,{\mathrm {e}}^{10}+\frac {5\,{\mathrm {e}}^5\,\left (800\,{\mathrm {e}}^5-600\,{\mathrm {e}}^{15}+40\,{\mathrm {e}}^5\,\left (15\,{\mathrm {e}}^{10}-23\right )\right )}{2}-60\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5)*(120*x^6 - 220*x^4 + 720*x^8 - 920*x^10 + 300*x^12) + exp(10)*(600*x^5 - 880*x^3 + 1800*x^7 - 2120
*x^9 + 600*x^11) + exp(15)*(960*x^4 - 1600*x^2 + 2400*x^6 - 2440*x^8 + 600*x^10) + exp(25)*(480*x^4 - 320*x^6
+ 60*x^8 - 320) + exp(20)*(480*x^3 - 1280*x + 1680*x^5 - 1400*x^7 + 300*x^9) - 20*x^5 + 120*x^9 - 160*x^11 + 6
0*x^13)/(5*x^7*exp(5) + 10*x^6*exp(10) + 10*x^5*exp(15) + 5*x^4*exp(20) + x^3*exp(25) + x^8),x)

[Out]

x^3*((800*exp(5))/3 - 200*exp(15) + (40*exp(5)*(15*exp(10) - 23))/3) + (160*exp(20) - x^5*(120*exp(5) + 40*exp
(25)) + 320*x*exp(15) + x^2*(240*exp(10) - 280*exp(20) + 60*exp(30) - 40*exp(40)) + x^3*(80*exp(5) - 640*exp(1
5) + 120*exp(25) - 120*exp(35)) - x^4*(480*exp(10) - 60*exp(20) + 120*exp(30) - 10))/(4*x^5*exp(5) + 6*x^4*exp
(10) + 4*x^3*exp(15) + x^2*exp(20) + x^6) + x*(1600*exp(15) - 60*exp(25) + 5*exp(5)*(520*exp(10) + 5*exp(5)*(8
00*exp(5) - 600*exp(15) + 40*exp(5)*(15*exp(10) - 23)) - 120) - 10*exp(10)*(800*exp(5) - 600*exp(15) + 40*exp(
5)*(15*exp(10) - 23)) + 20*exp(5)*(3*exp(20) - 122*exp(10) + 36)) - 40*x^4 + 10*x^6 - x^2*(260*exp(10) + (5*ex
p(5)*(800*exp(5) - 600*exp(15) + 40*exp(5)*(15*exp(10) - 23)))/2 - 60)

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