3.81.62 \(\int \frac {e^4 (320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10})}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx\)

Optimal. Leaf size=26 \[ \frac {e^4 \left (1+\frac {x (3+x)}{-2+x}\right )^4}{\left (1+x+x^2\right )^4} \]

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Rubi [B]  time = 0.83, antiderivative size = 197, normalized size of antiderivative = 7.58, number of steps used = 23, number of rules used = 6, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 2074, 638, 614, 618, 204} \begin {gather*} \frac {162 e^4 (2722-3085 x)}{117649 \left (x^2+x+1\right )^2}+\frac {250560 e^4 (2 x+1)}{117649 \left (x^2+x+1\right )}-\frac {960 e^4 (4168 x+2039)}{823543 \left (x^2+x+1\right )}+\frac {53595 e^4 (2 x+1)}{16807 \left (x^2+x+1\right )^2}-\frac {648 e^4 (19-577 x)}{16807 \left (x^2+x+1\right )^3}-\frac {3159 e^4 (2 x+1)}{343 \left (x^2+x+1\right )^3}+\frac {729 e^4 (16-39 x)}{2401 \left (x^2+x+1\right )^4}-\frac {493440 e^4}{823543 (2-x)}-\frac {182400 e^4}{117649 (2-x)^2}-\frac {24000 e^4}{16807 (2-x)^3}+\frac {10000 e^4}{2401 (2-x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4*(320 - 1664*x + 1536*x^2 + 4032*x^3 - 6720*x^4 + 1248*x^5 + 912*x^6 - 1008*x^7 - 516*x^8 - 80*x^9 - 4
*x^10))/(-32 - 80*x - 160*x^2 - 120*x^3 - 50*x^4 + 119*x^5 + 95*x^6 + 75*x^7 - 60*x^8 - 25*x^9 - 31*x^10 + 25*
x^11 + 5*x^13 - 5*x^14 + x^15),x]

[Out]

(10000*E^4)/(2401*(2 - x)^4) - (24000*E^4)/(16807*(2 - x)^3) - (182400*E^4)/(117649*(2 - x)^2) - (493440*E^4)/
(823543*(2 - x)) + (729*E^4*(16 - 39*x))/(2401*(1 + x + x^2)^4) - (648*E^4*(19 - 577*x))/(16807*(1 + x + x^2)^
3) - (3159*E^4*(1 + 2*x))/(343*(1 + x + x^2)^3) + (162*E^4*(2722 - 3085*x))/(117649*(1 + x + x^2)^2) + (53595*
E^4*(1 + 2*x))/(16807*(1 + x + x^2)^2) + (250560*E^4*(1 + 2*x))/(117649*(1 + x + x^2)) - (960*E^4*(2039 + 4168
*x))/(823543*(1 + x + x^2))

Rule 12

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

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} &=e^4 \int \frac {320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10}}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx\\ &=e^4 \int \left (-\frac {40000}{2401 (-2+x)^5}-\frac {72000}{16807 (-2+x)^4}+\frac {364800}{117649 (-2+x)^3}-\frac {493440}{823543 (-2+x)^2}-\frac {2916 (94+71 x)}{2401 \left (1+x+x^2\right )^5}+\frac {5832 (391+205 x)}{16807 \left (1+x+x^2\right )^4}-\frac {972 (2964+2843 x)}{117649 \left (1+x+x^2\right )^3}-\frac {2880 (2099+30 x)}{823543 \left (1+x+x^2\right )^2}+\frac {493440}{823543 \left (1+x+x^2\right )}\right ) \, dx\\ &=\frac {10000 e^4}{2401 (2-x)^4}-\frac {24000 e^4}{16807 (2-x)^3}-\frac {182400 e^4}{117649 (2-x)^2}-\frac {493440 e^4}{823543 (2-x)}-\frac {\left (2880 e^4\right ) \int \frac {2099+30 x}{\left (1+x+x^2\right )^2} \, dx}{823543}-\frac {\left (972 e^4\right ) \int \frac {2964+2843 x}{\left (1+x+x^2\right )^3} \, dx}{117649}+\frac {\left (5832 e^4\right ) \int \frac {391+205 x}{\left (1+x+x^2\right )^4} \, dx}{16807}+\frac {\left (493440 e^4\right ) \int \frac {1}{1+x+x^2} \, dx}{823543}-\frac {\left (2916 e^4\right ) \int \frac {94+71 x}{\left (1+x+x^2\right )^5} \, dx}{2401}\\ &=\frac {10000 e^4}{2401 (2-x)^4}-\frac {24000 e^4}{16807 (2-x)^3}-\frac {182400 e^4}{117649 (2-x)^2}-\frac {493440 e^4}{823543 (2-x)}+\frac {729 e^4 (16-39 x)}{2401 \left (1+x+x^2\right )^4}-\frac {648 e^4 (19-577 x)}{16807 \left (1+x+x^2\right )^3}+\frac {162 e^4 (2722-3085 x)}{117649 \left (1+x+x^2\right )^2}-\frac {960 e^4 (2039+4168 x)}{823543 \left (1+x+x^2\right )}-\frac {\left (986880 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )}{823543}-\frac {\left (4001280 e^4\right ) \int \frac {1}{1+x+x^2} \, dx}{823543}-\frac {\left (1499310 e^4\right ) \int \frac {1}{\left (1+x+x^2\right )^2} \, dx}{117649}-\frac {1}{343} \left (28431 e^4\right ) \int \frac {1}{\left (1+x+x^2\right )^4} \, dx+\frac {\left (1869480 e^4\right ) \int \frac {1}{\left (1+x+x^2\right )^3} \, dx}{16807}\\ &=\frac {10000 e^4}{2401 (2-x)^4}-\frac {24000 e^4}{16807 (2-x)^3}-\frac {182400 e^4}{117649 (2-x)^2}-\frac {493440 e^4}{823543 (2-x)}+\frac {729 e^4 (16-39 x)}{2401 \left (1+x+x^2\right )^4}-\frac {648 e^4 (19-577 x)}{16807 \left (1+x+x^2\right )^3}-\frac {3159 e^4 (1+2 x)}{343 \left (1+x+x^2\right )^3}+\frac {162 e^4 (2722-3085 x)}{117649 \left (1+x+x^2\right )^2}+\frac {311580 e^4 (1+2 x)}{16807 \left (1+x+x^2\right )^2}-\frac {499770 e^4 (1+2 x)}{117649 \left (1+x+x^2\right )}-\frac {960 e^4 (2039+4168 x)}{823543 \left (1+x+x^2\right )}+\frac {328960 \sqrt {3} e^4 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{823543}-\frac {\left (999540 e^4\right ) \int \frac {1}{1+x+x^2} \, dx}{117649}+\frac {\left (8002560 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )}{823543}-\frac {1}{343} \left (31590 e^4\right ) \int \frac {1}{\left (1+x+x^2\right )^3} \, dx+\frac {\left (1869480 e^4\right ) \int \frac {1}{\left (1+x+x^2\right )^2} \, dx}{16807}\\ &=\frac {10000 e^4}{2401 (2-x)^4}-\frac {24000 e^4}{16807 (2-x)^3}-\frac {182400 e^4}{117649 (2-x)^2}-\frac {493440 e^4}{823543 (2-x)}+\frac {729 e^4 (16-39 x)}{2401 \left (1+x+x^2\right )^4}-\frac {648 e^4 (19-577 x)}{16807 \left (1+x+x^2\right )^3}-\frac {3159 e^4 (1+2 x)}{343 \left (1+x+x^2\right )^3}+\frac {162 e^4 (2722-3085 x)}{117649 \left (1+x+x^2\right )^2}+\frac {53595 e^4 (1+2 x)}{16807 \left (1+x+x^2\right )^2}+\frac {3862350 e^4 (1+2 x)}{117649 \left (1+x+x^2\right )}-\frac {960 e^4 (2039+4168 x)}{823543 \left (1+x+x^2\right )}-\frac {334080 \sqrt {3} e^4 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{117649}+\frac {\left (1999080 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )}{117649}+\frac {\left (1246320 e^4\right ) \int \frac {1}{1+x+x^2} \, dx}{16807}-\frac {1}{343} \left (31590 e^4\right ) \int \frac {1}{\left (1+x+x^2\right )^2} \, dx\\ &=\frac {10000 e^4}{2401 (2-x)^4}-\frac {24000 e^4}{16807 (2-x)^3}-\frac {182400 e^4}{117649 (2-x)^2}-\frac {493440 e^4}{823543 (2-x)}+\frac {729 e^4 (16-39 x)}{2401 \left (1+x+x^2\right )^4}-\frac {648 e^4 (19-577 x)}{16807 \left (1+x+x^2\right )^3}-\frac {3159 e^4 (1+2 x)}{343 \left (1+x+x^2\right )^3}+\frac {162 e^4 (2722-3085 x)}{117649 \left (1+x+x^2\right )^2}+\frac {53595 e^4 (1+2 x)}{16807 \left (1+x+x^2\right )^2}+\frac {250560 e^4 (1+2 x)}{117649 \left (1+x+x^2\right )}-\frac {960 e^4 (2039+4168 x)}{823543 \left (1+x+x^2\right )}-\frac {142920 \sqrt {3} e^4 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{16807}-\frac {1}{343} \left (21060 e^4\right ) \int \frac {1}{1+x+x^2} \, dx-\frac {\left (2492640 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )}{16807}\\ &=\frac {10000 e^4}{2401 (2-x)^4}-\frac {24000 e^4}{16807 (2-x)^3}-\frac {182400 e^4}{117649 (2-x)^2}-\frac {493440 e^4}{823543 (2-x)}+\frac {729 e^4 (16-39 x)}{2401 \left (1+x+x^2\right )^4}-\frac {648 e^4 (19-577 x)}{16807 \left (1+x+x^2\right )^3}-\frac {3159 e^4 (1+2 x)}{343 \left (1+x+x^2\right )^3}+\frac {162 e^4 (2722-3085 x)}{117649 \left (1+x+x^2\right )^2}+\frac {53595 e^4 (1+2 x)}{16807 \left (1+x+x^2\right )^2}+\frac {250560 e^4 (1+2 x)}{117649 \left (1+x+x^2\right )}-\frac {960 e^4 (2039+4168 x)}{823543 \left (1+x+x^2\right )}+\frac {14040}{343} \sqrt {3} e^4 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )+\frac {1}{343} \left (42120 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {10000 e^4}{2401 (2-x)^4}-\frac {24000 e^4}{16807 (2-x)^3}-\frac {182400 e^4}{117649 (2-x)^2}-\frac {493440 e^4}{823543 (2-x)}+\frac {729 e^4 (16-39 x)}{2401 \left (1+x+x^2\right )^4}-\frac {648 e^4 (19-577 x)}{16807 \left (1+x+x^2\right )^3}-\frac {3159 e^4 (1+2 x)}{343 \left (1+x+x^2\right )^3}+\frac {162 e^4 (2722-3085 x)}{117649 \left (1+x+x^2\right )^2}+\frac {53595 e^4 (1+2 x)}{16807 \left (1+x+x^2\right )^2}+\frac {250560 e^4 (1+2 x)}{117649 \left (1+x+x^2\right )}-\frac {960 e^4 (2039+4168 x)}{823543 \left (1+x+x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.04 \begin {gather*} \frac {e^4 \left (-2+4 x+x^2\right )^4}{\left (2+x+x^2-x^3\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(320 - 1664*x + 1536*x^2 + 4032*x^3 - 6720*x^4 + 1248*x^5 + 912*x^6 - 1008*x^7 - 516*x^8 - 80*x
^9 - 4*x^10))/(-32 - 80*x - 160*x^2 - 120*x^3 - 50*x^4 + 119*x^5 + 95*x^6 + 75*x^7 - 60*x^8 - 25*x^9 - 31*x^10
+ 25*x^11 + 5*x^13 - 5*x^14 + x^15),x]

[Out]

(E^4*(-2 + 4*x + x^2)^4)/(2 + x + x^2 - x^3)^4

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fricas [B]  time = 1.35, size = 94, normalized size = 3.62 \begin {gather*} \frac {{\left (x^{8} + 16 \, x^{7} + 88 \, x^{6} + 160 \, x^{5} - 104 \, x^{4} - 320 \, x^{3} + 352 \, x^{2} - 128 \, x + 16\right )} e^{4}}{x^{12} - 4 \, x^{11} + 2 \, x^{10} + 19 \, x^{8} - 8 \, x^{7} - 14 \, x^{6} - 44 \, x^{5} + x^{4} + 24 \, x^{3} + 56 \, x^{2} + 32 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^10-80*x^9-516*x^8-1008*x^7+912*x^6+1248*x^5-6720*x^4+4032*x^3+1536*x^2-1664*x+320)*exp(1)^4/(x
^15-5*x^14+5*x^13+25*x^11-31*x^10-25*x^9-60*x^8+75*x^7+95*x^6+119*x^5-50*x^4-120*x^3-160*x^2-80*x-32),x, algor
ithm="fricas")

[Out]

(x^8 + 16*x^7 + 88*x^6 + 160*x^5 - 104*x^4 - 320*x^3 + 352*x^2 - 128*x + 16)*e^4/(x^12 - 4*x^11 + 2*x^10 + 19*
x^8 - 8*x^7 - 14*x^6 - 44*x^5 + x^4 + 24*x^3 + 56*x^2 + 32*x + 16)

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giac [B]  time = 0.27, size = 56, normalized size = 2.15 \begin {gather*} \frac {{\left (x^{8} + 16 \, x^{7} + 88 \, x^{6} + 160 \, x^{5} - 104 \, x^{4} - 320 \, x^{3} + 352 \, x^{2} - 128 \, x + 16\right )} e^{4}}{{\left (x^{3} - x^{2} - x - 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^10-80*x^9-516*x^8-1008*x^7+912*x^6+1248*x^5-6720*x^4+4032*x^3+1536*x^2-1664*x+320)*exp(1)^4/(x
^15-5*x^14+5*x^13+25*x^11-31*x^10-25*x^9-60*x^8+75*x^7+95*x^6+119*x^5-50*x^4-120*x^3-160*x^2-80*x-32),x, algor
ithm="giac")

[Out]

(x^8 + 16*x^7 + 88*x^6 + 160*x^5 - 104*x^4 - 320*x^3 + 352*x^2 - 128*x + 16)*e^4/(x^3 - x^2 - x - 2)^4

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maple [B]  time = 0.13, size = 81, normalized size = 3.12

method	result	size
default	\(4 \,{\mathrm e}^{4} \left (\frac {2500}{2401 \left (x -2\right )^{4}}+\frac {6000}{16807 \left (x -2\right )^{3}}-\frac {45600}{117649 \left (x -2\right )^{2}}+\frac {123360}{823543 \left (x -2\right )}+\frac {-\frac {123360}{823543} x^{7}-\frac {420960}{823543} x^{6}-\frac {454320}{823543} x^{5}+\frac {4545543}{3294172} x^{4}+\frac {3863418}{823543} x^{3}+\frac {461016}{117649} x^{2}-\frac {678192}{823543} x +\frac {330522}{823543}}{\left (x^{2}+x +1\right )^{4}}\right )\)	\(81\)
norman	\(\frac {160 x^{5} {\mathrm e}^{4}+88 x^{6} {\mathrm e}^{4}+16 x^{7} {\mathrm e}^{4}-104 x^{4} {\mathrm e}^{4}+x^{8} {\mathrm e}^{4}-320 x^{3} {\mathrm e}^{4}-128 x \,{\mathrm e}^{4}+352 x^{2} {\mathrm e}^{4}+16 \,{\mathrm e}^{4}}{\left (x^{3}-x^{2}-x -2\right )^{4}}\)	\(93\)
risch	\(\frac {\left (x^{8}+16 x^{7}+88 x^{6}+160 x^{5}-104 x^{4}-320 x^{3}+352 x^{2}-128 x +16\right ) {\mathrm e}^{4}}{x^{12}-4 x^{11}+2 x^{10}+19 x^{8}-8 x^{7}-14 x^{6}-44 x^{5}+x^{4}+24 x^{3}+56 x^{2}+32 x +16}\)	\(95\)
gosper	\(\frac {\left (x^{8}+16 x^{7}+88 x^{6}+160 x^{5}-104 x^{4}-320 x^{3}+352 x^{2}-128 x +16\right ) {\mathrm e}^{4}}{x^{12}-4 x^{11}+2 x^{10}+19 x^{8}-8 x^{7}-14 x^{6}-44 x^{5}+x^{4}+24 x^{3}+56 x^{2}+32 x +16}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^10-80*x^9-516*x^8-1008*x^7+912*x^6+1248*x^5-6720*x^4+4032*x^3+1536*x^2-1664*x+320)*exp(1)^4/(x^15-5*
x^14+5*x^13+25*x^11-31*x^10-25*x^9-60*x^8+75*x^7+95*x^6+119*x^5-50*x^4-120*x^3-160*x^2-80*x-32),x,method=_RETU
RNVERBOSE)

[Out]

4*exp(1)^4*(2500/2401/(x-2)^4+6000/16807/(x-2)^3-45600/117649/(x-2)^2+123360/823543/(x-2)+3/823543*(-41120*x^7
-140320*x^6-151440*x^5+1515181/4*x^4+1287806*x^3+1075704*x^2-226064*x+110174)/(x^2+x+1)^4)

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maxima [B]  time = 0.35, size = 94, normalized size = 3.62 \begin {gather*} \frac {{\left (x^{8} + 16 \, x^{7} + 88 \, x^{6} + 160 \, x^{5} - 104 \, x^{4} - 320 \, x^{3} + 352 \, x^{2} - 128 \, x + 16\right )} e^{4}}{x^{12} - 4 \, x^{11} + 2 \, x^{10} + 19 \, x^{8} - 8 \, x^{7} - 14 \, x^{6} - 44 \, x^{5} + x^{4} + 24 \, x^{3} + 56 \, x^{2} + 32 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^10-80*x^9-516*x^8-1008*x^7+912*x^6+1248*x^5-6720*x^4+4032*x^3+1536*x^2-1664*x+320)*exp(1)^4/(x
^15-5*x^14+5*x^13+25*x^11-31*x^10-25*x^9-60*x^8+75*x^7+95*x^6+119*x^5-50*x^4-120*x^3-160*x^2-80*x-32),x, algor
ithm="maxima")

[Out]

(x^8 + 16*x^7 + 88*x^6 + 160*x^5 - 104*x^4 - 320*x^3 + 352*x^2 - 128*x + 16)*e^4/(x^12 - 4*x^11 + 2*x^10 + 19*
x^8 - 8*x^7 - 14*x^6 - 44*x^5 + x^4 + 24*x^3 + 56*x^2 + 32*x + 16)

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mupad [B]  time = 0.22, size = 111, normalized size = 4.27 \begin {gather*} \frac {493440\,{\mathrm {e}}^4}{823543\,\left (x-2\right )}-\frac {182400\,{\mathrm {e}}^4}{117649\,{\left (x-2\right )}^2}+\frac {24000\,{\mathrm {e}}^4}{16807\,{\left (x-2\right )}^3}+\frac {10000\,{\mathrm {e}}^4}{2401\,{\left (x-2\right )}^4}-\frac {729\,{\mathrm {e}}^4\,\left (39\,x-16\right )}{2401\,{\left (x^2+x+1\right )}^4}-\frac {1920\,{\mathrm {e}}^4\,\left (257\,x+106\right )}{823543\,\left (x^2+x+1\right )}+\frac {81\,{\mathrm {e}}^4\,\left (794\,x-2063\right )}{16807\,{\left (x^2+x+1\right )}^3}+\frac {27\,{\mathrm {e}}^4\,\left (9280\,x+30227\right )}{117649\,{\left (x^2+x+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4)*(1664*x - 1536*x^2 - 4032*x^3 + 6720*x^4 - 1248*x^5 - 912*x^6 + 1008*x^7 + 516*x^8 + 80*x^9 + 4*x^
10 - 320))/(80*x + 160*x^2 + 120*x^3 + 50*x^4 - 119*x^5 - 95*x^6 - 75*x^7 + 60*x^8 + 25*x^9 + 31*x^10 - 25*x^1
1 - 5*x^13 + 5*x^14 - x^15 + 32),x)

[Out]

(493440*exp(4))/(823543*(x - 2)) - (182400*exp(4))/(117649*(x - 2)^2) + (24000*exp(4))/(16807*(x - 2)^3) + (10
000*exp(4))/(2401*(x - 2)^4) - (729*exp(4)*(39*x - 16))/(2401*(x + x^2 + 1)^4) - (1920*exp(4)*(257*x + 106))/(
823543*(x + x^2 + 1)) + (81*exp(4)*(794*x - 2063))/(16807*(x + x^2 + 1)^3) + (27*exp(4)*(9280*x + 30227))/(117
649*(x + x^2 + 1)^2)

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sympy [B]  time = 1.02, size = 122, normalized size = 4.69 \begin {gather*} - \frac {- x^{8} e^{4} - 16 x^{7} e^{4} - 88 x^{6} e^{4} - 160 x^{5} e^{4} + 104 x^{4} e^{4} + 320 x^{3} e^{4} - 352 x^{2} e^{4} + 128 x e^{4} - 16 e^{4}}{x^{12} - 4 x^{11} + 2 x^{10} + 19 x^{8} - 8 x^{7} - 14 x^{6} - 44 x^{5} + x^{4} + 24 x^{3} + 56 x^{2} + 32 x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**10-80*x**9-516*x**8-1008*x**7+912*x**6+1248*x**5-6720*x**4+4032*x**3+1536*x**2-1664*x+320)*ex
p(1)**4/(x**15-5*x**14+5*x**13+25*x**11-31*x**10-25*x**9-60*x**8+75*x**7+95*x**6+119*x**5-50*x**4-120*x**3-160
*x**2-80*x-32),x)

[Out]

-(-x**8*exp(4) - 16*x**7*exp(4) - 88*x**6*exp(4) - 160*x**5*exp(4) + 104*x**4*exp(4) + 320*x**3*exp(4) - 352*x
**2*exp(4) + 128*x*exp(4) - 16*exp(4))/(x**12 - 4*x**11 + 2*x**10 + 19*x**8 - 8*x**7 - 14*x**6 - 44*x**5 + x**
4 + 24*x**3 + 56*x**2 + 32*x + 16)

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