3.23.15 \(\int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6)+e^6 (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8)+e^3 (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10})}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx\)

Optimal. Leaf size=35 \[ \left (x+5 x^2\right )^2 \left (5+x+\frac {3}{\frac {e^3}{x}+x}\right )^2-\log (-x) \]

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Rubi [B]  time = 0.35, antiderivative size = 181, normalized size of antiderivative = 5.17, number of steps used = 8, number of rules used = 4, integrand size = 175, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2074, 639, 199, 203} \begin {gather*} 25 x^6+260 x^5+876 x^4+1070 x^3+2 \left (278-75 e^3\right ) x^2+\frac {135 e^3 x}{x^2+e^3}-\frac {3 e^3 \left (5 \left (23-54 e^3\right ) x+50 e^6-327 e^3+6\right )}{x^2+e^3}+\frac {9 e^6 \left (10 x-25 e^3+1\right )}{\left (x^2+e^3\right )^2}+30 \left (4-27 e^3\right ) x-\log (x)+30 e^{3/2} \left (7-27 e^3\right ) \tan ^{-1}\left (\frac {x}{e^{3/2}}\right )-15 e^{3/2} \left (23-54 e^3\right ) \tan ^{-1}\left (\frac {x}{e^{3/2}}\right )+135 e^{3/2} \tan ^{-1}\left (\frac {x}{e^{3/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^6 + 120*x^7 + 1112*x^8 + 3210*x^9 + 3504*x^10 + 1300*x^11 + 150*x^12 + E^9*(-1 + 50*x^2 + 780*x^3 + 29
04*x^4 + 1300*x^5 + 150*x^6) + E^6*(-3*x^2 + 90*x^3 + 1374*x^4 + 6390*x^5 + 9612*x^6 + 3900*x^7 + 450*x^8) + E
^3*(33*x^4 + 570*x^5 + 3336*x^6 + 8820*x^7 + 10212*x^8 + 3900*x^9 + 450*x^10))/(E^9*x + 3*E^6*x^3 + 3*E^3*x^5
+ x^7),x]

[Out]

30*(4 - 27*E^3)*x + 2*(278 - 75*E^3)*x^2 + 1070*x^3 + 876*x^4 + 260*x^5 + 25*x^6 + (9*E^6*(1 - 25*E^3 + 10*x))
/(E^3 + x^2)^2 + (135*E^3*x)/(E^3 + x^2) - (3*E^3*(6 - 327*E^3 + 50*E^6 + 5*(23 - 54*E^3)*x))/(E^3 + x^2) + 13
5*E^(3/2)*ArcTan[x/E^(3/2)] - 15*E^(3/2)*(23 - 54*E^3)*ArcTan[x/E^(3/2)] + 30*E^(3/2)*(7 - 27*E^3)*ArcTan[x/E^
(3/2)] - Log[x]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

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

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& 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} &=\int \left (-30 \left (-4+27 e^3\right )-\frac {1}{x}-4 \left (-278+75 e^3\right ) x+3210 x^2+3504 x^3+1300 x^4+150 x^5+\frac {36 \left (10 e^9-e^6 \left (1-25 e^3\right ) x\right )}{\left (e^3+x^2\right )^3}+\frac {6 \left (-5 e^6 \left (23-54 e^3\right )+e^3 \left (6-327 e^3+50 e^6\right ) x\right )}{\left (e^3+x^2\right )^2}-\frac {30 e^3 \left (-7+27 e^3\right )}{e^3+x^2}\right ) \, dx\\ &=30 \left (4-27 e^3\right ) x+2 \left (278-75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6-\log (x)+6 \int \frac {-5 e^6 \left (23-54 e^3\right )+e^3 \left (6-327 e^3+50 e^6\right ) x}{\left (e^3+x^2\right )^2} \, dx+36 \int \frac {10 e^9-e^6 \left (1-25 e^3\right ) x}{\left (e^3+x^2\right )^3} \, dx+\left (30 e^3 \left (7-27 e^3\right )\right ) \int \frac {1}{e^3+x^2} \, dx\\ &=30 \left (4-27 e^3\right ) x+2 \left (278-75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6+\frac {9 e^6 \left (1-25 e^3+10 x\right )}{\left (e^3+x^2\right )^2}-\frac {3 e^3 \left (6-327 e^3+50 e^6+5 \left (23-54 e^3\right ) x\right )}{e^3+x^2}+30 e^{3/2} \left (7-27 e^3\right ) \tan ^{-1}\left (\frac {x}{e^{3/2}}\right )-\log (x)+\left (270 e^6\right ) \int \frac {1}{\left (e^3+x^2\right )^2} \, dx-\left (15 e^3 \left (23-54 e^3\right )\right ) \int \frac {1}{e^3+x^2} \, dx\\ &=30 \left (4-27 e^3\right ) x+2 \left (278-75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6+\frac {9 e^6 \left (1-25 e^3+10 x\right )}{\left (e^3+x^2\right )^2}+\frac {135 e^3 x}{e^3+x^2}-\frac {3 e^3 \left (6-327 e^3+50 e^6+5 \left (23-54 e^3\right ) x\right )}{e^3+x^2}-15 e^{3/2} \left (23-54 e^3\right ) \tan ^{-1}\left (\frac {x}{e^{3/2}}\right )+30 e^{3/2} \left (7-27 e^3\right ) \tan ^{-1}\left (\frac {x}{e^{3/2}}\right )-\log (x)+\left (135 e^3\right ) \int \frac {1}{e^3+x^2} \, dx\\ &=30 \left (4-27 e^3\right ) x+2 \left (278-75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6+\frac {9 e^6 \left (1-25 e^3+10 x\right )}{\left (e^3+x^2\right )^2}+\frac {135 e^3 x}{e^3+x^2}-\frac {3 e^3 \left (6-327 e^3+50 e^6+5 \left (23-54 e^3\right ) x\right )}{e^3+x^2}+135 e^{3/2} \tan ^{-1}\left (\frac {x}{e^{3/2}}\right )-15 e^{3/2} \left (23-54 e^3\right ) \tan ^{-1}\left (\frac {x}{e^{3/2}}\right )+30 e^{3/2} \left (7-27 e^3\right ) \tan ^{-1}\left (\frac {x}{e^{3/2}}\right )-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 105, normalized size = 3.00 \begin {gather*} -30 \left (-4+27 e^3\right ) x-2 \left (-278+75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6+\frac {9 e^6 \left (1-25 e^3+10 x\right )}{\left (e^3+x^2\right )^2}-\frac {3 e^3 \left (6+50 e^6+70 x-3 e^3 (109+90 x)\right )}{e^3+x^2}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^6 + 120*x^7 + 1112*x^8 + 3210*x^9 + 3504*x^10 + 1300*x^11 + 150*x^12 + E^9*(-1 + 50*x^2 + 780*x^
3 + 2904*x^4 + 1300*x^5 + 150*x^6) + E^6*(-3*x^2 + 90*x^3 + 1374*x^4 + 6390*x^5 + 9612*x^6 + 3900*x^7 + 450*x^
8) + E^3*(33*x^4 + 570*x^5 + 3336*x^6 + 8820*x^7 + 10212*x^8 + 3900*x^9 + 450*x^10))/(E^9*x + 3*E^6*x^3 + 3*E^
3*x^5 + x^7),x]

[Out]

-30*(-4 + 27*E^3)*x - 2*(-278 + 75*E^3)*x^2 + 1070*x^3 + 876*x^4 + 260*x^5 + 25*x^6 + (9*E^6*(1 - 25*E^3 + 10*
x))/(E^3 + x^2)^2 - (3*E^3*(6 + 50*E^6 + 70*x - 3*E^3*(109 + 90*x)))/(E^3 + x^2) - Log[x]

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fricas [B]  time = 0.70, size = 149, normalized size = 4.26 \begin {gather*} \frac {25 \, x^{10} + 260 \, x^{9} + 876 \, x^{8} + 1070 \, x^{7} + 556 \, x^{6} + 120 \, x^{5} - 12 \, {\left (25 \, x^{2} - 63\right )} e^{9} + {\left (25 \, x^{6} + 260 \, x^{5} + 576 \, x^{4} + 260 \, x^{3} + 1537 \, x^{2} - 9\right )} e^{6} + 2 \, {\left (25 \, x^{8} + 260 \, x^{7} + 801 \, x^{6} + 665 \, x^{5} + 556 \, x^{4} + 15 \, x^{3} - 9 \, x^{2}\right )} e^{3} - {\left (x^{4} + 2 \, x^{2} e^{3} + e^{6}\right )} \log \relax (x) - 150 \, e^{12}}{x^{4} + 2 \, x^{2} e^{3} + e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+3900*x^7+9612*x^6+6390*x^5+1374*x^4+
90*x^3-3*x^2)*exp(3)^2+(450*x^10+3900*x^9+10212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^1
1+3504*x^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5*exp(3)+x^7),x, algorithm="fricas")

[Out]

(25*x^10 + 260*x^9 + 876*x^8 + 1070*x^7 + 556*x^6 + 120*x^5 - 12*(25*x^2 - 63)*e^9 + (25*x^6 + 260*x^5 + 576*x
^4 + 260*x^3 + 1537*x^2 - 9)*e^6 + 2*(25*x^8 + 260*x^7 + 801*x^6 + 665*x^5 + 556*x^4 + 15*x^3 - 9*x^2)*e^3 - (
x^4 + 2*x^2*e^3 + e^6)*log(x) - 150*e^12)/(x^4 + 2*x^2*e^3 + e^6)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+3900*x^7+9612*x^6+6390*x^5+1374*x^4+
90*x^3-3*x^2)*exp(3)^2+(450*x^10+3900*x^9+10212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^1
1+3504*x^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5*exp(3)+x^7),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.19, size = 116, normalized size = 3.31

method	result	size
risch	\(25 x^{6}+260 x^{5}+876 x^{4}-150 x^{2} {\mathrm e}^{3}+1070 x^{3}-810 x \,{\mathrm e}^{3}+556 x^{2}+120 x +\frac {30 \,{\mathrm e}^{3} \left (27 \,{\mathrm e}^{3}-7\right ) x^{3}+\left (-150 \,{\mathrm e}^{9}+981 \,{\mathrm e}^{6}-18 \,{\mathrm e}^{3}\right ) x^{2}+\left (810 \,{\mathrm e}^{9}-120 \,{\mathrm e}^{6}\right ) x -150 \,{\mathrm e}^{12}+756 \,{\mathrm e}^{9}-9 \,{\mathrm e}^{6}}{x^{4}+2 x^{2} {\mathrm e}^{3}+{\mathrm e}^{6}}-\ln \relax (x )\)	\(116\)
norman	\(\frac {\left (876+50 \,{\mathrm e}^{3}\right ) x^{8}+\left (1070+520 \,{\mathrm e}^{3}\right ) x^{7}+\left (260 \,{\mathrm e}^{6}+30 \,{\mathrm e}^{3}\right ) x^{3}+\left (25 \,{\mathrm e}^{6}+1602 \,{\mathrm e}^{3}+556\right ) x^{6}+\left (260 \,{\mathrm e}^{6}+1330 \,{\mathrm e}^{3}+120\right ) x^{5}+\left (-1452 \,{\mathrm e}^{9}-687 \,{\mathrm e}^{6}-18 \,{\mathrm e}^{3}\right ) x^{2}+260 x^{9}+25 x^{10}-726 \,{\mathrm e}^{12}-356 \,{\mathrm e}^{9}-9 \,{\mathrm e}^{6}}{\left (x^{2}+{\mathrm e}^{3}\right )^{2}}-\ln \relax (x )\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+3900*x^7+9612*x^6+6390*x^5+1374*x^4+90*x^3
-3*x^2)*exp(3)^2+(450*x^10+3900*x^9+10212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^11+3504
*x^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5*exp(3)+x^7),x,method=_RETURNVERBOSE)

[Out]

25*x^6+260*x^5+876*x^4-150*x^2*exp(3)+1070*x^3-810*x*exp(3)+556*x^2+120*x+(30*exp(3)*(27*exp(3)-7)*x^3+(-150*e
xp(9)+981*exp(6)-18*exp(3))*x^2+(810*exp(9)-120*exp(6))*x-150*exp(12)+756*exp(9)-9*exp(6))/(x^4+2*x^2*exp(3)+e
xp(6))-ln(x)

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maxima [B]  time = 0.38, size = 119, normalized size = 3.40 \begin {gather*} 25 \, x^{6} + 260 \, x^{5} + 876 \, x^{4} + 1070 \, x^{3} - 2 \, x^{2} {\left (75 \, e^{3} - 278\right )} - 30 \, x {\left (27 \, e^{3} - 4\right )} + \frac {3 \, {\left (10 \, x^{3} {\left (27 \, e^{6} - 7 \, e^{3}\right )} - x^{2} {\left (50 \, e^{9} - 327 \, e^{6} + 6 \, e^{3}\right )} + 10 \, x {\left (27 \, e^{9} - 4 \, e^{6}\right )} - 50 \, e^{12} + 252 \, e^{9} - 3 \, e^{6}\right )}}{x^{4} + 2 \, x^{2} e^{3} + e^{6}} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+3900*x^7+9612*x^6+6390*x^5+1374*x^4+
90*x^3-3*x^2)*exp(3)^2+(450*x^10+3900*x^9+10212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^1
1+3504*x^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5*exp(3)+x^7),x, algorithm="maxima")

[Out]

25*x^6 + 260*x^5 + 876*x^4 + 1070*x^3 - 2*x^2*(75*e^3 - 278) - 30*x*(27*e^3 - 4) + 3*(10*x^3*(27*e^6 - 7*e^3)
- x^2*(50*e^9 - 327*e^6 + 6*e^3) + 10*x*(27*e^9 - 4*e^6) - 50*e^12 + 252*e^9 - 3*e^6)/(x^4 + 2*x^2*e^3 + e^6)
- log(x)

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mupad [B]  time = 0.29, size = 116, normalized size = 3.31 \begin {gather*} 1070\,x^3-x^2\,\left (150\,{\mathrm {e}}^3-556\right )-\frac {\left (210\,{\mathrm {e}}^3-810\,{\mathrm {e}}^6\right )\,x^3+\left (18\,{\mathrm {e}}^3-981\,{\mathrm {e}}^6+150\,{\mathrm {e}}^9\right )\,x^2+\left (120\,{\mathrm {e}}^6-810\,{\mathrm {e}}^9\right )\,x+9\,{\mathrm {e}}^6-756\,{\mathrm {e}}^9+150\,{\mathrm {e}}^{12}}{x^4+2\,{\mathrm {e}}^3\,x^2+{\mathrm {e}}^6}-\ln \relax (x)+876\,x^4+260\,x^5+25\,x^6-x\,\left (810\,{\mathrm {e}}^3-120\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(9)*(50*x^2 + 780*x^3 + 2904*x^4 + 1300*x^5 + 150*x^6 - 1) + exp(6)*(90*x^3 - 3*x^2 + 1374*x^4 + 6390*
x^5 + 9612*x^6 + 3900*x^7 + 450*x^8) + exp(3)*(33*x^4 + 570*x^5 + 3336*x^6 + 8820*x^7 + 10212*x^8 + 3900*x^9 +
450*x^10) - x^6 + 120*x^7 + 1112*x^8 + 3210*x^9 + 3504*x^10 + 1300*x^11 + 150*x^12)/(x*exp(9) + 3*x^5*exp(3)
+ 3*x^3*exp(6) + x^7),x)

[Out]

1070*x^3 - x^2*(150*exp(3) - 556) - (9*exp(6) - 756*exp(9) + 150*exp(12) + x^3*(210*exp(3) - 810*exp(6)) + x*(
120*exp(6) - 810*exp(9)) + x^2*(18*exp(3) - 981*exp(6) + 150*exp(9)))/(exp(6) + 2*x^2*exp(3) + x^4) - log(x) +
876*x^4 + 260*x^5 + 25*x^6 - x*(810*exp(3) - 120)

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sympy [B]  time = 3.37, size = 116, normalized size = 3.31 \begin {gather*} 25 x^{6} + 260 x^{5} + 876 x^{4} + 1070 x^{3} + x^{2} \left (556 - 150 e^{3}\right ) + x \left (120 - 810 e^{3}\right ) - \log {\relax (x )} + \frac {x^{3} \left (- 210 e^{3} + 810 e^{6}\right ) + x^{2} \left (- 150 e^{9} - 18 e^{3} + 981 e^{6}\right ) + x \left (- 120 e^{6} + 810 e^{9}\right ) - 150 e^{12} - 9 e^{6} + 756 e^{9}}{x^{4} + 2 x^{2} e^{3} + e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((150*x**6+1300*x**5+2904*x**4+780*x**3+50*x**2-1)*exp(3)**3+(450*x**8+3900*x**7+9612*x**6+6390*x**5
+1374*x**4+90*x**3-3*x**2)*exp(3)**2+(450*x**10+3900*x**9+10212*x**8+8820*x**7+3336*x**6+570*x**5+33*x**4)*exp
(3)+150*x**12+1300*x**11+3504*x**10+3210*x**9+1112*x**8+120*x**7-x**6)/(x*exp(3)**3+3*x**3*exp(3)**2+3*x**5*ex
p(3)+x**7),x)

[Out]

25*x**6 + 260*x**5 + 876*x**4 + 1070*x**3 + x**2*(556 - 150*exp(3)) + x*(120 - 810*exp(3)) - log(x) + (x**3*(-
210*exp(3) + 810*exp(6)) + x**2*(-150*exp(9) - 18*exp(3) + 981*exp(6)) + x*(-120*exp(6) + 810*exp(9)) - 150*ex
p(12) - 9*exp(6) + 756*exp(9))/(x**4 + 2*x**2*exp(3) + exp(6))

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