3.1.94 \(\int \frac {1}{(3-x+2 x^2)^{5/2} (2+3 x+5 x^2)} \, dx\)
Optimal. Leaf size=199 \[ \frac {3603-658 x}{128018 \sqrt {2 x^2-x+3}}+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}+\frac {1}{484} \sqrt {\frac {1}{682} \left (25000 \sqrt {2}-15457\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (25000 \sqrt {2}-15457\right )}} \left (\left (247+345 \sqrt {2}\right ) x-98 \sqrt {2}+443\right )}{\sqrt {2 x^2-x+3}}\right )-\frac {1}{484} \sqrt {\frac {1}{682} \left (15457+25000 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (15457+25000 \sqrt {2}\right )}} \left (\left (247-345 \sqrt {2}\right ) x+98 \sqrt {2}+443\right )}{\sqrt {2 x^2-x+3}}\right ) \]
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Rubi [A] time = 0.46, antiderivative size = 199, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used =
{974, 1060, 1035, 1029, 206, 204} \begin {gather*} \frac {3603-658 x}{128018 \sqrt {2 x^2-x+3}}+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}+\frac {1}{484} \sqrt {\frac {1}{682} \left (25000 \sqrt {2}-15457\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (25000 \sqrt {2}-15457\right )}} \left (\left (247+345 \sqrt {2}\right ) x-98 \sqrt {2}+443\right )}{\sqrt {2 x^2-x+3}}\right )-\frac {1}{484} \sqrt {\frac {1}{682} \left (15457+25000 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (15457+25000 \sqrt {2}\right )}} \left (\left (247-345 \sqrt {2}\right ) x+98 \sqrt {2}+443\right )}{\sqrt {2 x^2-x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)),x]
[Out]
(13 - 6*x)/(759*(3 - x + 2*x^2)^(3/2)) + (3603 - 658*x)/(128018*Sqrt[3 - x + 2*x^2]) + (Sqrt[(-15457 + 25000*S
qrt[2])/682]*ArcTan[(Sqrt[11/(31*(-15457 + 25000*Sqrt[2]))]*(443 - 98*Sqrt[2] + (247 + 345*Sqrt[2])*x))/Sqrt[3
- x + 2*x^2]])/484 - (Sqrt[(15457 + 25000*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(15457 + 25000*Sqrt[2]))]*(443 +
98*Sqrt[2] + (247 - 345*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/484
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 974
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((2*a
*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
1)*(d + e*x + f*x^2)^(q + 1))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] - Dist[1/
((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*
x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e
- b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Rule 1029
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]
Rule 1035
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x
+ c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e
^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]
Rule 1060
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
(b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Rubi steps
\begin {align*} \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx &=\frac {13-6 x}{759 \left (3-x+2 x^2\right )^{3/2}}-\frac {\int \frac {-2772-\frac {3003 x}{2}+660 x^2}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{8349}\\ &=\frac {13-6 x}{759 \left (3-x+2 x^2\right )^{3/2}}+\frac {3603-658 x}{128018 \sqrt {3-x+2 x^2}}-\frac {\int \frac {-\frac {5184729}{2}-\frac {12481755 x}{4}}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{23235267}\\ &=\frac {13-6 x}{759 \left (3-x+2 x^2\right )^{3/2}}+\frac {3603-658 x}{128018 \sqrt {3-x+2 x^2}}+\frac {\int \frac {-\frac {2112297}{4} \left (11-54 \sqrt {2}\right )-\frac {2112297}{4} \left (119-65 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{511175874 \sqrt {2}}-\frac {\int \frac {-\frac {2112297}{4} \left (11+54 \sqrt {2}\right )-\frac {2112297}{4} \left (119+65 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{511175874 \sqrt {2}}\\ &=\frac {13-6 x}{759 \left (3-x+2 x^2\right )^{3/2}}+\frac {3603-658 x}{128018 \sqrt {3-x+2 x^2}}-\frac {1}{32} \left (17457 \left (50000-15457 \sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {138315757102479}{16} \left (15457-25000 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {2112297}{4} \left (443-98 \sqrt {2}\right )+\frac {2112297}{4} \left (247+345 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{32} \left (17457 \left (50000+15457 \sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {138315757102479}{16} \left (15457+25000 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {2112297}{4} \left (443+98 \sqrt {2}\right )+\frac {2112297}{4} \left (247-345 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )\\ &=\frac {13-6 x}{759 \left (3-x+2 x^2\right )^{3/2}}+\frac {3603-658 x}{128018 \sqrt {3-x+2 x^2}}+\frac {1}{484} \sqrt {\frac {1}{682} \left (-15457+25000 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-15457+25000 \sqrt {2}\right )}} \left (443-98 \sqrt {2}+\left (247+345 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{484} \sqrt {\frac {1}{682} \left (15457+25000 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (15457+25000 \sqrt {2}\right )}} \left (443+98 \sqrt {2}+\left (247-345 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.87, size = 218, normalized size = 1.10 \begin {gather*} -\frac {\sqrt {\frac {1}{682} \left (13+i \sqrt {31}\right )} \left (119 \sqrt {31}+247 i\right ) \tanh ^{-1}\left (\frac {\left (-22-4 i \sqrt {31}\right ) x+i \sqrt {31}+63}{2 \sqrt {286+22 i \sqrt {31}} \sqrt {2 x^2-x+3}}\right )}{9680}+\frac {\sqrt {\frac {1}{682} \left (13-i \sqrt {31}\right )} \left (119 \sqrt {31}-247 i\right ) \tanh ^{-1}\left (\frac {\left (22-4 i \sqrt {31}\right ) x+i \sqrt {31}-63}{2 \sqrt {286-22 i \sqrt {31}} \sqrt {2 x^2-x+3}}\right )}{9680}+\frac {-3948 x^3+23592 x^2-19767 x+39005}{384054 \left (2 x^2-x+3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)),x]
[Out]
(39005 - 19767*x + 23592*x^2 - 3948*x^3)/(384054*(3 - x + 2*x^2)^(3/2)) - (Sqrt[(13 + I*Sqrt[31])/682]*(247*I
+ 119*Sqrt[31])*ArcTanh[(63 + I*Sqrt[31] + (-22 - (4*I)*Sqrt[31])*x)/(2*Sqrt[286 + (22*I)*Sqrt[31]]*Sqrt[3 - x
+ 2*x^2])])/9680 + (Sqrt[(13 - I*Sqrt[31])/682]*(-247*I + 119*Sqrt[31])*ArcTanh[(-63 + I*Sqrt[31] + (22 - (4*
I)*Sqrt[31])*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])])/9680
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IntegrateAlgebraic [C] time = 0.83, size = 209, normalized size = 1.05 \begin {gather*} \frac {1}{484} \text {RootSum}\left [-5 \text {$\#$1}^4+6 \sqrt {2} \text {$\#$1}^3+17 \text {$\#$1}^2-26 \sqrt {2} \text {$\#$1}-56\&,\frac {-65 \text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {2 x^2-x+3}-\sqrt {2} x\right )+108 \sqrt {2} \text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {2 x^2-x+3}-\sqrt {2} x\right )+249 \log \left (-\text {$\#$1}+\sqrt {2 x^2-x+3}-\sqrt {2} x\right )}{-10 \text {$\#$1}^3+9 \sqrt {2} \text {$\#$1}^2+17 \text {$\#$1}-13 \sqrt {2}}\&\right ]+\frac {-3948 x^3+23592 x^2-19767 x+39005}{384054 \left (2 x^2-x+3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)),x]
[Out]
(39005 - 19767*x + 23592*x^2 - 3948*x^3)/(384054*(3 - x + 2*x^2)^(3/2)) + RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^
2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (249*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 108*Sqrt[2]*Log[-(Sqrt[2]*
x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 65*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[2] + 17*#1
+ 9*Sqrt[2]*#1^2 - 10*#1^3) & ]/484
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fricas [B] time = 1.24, size = 2133, normalized size = 10.72
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x, algorithm="fricas")
[Out]
1/370971467791584000*(1123856268*sqrt(341)*200^(1/4)*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*sqrt(-77285000
0*sqrt(2) + 2500000000)*arctan(-1/7889389562500*(71300*sqrt(341)*sqrt(2*x^2 - x + 3)*(11*200^(3/4)*(347404*x^7
- 907814*x^6 + 2112962*x^5 - 2166688*x^4 + 787344*x^3 + 304128*x^2 - sqrt(2)*(35898*x^7 - 441939*x^6 + 782418
*x^5 - 2117233*x^4 + 1272680*x^3 - 1081800*x^2 - 518400*x + 1043712) - 2087424*x + 518400) + 5*200^(1/4)*(7127
57*x^7 - 10233303*x^6 + 48529768*x^5 - 94500260*x^4 + 113086944*x^3 - 22282848*x^2 - sqrt(2)*(158647*x^7 - 293
5272*x^6 + 19428740*x^5 - 55765712*x^4 + 78380640*x^3 - 84096000*x^2 - 37407744*x + 53208576) - 106417152*x +
37407744))*sqrt(-772850000*sqrt(2) + 2500000000) + 22395686500000*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 7
04270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 -
783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - sqrt(310/5711
)*(sqrt(341)*sqrt(2*x^2 - x + 3)*(11*200^(3/4)*(1665224*x^7 - 2325796*x^6 + 7065036*x^5 - 196416*x^4 - 2176416
*x^3 + 8895744*x^2 + sqrt(2)*(167914*x^7 - 195429*x^6 + 331239*x^5 + 1685680*x^4 - 3693960*x^3 + 4195584*x^2 -
4195584*x) - 8895744*x) + 5*200^(1/4)*(3246491*x^7 - 41888524*x^6 + 159670660*x^5 - 190080576*x^4 + 180496224
*x^3 + 376648704*x^2 - 2*sqrt(2)*(40239*x^7 - 558044*x^6 + 2804660*x^5 - 9524160*x^4 + 34843680*x^3 - 74006784
*x^2 + 74006784*x) - 376648704*x))*sqrt(-772850000*sqrt(2) + 2500000000) + 314105000*sqrt(31)*sqrt(2)*(123408*
x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 11
8051*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) + 14277
500*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 16895692
8*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820
224*x))*sqrt(-(sqrt(341)*200^(1/4)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(281*x - 444) + 163*x - 725)*sqrt(-77
2850000*sqrt(2) + 2500000000) - 4337504500*x^2 - 3894902000*sqrt(2)*(2*x^2 - x + 3) + 13366595500*x - 17704100
000)/x^2) + 254496437500*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 -
249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 10070*x^5 + 15569*x^4 - 5568*x^3
+ 1080*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5
- 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 1123856268*sqrt(341)*200^(1/4)*sqrt(
2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*sqrt(-772850000*sqrt(2) + 2500000000)*arctan(-1/7889389562500*(71300*sqr
t(341)*sqrt(2*x^2 - x + 3)*(11*200^(3/4)*(347404*x^7 - 907814*x^6 + 2112962*x^5 - 2166688*x^4 + 787344*x^3 + 3
04128*x^2 - sqrt(2)*(35898*x^7 - 441939*x^6 + 782418*x^5 - 2117233*x^4 + 1272680*x^3 - 1081800*x^2 - 518400*x
+ 1043712) - 2087424*x + 518400) + 5*200^(1/4)*(712757*x^7 - 10233303*x^6 + 48529768*x^5 - 94500260*x^4 + 1130
86944*x^3 - 22282848*x^2 - sqrt(2)*(158647*x^7 - 2935272*x^6 + 19428740*x^5 - 55765712*x^4 + 78380640*x^3 - 84
096000*x^2 - 37407744*x + 53208576) - 106417152*x + 37407744))*sqrt(-772850000*sqrt(2) + 2500000000) - 2239568
6500000*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98496
*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 5460
48*x - 539136) + 1154304*x - 456192) - sqrt(310/5711)*(sqrt(341)*sqrt(2*x^2 - x + 3)*(11*200^(3/4)*(1665224*x^
7 - 2325796*x^6 + 7065036*x^5 - 196416*x^4 - 2176416*x^3 + 8895744*x^2 + sqrt(2)*(167914*x^7 - 195429*x^6 + 33
1239*x^5 + 1685680*x^4 - 3693960*x^3 + 4195584*x^2 - 4195584*x) - 8895744*x) + 5*200^(1/4)*(3246491*x^7 - 4188
8524*x^6 + 159670660*x^5 - 190080576*x^4 + 180496224*x^3 + 376648704*x^2 - 2*sqrt(2)*(40239*x^7 - 558044*x^6 +
2804660*x^5 - 9524160*x^4 + 34843680*x^3 - 74006784*x^2 + 74006784*x) - 376648704*x))*sqrt(-772850000*sqrt(2)
+ 2500000000) - 314105000*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4
+ 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952
*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 14277500*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90
866808*x^5 + 108781920*x^4 - 74219328*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517*x^6 - 1536*x^5
+ 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt((sqrt(341)*200^(1/4)*sqrt(31)*sqrt(2*x^2 - x
+ 3)*(sqrt(2)*(281*x - 444) + 163*x - 725)*sqrt(-772850000*sqrt(2) + 2500000000) + 4337504500*x^2 + 3894902000
*sqrt(2)*(2*x^2 - x + 3) - 13366595500*x + 17704100000)/x^2) - 254496437500*sqrt(31)*(2828123*x^8 - 9696916*x^
7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 269
2*x^7 + 9789*x^6 - 10070*x^5 + 15569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(25
85191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x
+ 18579456)) + 1587*sqrt(341)*200^(1/4)*sqrt(31)*(200000*x^4 - 200000*x^3 + 650000*x^2 + 15457*sqrt(2)*(4*x^4
- 4*x^3 + 13*x^2 - 6*x + 9) - 300000*x + 450000)*sqrt(-772850000*sqrt(2) + 2500000000)*log(77500000/5711*(sqrt
(341)*200^(1/4)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(281*x - 444) + 163*x - 725)*sqrt(-772850000*sqrt(2) + 2
500000000) + 4337504500*x^2 + 3894902000*sqrt(2)*(2*x^2 - x + 3) - 13366595500*x + 17704100000)/x^2) - 1587*sq
rt(341)*200^(1/4)*sqrt(31)*(200000*x^4 - 200000*x^3 + 650000*x^2 + 15457*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x
+ 9) - 300000*x + 450000)*sqrt(-772850000*sqrt(2) + 2500000000)*log(-77500000/5711*(sqrt(341)*200^(1/4)*sqrt(
31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(281*x - 444) + 163*x - 725)*sqrt(-772850000*sqrt(2) + 2500000000) - 43375045
00*x^2 - 3894902000*sqrt(2)*(2*x^2 - x + 3) + 13366595500*x - 17704100000)/x^2) - 965935696000*(3948*x^3 - 235
92*x^2 + 19767*x - 39005)*sqrt(2*x^2 - x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Fran
cis algorithm failure for[-1.0,infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,infinity
,infinity]Francis algorithm failure for[-1.0,infinity,infinity,infinity,infinity]proot error [1.0,infinity,inf
inity,infinity,infinity]Francis algorithm failure for[-1.0,infinity,infinity,infinity,infinity]proot error [1.
0,infinity,infinity,infinity,infinity]Francis algorithm failure for[-1.0,infinity,infinity,infinity,infinity]p
root error [1.0,infinity,infinity,infinity,infinity]Evaluation time: 16.02Done
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maple [B] time = 0.04, size = 751, normalized size = 3.77 \begin {gather*} \frac {\sqrt {\frac {8 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+\frac {3 \sqrt {2}\, \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (-993674 \sqrt {2}\, \arctanh \left (\frac {31 \sqrt {\frac {8 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+\frac {3 \sqrt {2}\, \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )-42685698 \arctanh \left (\frac {31 \sqrt {\frac {8 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+\frac {3 \sqrt {2}\, \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )+10111 \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \sqrt {-775687+549362 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+\frac {10368 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+22379 \sqrt {2}+32016\right ) \left (x +\sqrt {2}-1\right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (x +\sqrt {2}-1\right )^{4}}{\left (-x +\sqrt {2}+1\right )^{4}}+\frac {82 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+23\right ) \left (-x +\sqrt {2}+1\right )}\right )+13910 \sqrt {-8866+6820 \sqrt {2}}\, \sqrt {-775687+549362 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+\frac {10368 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+22379 \sqrt {2}+32016\right ) \left (x +\sqrt {2}-1\right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (x +\sqrt {2}-1\right )^{4}}{\left (-x +\sqrt {2}+1\right )^{4}}+\frac {82 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+23\right ) \left (-x +\sqrt {2}+1\right )}\right )\right )}{10232728 \sqrt {\frac {\frac {8 \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+\frac {3 \sqrt {2}\, \left (x +\sqrt {2}-1\right )^{2}}{\left (-x +\sqrt {2}+1\right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {x +\sqrt {2}-1}{-x +\sqrt {2}+1}\right )^{2}}}\, \left (1+\frac {x +\sqrt {2}-1}{-x +\sqrt {2}+1}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}+\frac {13}{484 \sqrt {2 x^{2}-x +3}}-\frac {329 \left (4 x -1\right )}{256036 \sqrt {2 x^{2}-x +3}}+\frac {1}{66 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {4 x -1}{506 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x)
[Out]
1/10232728*(8*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+3*2^(1/2)*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+8-3*2^(1/2))^(1/2)*2
^(1/2)*(10111*2^(1/2)*(-8866+6820*2^(1/2))^(1/2)*(-775687+549362*2^(1/2))^(1/2)*arctan(1/11692487*(-775687+549
362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-23*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+24*2^(1/2)-41))^(1/2)*(6485*2^(1/2
)*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+10368*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+22379*2^(1/2)+32016)/(23*(x+2^(1/2)-
1)^4/(-x+2^(1/2)+1)^4+82*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+23)*(x+2^(1/2)-1)/(-x+2^(1/2)+1)*(8+3*2^(1/2)))+1391
0*(-8866+6820*2^(1/2))^(1/2)*(-775687+549362*2^(1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(
-23*(8+3*2^(1/2))*(-23*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+24*2^(1/2)-41))^(1/2)*(6485*2^(1/2)*(x+2^(1/2)-1)^2/(-
x+2^(1/2)+1)^2+10368*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+22379*2^(1/2)+32016)/(23*(x+2^(1/2)-1)^4/(-x+2^(1/2)+1)^
4+82*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+23)*(x+2^(1/2)-1)/(-x+2^(1/2)+1)*(8+3*2^(1/2)))-993674*arctanh(31/2*(8*(
x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+3*2^(1/2)*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1
/2))^(1/2))*2^(1/2)-42685698*arctanh(31/2*(8*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+3*2^(1/2)*(x+2^(1/2)-1)^2/(-x+2^
(1/2)+1)^2+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1/2))^(1/2)))/((8*(x+2^(1/2)-1)^2/(-x+2^(1/2)+1)^2+3*2^(1/2)*(x+2
^(1/2)-1)^2/(-x+2^(1/2)+1)^2+8-3*2^(1/2))/(1+(x+2^(1/2)-1)/(-x+2^(1/2)+1))^2)^(1/2)/(1+(x+2^(1/2)-1)/(-x+2^(1/
2)+1))/(8+3*2^(1/2))/(-8866+6820*2^(1/2))^(1/2)+13/484/(2*x^2-x+3)^(1/2)-329/256036*(4*x-1)/(2*x^2-x+3)^(1/2)+
1/66/(2*x^2-x+3)^(3/2)-1/506*(4*x-1)/(2*x^2-x+3)^(3/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)*(2*x^2 - x + 3)^(5/2)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (2\,x^2-x+3\right )}^{5/2}\,\left (5\,x^2+3\,x+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((2*x^2 - x + 3)^(5/2)*(3*x + 5*x^2 + 2)),x)
[Out]
int(1/((2*x^2 - x + 3)^(5/2)*(3*x + 5*x^2 + 2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}} \left (5 x^{2} + 3 x + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*x**2-x+3)**(5/2)/(5*x**2+3*x+2),x)
[Out]
Integral(1/((2*x**2 - x + 3)**(5/2)*(5*x**2 + 3*x + 2)), x)
________________________________________________________________________________________