3.83 \(\int \frac{1}{\sqrt{3-x+2 x^2} (2+3 x+5 x^2)} \, dx\)
Optimal. Leaf size=148 \[ \sqrt{\frac{1}{682} \left (13+10 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (13+10 \sqrt{2}\right )}} \left (\left (13+10 \sqrt{2}\right ) x+3 \sqrt{2}+7\right )}{\sqrt{2 x^2-x+3}}\right )-\sqrt{\frac{1}{682} \left (10 \sqrt{2}-13\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (10 \sqrt{2}-13\right )}} \left (\left (13-10 \sqrt{2}\right ) x-3 \sqrt{2}+7\right )}{\sqrt{2 x^2-x+3}}\right ) \]
[Out]
Sqrt[(13 + 10*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(13 + 10*Sqrt[2]))]*(7 + 3*Sqrt[2] + (13 + 10*Sqrt[2])*x))/Sqr
t[3 - x + 2*x^2]] - Sqrt[(-13 + 10*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-13 + 10*Sqrt[2]))]*(7 - 3*Sqrt[2] + (1
3 - 10*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]]
________________________________________________________________________________________
Rubi [A] time = 0.314359, antiderivative size = 148, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used =
{986, 1029, 204, 206} \[ \sqrt{\frac{1}{682} \left (13+10 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (13+10 \sqrt{2}\right )}} \left (\left (13+10 \sqrt{2}\right ) x+3 \sqrt{2}+7\right )}{\sqrt{2 x^2-x+3}}\right )-\sqrt{\frac{1}{682} \left (10 \sqrt{2}-13\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (10 \sqrt{2}-13\right )}} \left (\left (13-10 \sqrt{2}\right ) x-3 \sqrt{2}+7\right )}{\sqrt{2 x^2-x+3}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)),x]
[Out]
Sqrt[(13 + 10*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(13 + 10*Sqrt[2]))]*(7 + 3*Sqrt[2] + (13 + 10*Sqrt[2])*x))/Sqr
t[3 - x + 2*x^2]] - Sqrt[(-13 + 10*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-13 + 10*Sqrt[2]))]*(7 - 3*Sqrt[2] + (1
3 - 10*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]]
Rule 986
Int[1/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt
[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[(c*d - a*f + q + (c*e - b*f)*x)/((a + b*x + c
*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[(c*d - a*f - q + (c*e - b*f)*x)/((a + b*x + c*x^2)*Sq
rt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] &&
NeQ[c*e - b*f, 0] && NegQ[b^2 - 4*a*c]
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 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])
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx &=-\frac{\int \frac{11-11 \sqrt{2}-11 x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{22 \sqrt{2}}+\frac{\int \frac{11+11 \sqrt{2}-11 x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{22 \sqrt{2}}\\ &=-\left (\frac{1}{2} \left (11 \left (20-13 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3751 \left (13-10 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{11 \left (7-3 \sqrt{2}\right )+11 \left (13-10 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )\right )-\frac{1}{2} \left (11 \left (20+13 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3751 \left (13+10 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{11 \left (7+3 \sqrt{2}\right )+11 \left (13+10 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )\\ &=\sqrt{\frac{1}{682} \left (13+10 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (13+10 \sqrt{2}\right )}} \left (7+3 \sqrt{2}+\left (13+10 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )-\sqrt{\frac{1}{682} \left (-13+10 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-13+10 \sqrt{2}\right )}} \left (7-3 \sqrt{2}+\left (13-10 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.34685, size = 176, normalized size = 1.19 \[ -\frac{\sqrt{13+i \sqrt{31}} \left (\sqrt{31}+13 i\right ) \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )+\sqrt{13-i \sqrt{31}} \left (\sqrt{31}-13 i\right ) \tanh ^{-1}\left (\frac{\left (-22+4 i \sqrt{31}\right ) x-i \sqrt{31}+63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{20 \sqrt{682}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)),x]
[Out]
-(Sqrt[13 + I*Sqrt[31]]*(13*I + 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])] + Sqrt[13 - I*Sqrt[31]]*(-13*I + 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])])/(20*Sqrt[682])
________________________________________________________________________________________
Maple [B] time = 0.108, size = 684, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(5*x^2+3*x+2)/(2*x^2-x+3)^(1/2),x)
[Out]
1/21142*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)*2^(1/2
)*(369*(-775687+549362*2^(1/2))^(1/2)*2^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2^(
1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-23*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*(2^(1/2)-1+x)^
2/(2^(1/2)+1-x)^2*2^(1/2)+10368*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+22379*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^(1
/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+23)*(8+3*2^(1/2))*(2^(1/2)-1+x)/(2^(1/2)+1-x))+520*(-775687+5493
62*2^(1/2))^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2
))*(-23*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+10
368*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+22379*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+x)^
2/(2^(1/2)+1-x)^2+23)*(8+3*2^(1/2))*(2^(1/2)-1+x)/(2^(1/2)+1-x))+465124*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/
2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1/2))^(1/2))*2^(1/2)-866
822*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/
2)/(-8866+6820*2^(1/2))^(1/2)))/((8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+
8-3*2^(1/2))/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))^2)^(1/2)/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))/(8+3*2^(1/2))/(-8866+682
0*2^(1/2))^(1/2)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt{2 \, x^{2} - x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5*x^2+3*x+2)/(2*x^2-x+3)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)*sqrt(2*x^2 - x + 3)), x)
________________________________________________________________________________________
Fricas [B] time = 4.6852, size = 7101, normalized size = 47.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5*x^2+3*x+2)/(2*x^2-x+3)^(1/2),x, algorithm="fricas")
[Out]
-1/845680*sqrt(341)*200^(1/4)*sqrt(31)*sqrt(5)*sqrt(13*sqrt(2) + 20)*(13*sqrt(2) - 20)*log(1240*(sqrt(341)*200
^(1/4)*sqrt(31)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(4*x - 1) - 3*x - 5)*sqrt(13*sqrt(2) + 20) + 7595*x^2 + 6
820*sqrt(2)*(2*x^2 - x + 3) - 23405*x + 31000)/x^2) + 1/845680*sqrt(341)*200^(1/4)*sqrt(31)*sqrt(5)*sqrt(13*sq
rt(2) + 20)*(13*sqrt(2) - 20)*log(-1240*(sqrt(341)*200^(1/4)*sqrt(31)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(4*
x - 1) - 3*x - 5)*sqrt(13*sqrt(2) + 20) - 7595*x^2 - 6820*sqrt(2)*(2*x^2 - x + 3) + 23405*x - 31000)/x^2) - 1/
6820*sqrt(341)*200^(1/4)*sqrt(5)*sqrt(2)*sqrt(13*sqrt(2) + 20)*arctan(1/2762875*(14260*sqrt(341)*sqrt(5)*sqrt(
2*x^2 - x + 3)*(11*200^(3/4)*(8056*x^7 - 28976*x^6 + 61838*x^5 - 93342*x^4 + 45376*x^3 - 18288*x^2 - sqrt(2)*(
4702*x^7 - 19541*x^6 + 40352*x^5 - 68777*x^4 + 35480*x^3 - 19080*x^2 - 34560*x + 27648) - 55296*x + 34560) + 5
*200^(1/4)*(18463*x^7 - 280047*x^6 + 1453472*x^5 - 3238500*x^4 + 4140576*x^3 - 2378592*x^2 - sqrt(2)*(11418*x^
7 - 177633*x^6 + 957180*x^5 - 2237548*x^4 + 2920320*x^3 - 2005920*x^2 - 1990656*x + 1534464) - 3068928*x + 199
0656))*sqrt(13*sqrt(2) + 20) + 7843000*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1
549144*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) - 2*sqrt(310)*(sqrt(341)*sqrt(5)*sqrt(2*x^
2 - x + 3)*(11*200^(3/4)*(30876*x^7 - 44014*x^6 + 139674*x^5 - 42464*x^4 + 38736*x^3 + 89856*x^2 - sqrt(2)*(15
454*x^7 - 22399*x^6 + 73509*x^5 - 37360*x^4 + 52200*x^3 + 13824*x^2 - 13824*x) - 89856*x) + 5*200^(1/4)*(69479
*x^7 - 898236*x^6 + 3454740*x^5 - 4394304*x^4 + 5347296*x^3 + 4478976*x^2 - sqrt(2)*(38627*x^7 - 500012*x^6 +
1934180*x^5 - 2560320*x^4 + 3506400*x^3 + 1202688*x^2 - 1202688*x) - 4478976*x))*sqrt(13*sqrt(2) + 20) + 550*s
qrt(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) + 25*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219
328*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(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(4*x - 1) -
3*x - 5)*sqrt(13*sqrt(2) + 20) - 7595*x^2 - 6820*sqrt(2)*(2*x^2 - x + 3) + 23405*x - 31000)/x^2) + 89125*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)) - 1/6820*sqrt(341)*200^(1/4)*sqrt(5)*sqrt(2)*sqrt(13*sqrt(2) + 20)*
arctan(1/2762875*(14260*sqrt(341)*sqrt(5)*sqrt(2*x^2 - x + 3)*(11*200^(3/4)*(8056*x^7 - 28976*x^6 + 61838*x^5
- 93342*x^4 + 45376*x^3 - 18288*x^2 - sqrt(2)*(4702*x^7 - 19541*x^6 + 40352*x^5 - 68777*x^4 + 35480*x^3 - 1908
0*x^2 - 34560*x + 27648) - 55296*x + 34560) + 5*200^(1/4)*(18463*x^7 - 280047*x^6 + 1453472*x^5 - 3238500*x^4
+ 4140576*x^3 - 2378592*x^2 - sqrt(2)*(11418*x^7 - 177633*x^6 + 957180*x^5 - 2237548*x^4 + 2920320*x^3 - 20059
20*x^2 - 1990656*x + 1534464) - 3068928*x + 1990656))*sqrt(13*sqrt(2) + 20) - 7843000*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 - 10233
5*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456
192) - 2*sqrt(310)*(sqrt(341)*sqrt(5)*sqrt(2*x^2 - x + 3)*(11*200^(3/4)*(30876*x^7 - 44014*x^6 + 139674*x^5 -
42464*x^4 + 38736*x^3 + 89856*x^2 - sqrt(2)*(15454*x^7 - 22399*x^6 + 73509*x^5 - 37360*x^4 + 52200*x^3 + 13824
*x^2 - 13824*x) - 89856*x) + 5*200^(1/4)*(69479*x^7 - 898236*x^6 + 3454740*x^5 - 4394304*x^4 + 5347296*x^3 + 4
478976*x^2 - sqrt(2)*(38627*x^7 - 500012*x^6 + 1934180*x^5 - 2560320*x^4 + 3506400*x^3 + 1202688*x^2 - 1202688
*x) - 4478976*x))*sqrt(13*sqrt(2) + 20) - 550*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 329307
2*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 + 10
53960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 25*sqrt(31)*(254591*x^8 - 4815126*x^7 + 3230
3580*x^6 - 90866808*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)*sq
rt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(4*x - 1) - 3*x - 5)*sqrt(13*sqrt(2) + 20) + 7595*x^2 + 6820*sqrt(2)*(2*x^2
- x + 3) - 23405*x + 31000)/x^2) - 89125*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 + 155
69*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))
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5*x**2+3*x+2)/(2*x**2-x+3)**(1/2),x)
[Out]
Integral(1/(sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)), x)
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5*x^2+3*x+2)/(2*x^2-x+3)^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError