3.62 \(\int \frac{\sqrt{3-x+2 x^2}}{2+3 x+5 x^2} \, dx\)
Optimal. Leaf size=174 \[ \frac{1}{5} \sqrt{\frac{11}{31} \left (13+10 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (13+10 \sqrt{2}\right )}} \left (\left (20+13 \sqrt{2}\right ) x+7 \sqrt{2}+6\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{5} \sqrt{\frac{11}{31} \left (10 \sqrt{2}-13\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (10 \sqrt{2}-13\right )}} \left (\left (20-13 \sqrt{2}\right ) x-7 \sqrt{2}+6\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{5} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right ) \]
[Out]
-(Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/5 + (Sqrt[(11*(13 + 10*Sqrt[2]))/31]*ArcT
an[(Sqrt[11/(62*(13 + 10*Sqrt[2]))]*(6 + 7*Sqrt[2] + (20 + 13*Sqrt[2])*x))/Sqrt[
3 - x + 2*x^2]])/5 - (Sqrt[(11*(-13 + 10*Sqrt[2]))/31]*ArcTanh[(Sqrt[11/(62*(-13
+ 10*Sqrt[2]))]*(6 - 7*Sqrt[2] + (20 - 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/5
_______________________________________________________________________________________
Rubi [A] time = 0.861891, antiderivative size = 174, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259
\[ \frac{1}{5} \sqrt{\frac{11}{31} \left (13+10 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (13+10 \sqrt{2}\right )}} \left (\left (20+13 \sqrt{2}\right ) x+7 \sqrt{2}+6\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{5} \sqrt{\frac{11}{31} \left (10 \sqrt{2}-13\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (10 \sqrt{2}-13\right )}} \left (\left (20-13 \sqrt{2}\right ) x-7 \sqrt{2}+6\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{5} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2),x]
[Out]
-(Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/5 + (Sqrt[(11*(13 + 10*Sqrt[2]))/31]*ArcT
an[(Sqrt[11/(62*(13 + 10*Sqrt[2]))]*(6 + 7*Sqrt[2] + (20 + 13*Sqrt[2])*x))/Sqrt[
3 - x + 2*x^2]])/5 - (Sqrt[(11*(-13 + 10*Sqrt[2]))/31]*ArcTanh[(Sqrt[11/(62*(-13
+ 10*Sqrt[2]))]*(6 - 7*Sqrt[2] + (20 - 13*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/5
_______________________________________________________________________________________
Rubi in Sympy [A] time = 75.4788, size = 204, normalized size = 1.17 \[ \frac{\sqrt{341} \left (726 + 847 \sqrt{2}\right ) \left (242 \sqrt{2} + 484\right ) \operatorname{atan}{\left (\frac{\sqrt{682} \left (x \left (1573 \sqrt{2} + 2420\right ) + 726 + 847 \sqrt{2}\right )}{7502 \sqrt{13 + 10 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{9077420 \sqrt{13 + 10 \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{5} + \frac{\sqrt{341} \left (- 847 \sqrt{2} + 726\right ) \left (- 242 \sqrt{2} + 484\right ) \operatorname{atanh}{\left (\frac{\sqrt{682} \left (x \left (- 1573 \sqrt{2} + 2420\right ) - 847 \sqrt{2} + 726\right )}{7502 \sqrt{-13 + 10 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{9077420 \sqrt{-13 + 10 \sqrt{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*x**2-x+3)**(1/2)/(5*x**2+3*x+2),x)
[Out]
sqrt(341)*(726 + 847*sqrt(2))*(242*sqrt(2) + 484)*atan(sqrt(682)*(x*(1573*sqrt(2
) + 2420) + 726 + 847*sqrt(2))/(7502*sqrt(13 + 10*sqrt(2))*sqrt(2*x**2 - x + 3))
)/(9077420*sqrt(13 + 10*sqrt(2))) + sqrt(2)*atanh(sqrt(2)*(4*x - 1)/(4*sqrt(2*x*
*2 - x + 3)))/5 + sqrt(341)*(-847*sqrt(2) + 726)*(-242*sqrt(2) + 484)*atanh(sqrt
(682)*(x*(-1573*sqrt(2) + 2420) - 847*sqrt(2) + 726)/(7502*sqrt(-13 + 10*sqrt(2)
)*sqrt(2*x**2 - x + 3)))/(9077420*sqrt(-13 + 10*sqrt(2)))
_______________________________________________________________________________________
Mathematica [C] time = 6.41229, size = 1133, normalized size = 6.51 \[ \frac{1}{5} \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )-\frac{i \left (-13 i+\sqrt{31}\right ) \tan ^{-1}\left (\frac{31 \left (1100 \sqrt{31} x^4-8675 i x^4+2970 \sqrt{31} x^3-31860 i x^3+2706 \sqrt{31} x^2+4347 i x^2+3872 \sqrt{31} x-27836 i x+4224 \sqrt{31}+7588 i\right )}{-262775 i \sqrt{31} x^4+443300 x^4+10000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3-143180 i \sqrt{31} x^3+514910 x^3+3500 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2-308889 i \sqrt{31} x^2+340318 x^2+2500 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-46668 i \sqrt{31} x+1083016 x-1000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+35044 i \sqrt{31}+65472}\right )}{5 \sqrt{\frac{62}{11} \left (13+i \sqrt{31}\right )}}-\frac{i \left (13 i+\sqrt{31}\right ) \tanh ^{-1}\left (\frac{262775 \sqrt{31} x^4-443300 i x^4+55000 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3+143180 \sqrt{31} x^3-514910 i x^3-124500 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2+308889 \sqrt{31} x^2-340318 i x^2-72500 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x+46668 \sqrt{31} x-1083016 i x-63000 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}-35044 \sqrt{31}-65472 i}{34100 \sqrt{31} x^4+1493925 i x^4+92070 \sqrt{31} x^3-2052340 i x^3+83886 \sqrt{31} x^2+3090243 i x^2+120032 \sqrt{31} x+2352916 i x+130944 \sqrt{31}+1764772 i}\right )}{5 \sqrt{\frac{62}{11} \left (-13+i \sqrt{31}\right )}}+\frac{i \left (13 i+\sqrt{31}\right ) \log \left (\left (-10 i x+\sqrt{31}-3 i\right )^2 \left (10 i x+\sqrt{31}+3 i\right )^2\right )}{10 \sqrt{\frac{62}{11} \left (-13+i \sqrt{31}\right )}}-\frac{\left (-13 i+\sqrt{31}\right ) \log \left (\left (-10 i x+\sqrt{31}-3 i\right )^2 \left (10 i x+\sqrt{31}+3 i\right )^2\right )}{10 \sqrt{\frac{62}{11} \left (13+i \sqrt{31}\right )}}-\frac{i \left (13 i+\sqrt{31}\right ) \log \left (\left (5 x^2+3 x+2\right ) \left (44 \sqrt{31} x^2+327 i x^2-4 i \sqrt{682 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-22 \sqrt{31} x+469 i x+i \sqrt{682 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+66 \sqrt{31}-142 i\right )\right )}{10 \sqrt{\frac{62}{11} \left (-13+i \sqrt{31}\right )}}+\frac{\left (-13 i+\sqrt{31}\right ) \log \left (\left (5 x^2+3 x+2\right ) \left (44 \sqrt{31} x^2-817 i x^2+22 i \sqrt{22 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-22 \sqrt{31} x+1041 i x-63 i \sqrt{22 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+66 \sqrt{31}-1858 i\right )\right )}{10 \sqrt{\frac{62}{11} \left (13+i \sqrt{31}\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2),x]
[Out]
(Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/5 - ((I/5)*(-13*I + Sqrt[31])*ArcTan[(31*
(7588*I + 4224*Sqrt[31] - (27836*I)*x + 3872*Sqrt[31]*x + (4347*I)*x^2 + 2706*Sq
rt[31]*x^2 - (31860*I)*x^3 + 2970*Sqrt[31]*x^3 - (8675*I)*x^4 + 1100*Sqrt[31]*x^
4))/(65472 + (35044*I)*Sqrt[31] + 1083016*x - (46668*I)*Sqrt[31]*x + 340318*x^2
- (308889*I)*Sqrt[31]*x^2 + 514910*x^3 - (143180*I)*Sqrt[31]*x^3 + 443300*x^4 -
(262775*I)*Sqrt[31]*x^4 - (1000*I)*Sqrt[682*(13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^
2] + (2500*I)*Sqrt[682*(13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] + (3500*I)*Sqrt[
682*(13 + I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2] + (10000*I)*Sqrt[682*(13 + I*Sqrt
[31])]*x^3*Sqrt[3 - x + 2*x^2])])/Sqrt[(62*(13 + I*Sqrt[31]))/11] - ((I/5)*(13*I
+ Sqrt[31])*ArcTanh[(-65472*I - 35044*Sqrt[31] - (1083016*I)*x + 46668*Sqrt[31]
*x - (340318*I)*x^2 + 308889*Sqrt[31]*x^2 - (514910*I)*x^3 + 143180*Sqrt[31]*x^3
- (443300*I)*x^4 + 262775*Sqrt[31]*x^4 - 63000*Sqrt[22*(-13 + I*Sqrt[31])]*Sqrt
[3 - x + 2*x^2] - 72500*Sqrt[22*(-13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] - 1245
00*Sqrt[22*(-13 + I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2] + 55000*Sqrt[22*(-13 + I*
Sqrt[31])]*x^3*Sqrt[3 - x + 2*x^2])/(1764772*I + 130944*Sqrt[31] + (2352916*I)*x
+ 120032*Sqrt[31]*x + (3090243*I)*x^2 + 83886*Sqrt[31]*x^2 - (2052340*I)*x^3 +
92070*Sqrt[31]*x^3 + (1493925*I)*x^4 + 34100*Sqrt[31]*x^4)])/Sqrt[(62*(-13 + I*S
qrt[31]))/11] - ((-13*I + Sqrt[31])*Log[(-3*I + Sqrt[31] - (10*I)*x)^2*(3*I + Sq
rt[31] + (10*I)*x)^2])/(10*Sqrt[(62*(13 + I*Sqrt[31]))/11]) + ((I/10)*(13*I + Sq
rt[31])*Log[(-3*I + Sqrt[31] - (10*I)*x)^2*(3*I + Sqrt[31] + (10*I)*x)^2])/Sqrt[
(62*(-13 + I*Sqrt[31]))/11] - ((I/10)*(13*I + Sqrt[31])*Log[(2 + 3*x + 5*x^2)*(-
142*I + 66*Sqrt[31] + (469*I)*x - 22*Sqrt[31]*x + (327*I)*x^2 + 44*Sqrt[31]*x^2
+ I*Sqrt[682*(-13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2] - (4*I)*Sqrt[682*(-13 + I*S
qrt[31])]*x*Sqrt[3 - x + 2*x^2])])/Sqrt[(62*(-13 + I*Sqrt[31]))/11] + ((-13*I +
Sqrt[31])*Log[(2 + 3*x + 5*x^2)*(-1858*I + 66*Sqrt[31] + (1041*I)*x - 22*Sqrt[31
]*x - (817*I)*x^2 + 44*Sqrt[31]*x^2 - (63*I)*Sqrt[22*(13 + I*Sqrt[31])]*Sqrt[3 -
x + 2*x^2] + (22*I)*Sqrt[22*(13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2])])/(10*Sqr
t[(62*(13 + I*Sqrt[31]))/11])
_______________________________________________________________________________________
Maple [B] time = 0.187, size = 2065, normalized size = 11.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2),x)
[Out]
1/5*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+1/52855*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1
-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)*2^(1/2)*(-285
*2^(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-2
3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*2^(1/2)*(2^(1/2)-1
+x)^2/(2^(1/2)+1-x)^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))*(-8866+6820*2^(1/2))^(1/2)*(-775687+54936
2*2^(1/2))^(1/2)-386*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
)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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))*(-8866+6820*2^(1/2))^(1/2)*(-
775687+549362*2^(1/2))^(1/2)+274846*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x
)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(
1/2))^(1/2))*2^(1/2)+1543366*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*2
^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-
x)^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+6820*2^(1/2))^(1/2)+1/21142*(8*(2^(1/2)-1+x)^2/(
2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)*2^(1
/2)*(151*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)*(2
^(1/2)-1+x)^2/(2^(1/2)+1-x)^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))*(-8866+6820*2^(1/2))^(1/2)*(-7756
87+549362*2^(1/2))^(1/2)+218*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)*(648
5*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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))*(-8866+6820*2^(1/2))^
(1/2)*(-775687+549362*2^(1/2))^(1/2)+401698*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(
1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)/(-8866+
6820*2^(1/2))^(1/2))*2^(1/2)-63426*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)
^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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)*(2^(1/2)-1+x)^2/(2^(1
/2)+1-x)^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+6820*2^(1/2))^(1/2)+3/21142*(8*(2^(1/2)-1+
x)^2/(2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2
)*2^(1/2)*(369*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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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))*(-8866+6820*2^(1/2))^(1/2)*
(-775687+549362*2^(1/2))^(1/2)+520*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)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^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))*(-8866+6820*2^(
1/2))^(1/2)*(-775687+549362*2^(1/2))^(1/2)+465124*arctanh(31/2*(8*(2^(1/2)-1+x)^
2/(2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)/(
-8866+6820*2^(1/2))^(1/2))*2^(1/2)-866822*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/
2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+8-3*2^(1/2))^(1/2)/(-8866+68
20*2^(1/2))^(1/2)))/((8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*2^(1/2)*(2^(1/2)-1+x)^
2/(2^(1/2)+1-x)^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+6820*2^(1/2))^(1/2)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 \, x^{2} - x + 3}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2),x, algorithm="maxima")
[Out]
integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2), x)
_______________________________________________________________________________________
Fricas [A] time = 0.337173, size = 1397, normalized size = 8.03 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2),x, algorithm="fricas")
[Out]
-1/480500*sqrt(155)*sqrt(31)*sqrt(5)*(2*sqrt(155)*sqrt(31)*sqrt(5)*(13*sqrt(2) -
20)*sqrt((13*sqrt(2) - 20)/(260*sqrt(2) - 369))*log(-4*sqrt(2)*sqrt(2*x^2 - x +
3)*(4*x - 1) - 32*x^2 + 16*x - 25) - 5*242^(1/4)*sqrt(31)*(10*sqrt(2) - 13)*log
(-8/5*(2*242^(1/4)*sqrt(155)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(61198*x + 253
53) - 86551*x - 35845)*sqrt((13*sqrt(2) - 20)/(260*sqrt(2) - 369)) + 3464300*x^2
+ 220*sqrt(2)*(28280*x^2 - 9997*sqrt(2)*(2*x^2 - x + 3) - 14140*x + 42420) - 49
985*sqrt(2)*(49*x^2 - 151*x + 200) - 10675700*x + 14140000)/(9997*sqrt(2)*x^2 -
14140*x^2)) + 5*242^(1/4)*sqrt(31)*(10*sqrt(2) - 13)*log(8/5*(2*242^(1/4)*sqrt(1
55)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(61198*x + 25353) - 86551*x - 35845)*sq
rt((13*sqrt(2) - 20)/(260*sqrt(2) - 369)) - 3464300*x^2 - 220*sqrt(2)*(28280*x^2
- 9997*sqrt(2)*(2*x^2 - x + 3) - 14140*x + 42420) + 49985*sqrt(2)*(49*x^2 - 151
*x + 200) + 10675700*x - 14140000)/(9997*sqrt(2)*x^2 - 14140*x^2)) + 620*242^(1/
4)*arctan(31*(sqrt(155)*sqrt(5)*(10*sqrt(2)*(x - 6) - 13*x + 78)*sqrt((13*sqrt(2
) - 20)/(260*sqrt(2) - 369)) + 10*242^(1/4)*sqrt(2*x^2 - x + 3)*(3*sqrt(2) - 7))
/(2*sqrt(155)*sqrt(31)*sqrt(5)*sqrt(2/5)*(10*sqrt(2)*x - 13*x)*sqrt(-(2*242^(1/4
)*sqrt(155)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(61198*x + 25353) - 86551*x - 3
5845)*sqrt((13*sqrt(2) - 20)/(260*sqrt(2) - 369)) + 3464300*x^2 + 220*sqrt(2)*(2
8280*x^2 - 9997*sqrt(2)*(2*x^2 - x + 3) - 14140*x + 42420) - 49985*sqrt(2)*(49*x
^2 - 151*x + 200) - 10675700*x + 14140000)/(9997*sqrt(2)*x^2 - 14140*x^2))*sqrt(
(13*sqrt(2) - 20)/(260*sqrt(2) - 369)) + sqrt(155)*sqrt(31)*sqrt(5)*(10*sqrt(2)*
(19*x - 22) - 247*x + 286)*sqrt((13*sqrt(2) - 20)/(260*sqrt(2) - 369)) - 310*242
^(1/4)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2) - 1))) + 620*242^(1/4)*arctan(-31*(
sqrt(155)*sqrt(5)*(10*sqrt(2)*(x - 6) - 13*x + 78)*sqrt((13*sqrt(2) - 20)/(260*s
qrt(2) - 369)) - 10*242^(1/4)*sqrt(2*x^2 - x + 3)*(3*sqrt(2) - 7))/(2*sqrt(155)*
sqrt(31)*sqrt(5)*sqrt(2/5)*(10*sqrt(2)*x - 13*x)*sqrt((2*242^(1/4)*sqrt(155)*sqr
t(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(61198*x + 25353) - 86551*x - 35845)*sqrt((13*
sqrt(2) - 20)/(260*sqrt(2) - 369)) - 3464300*x^2 - 220*sqrt(2)*(28280*x^2 - 9997
*sqrt(2)*(2*x^2 - x + 3) - 14140*x + 42420) + 49985*sqrt(2)*(49*x^2 - 151*x + 20
0) + 10675700*x - 14140000)/(9997*sqrt(2)*x^2 - 14140*x^2))*sqrt((13*sqrt(2) - 2
0)/(260*sqrt(2) - 369)) + sqrt(155)*sqrt(31)*sqrt(5)*(10*sqrt(2)*(19*x - 22) - 2
47*x + 286)*sqrt((13*sqrt(2) - 20)/(260*sqrt(2) - 369)) + 310*242^(1/4)*sqrt(31)
*sqrt(2*x^2 - x + 3)*(sqrt(2) - 1))))/((10*sqrt(2) - 13)*sqrt((13*sqrt(2) - 20)/
(260*sqrt(2) - 369)))
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 x^{2} - x + 3}}{5 x^{2} + 3 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x**2-x+3)**(1/2)/(5*x**2+3*x+2),x)
[Out]
Integral(sqrt(2*x**2 - x + 3)/(5*x**2 + 3*x + 2), x)
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2),x, algorithm="giac")
[Out]
Exception raised: TypeError