3.76 \(\int \frac{\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx\)
Optimal. Leaf size=222 \[ -\frac{1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}-\frac{(226249-99620 x) \sqrt{2 x^2-x+3}}{80000}-\frac{121 \sqrt{\frac{11}{31} \left (25000 \sqrt{2}-15457\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (25000 \sqrt{2}-15457\right )}} \left (-\left (690+247 \sqrt{2}\right ) x-443 \sqrt{2}+196\right )}{\sqrt{2 x^2-x+3}}\right )}{3125}+\frac{121 \sqrt{\frac{11}{31} \left (15457+25000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (15457+25000 \sqrt{2}\right )}} \left (-\left (690-247 \sqrt{2}\right ) x+443 \sqrt{2}+196\right )}{\sqrt{2 x^2-x+3}}\right )}{3125}-\frac{7216203 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{800000 \sqrt{2}} \]
[Out]
-((226249 - 99620*x)*Sqrt[3 - x + 2*x^2])/80000 - ((103 - 60*x)*(3 - x + 2*x^2)^
(3/2))/600 - (7216203*ArcSinh[(1 - 4*x)/Sqrt[23]])/(800000*Sqrt[2]) - (121*Sqrt[
(11*(-15457 + 25000*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(-15457 + 25000*Sqrt[2]))]
*(196 - 443*Sqrt[2] - (690 + 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3125 + (121*
Sqrt[(11*(15457 + 25000*Sqrt[2]))/31]*ArcTanh[(Sqrt[11/(62*(15457 + 25000*Sqrt[2
]))]*(196 + 443*Sqrt[2] - (690 - 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3125
_______________________________________________________________________________________
Rubi [A] time = 1.11301, antiderivative size = 222, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333
\[ -\frac{1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}-\frac{(226249-99620 x) \sqrt{2 x^2-x+3}}{80000}-\frac{121 \sqrt{\frac{11}{31} \left (25000 \sqrt{2}-15457\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (25000 \sqrt{2}-15457\right )}} \left (-\left (690+247 \sqrt{2}\right ) x-443 \sqrt{2}+196\right )}{\sqrt{2 x^2-x+3}}\right )}{3125}+\frac{121 \sqrt{\frac{11}{31} \left (15457+25000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (15457+25000 \sqrt{2}\right )}} \left (-\left (690-247 \sqrt{2}\right ) x+443 \sqrt{2}+196\right )}{\sqrt{2 x^2-x+3}}\right )}{3125}-\frac{7216203 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{800000 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(3 - x + 2*x^2)^(5/2)/(2 + 3*x + 5*x^2),x]
[Out]
-((226249 - 99620*x)*Sqrt[3 - x + 2*x^2])/80000 - ((103 - 60*x)*(3 - x + 2*x^2)^
(3/2))/600 - (7216203*ArcSinh[(1 - 4*x)/Sqrt[23]])/(800000*Sqrt[2]) - (121*Sqrt[
(11*(-15457 + 25000*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(-15457 + 25000*Sqrt[2]))]
*(196 - 443*Sqrt[2] - (690 + 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3125 + (121*
Sqrt[(11*(15457 + 25000*Sqrt[2]))/31]*ArcTanh[(Sqrt[11/(62*(15457 + 25000*Sqrt[2
]))]*(196 + 443*Sqrt[2] - (690 - 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3125
_______________________________________________________________________________________
Rubi in Sympy [A] time = 129.99, size = 250, normalized size = 1.13 \[ - \frac{\left (- \frac{74715 x}{2} + \frac{678747}{8}\right ) \sqrt{2 x^{2} - x + 3}}{30000} - \frac{\left (- 30 x + \frac{103}{2}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}{300} + \frac{\sqrt{341} \left (- 311326224 \sqrt{2} + 137742528\right ) \left (- 15460896 \sqrt{2} + 151797888\right ) \operatorname{atan}{\left (\frac{\sqrt{682} \left (x \left (-484909920 - 173583696 \sqrt{2}\right ) - 311326224 \sqrt{2} + 137742528\right )}{43571616 \sqrt{-15457 + 25000 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{1581649660800000 \sqrt{-15457 + 25000 \sqrt{2}}} + \frac{7216203 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{1600000} + \frac{\sqrt{341} \left (137742528 + 311326224 \sqrt{2}\right ) \left (15460896 \sqrt{2} + 151797888\right ) \operatorname{atanh}{\left (\frac{\sqrt{682} \left (x \left (-484909920 + 173583696 \sqrt{2}\right ) + 137742528 + 311326224 \sqrt{2}\right )}{43571616 \sqrt{15457 + 25000 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{1581649660800000 \sqrt{15457 + 25000 \sqrt{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*x**2-x+3)**(5/2)/(5*x**2+3*x+2),x)
[Out]
-(-74715*x/2 + 678747/8)*sqrt(2*x**2 - x + 3)/30000 - (-30*x + 103/2)*(2*x**2 -
x + 3)**(3/2)/300 + sqrt(341)*(-311326224*sqrt(2) + 137742528)*(-15460896*sqrt(2
) + 151797888)*atan(sqrt(682)*(x*(-484909920 - 173583696*sqrt(2)) - 311326224*sq
rt(2) + 137742528)/(43571616*sqrt(-15457 + 25000*sqrt(2))*sqrt(2*x**2 - x + 3)))
/(1581649660800000*sqrt(-15457 + 25000*sqrt(2))) + 7216203*sqrt(2)*atanh(sqrt(2)
*(4*x - 1)/(4*sqrt(2*x**2 - x + 3)))/1600000 + sqrt(341)*(137742528 + 311326224*
sqrt(2))*(15460896*sqrt(2) + 151797888)*atanh(sqrt(682)*(x*(-484909920 + 1735836
96*sqrt(2)) + 137742528 + 311326224*sqrt(2))/(43571616*sqrt(15457 + 25000*sqrt(2
))*sqrt(2*x**2 - x + 3)))/(1581649660800000*sqrt(15457 + 25000*sqrt(2)))
_______________________________________________________________________________________
Mathematica [C] time = 6.45856, size = 1189, normalized size = 5.36 \[ \sqrt{2 x^2-x+3} \left (\frac{x^3}{5}-\frac{133 x^2}{300}+\frac{20603 x}{12000}-\frac{267449}{80000}\right )+\frac{7216203 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{800000 \sqrt{2}}+\frac{121 \left (247 i+119 \sqrt{31}\right ) \tan ^{-1}\left (\frac{-214634275 i \sqrt{31} x^4-1002301300 x^4-137500000 i \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3-285779980 i \sqrt{31} x^3+1188688490 x^3+311250000 i \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2-326488029 i \sqrt{31} x^2-1240038998 x^2+181250000 i \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x+277778652 i \sqrt{31} x+727715824 x+157500000 i \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}-46000516 i \sqrt{31}+910772808}{482890100 \sqrt{31} x^4+8825296925 i x^4+248749270 \sqrt{31} x^3-1371093740 i x^3+603640246 \sqrt{31} x^2+14142713923 i x^2+673090352 \sqrt{31} x+4941322076 i x+186603384 \sqrt{31}+4168906492 i}\right )}{3125 \sqrt{\frac{62}{11} \left (-13+i \sqrt{31}\right )}}-\frac{121 i \left (-247 i+119 \sqrt{31}\right ) \tan ^{-1}\left (\frac{31 \left (15577100 \sqrt{31} x^4-185896675 i x^4+8024170 \sqrt{31} x^3-200932460 i x^3+19472266 \sqrt{31} x^2-196135933 i x^2+21712592 \sqrt{31} x-39236196 i x+6019464 \sqrt{31}+26809468 i\right )}{-214634275 i \sqrt{31} x^4+1002301300 x^4+25000000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3-285779980 i \sqrt{31} x^3-1188688490 x^3+8750000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2-326488029 i \sqrt{31} x^2+1240038998 x^2+6250000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x+277778652 i \sqrt{31} x-727715824 x-2500000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}-46000516 i \sqrt{31}-910772808}\right )}{3125 \sqrt{\frac{62}{11} \left (13+i \sqrt{31}\right )}}+\frac{121 i \left (247 i+119 \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 )}{6250 \sqrt{\frac{62}{11} \left (-13+i \sqrt{31}\right )}}-\frac{121 \left (-247 i+119 \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 )}{6250 \sqrt{\frac{62}{11} \left (13+i \sqrt{31}\right )}}-\frac{121 i \left (247 i+119 \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 )}{6250 \sqrt{\frac{62}{11} \left (-13+i \sqrt{31}\right )}}+\frac{121 \left (-247 i+119 \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 )}{6250 \sqrt{\frac{62}{11} \left (13+i \sqrt{31}\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(3 - x + 2*x^2)^(5/2)/(2 + 3*x + 5*x^2),x]
[Out]
Sqrt[3 - x + 2*x^2]*(-267449/80000 + (20603*x)/12000 - (133*x^2)/300 + x^3/5) +
(7216203*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(800000*Sqrt[2]) + (121*(247*I + 119*Sqrt
[31])*ArcTan[(910772808 - (46000516*I)*Sqrt[31] + 727715824*x + (277778652*I)*Sq
rt[31]*x - 1240038998*x^2 - (326488029*I)*Sqrt[31]*x^2 + 1188688490*x^3 - (28577
9980*I)*Sqrt[31]*x^3 - 1002301300*x^4 - (214634275*I)*Sqrt[31]*x^4 + (157500000*
I)*Sqrt[22*(-13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2] + (181250000*I)*Sqrt[22*(-13
+ I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] + (311250000*I)*Sqrt[22*(-13 + I*Sqrt[31])]
*x^2*Sqrt[3 - x + 2*x^2] - (137500000*I)*Sqrt[22*(-13 + I*Sqrt[31])]*x^3*Sqrt[3
- x + 2*x^2])/(4168906492*I + 186603384*Sqrt[31] + (4941322076*I)*x + 673090352*
Sqrt[31]*x + (14142713923*I)*x^2 + 603640246*Sqrt[31]*x^2 - (1371093740*I)*x^3 +
248749270*Sqrt[31]*x^3 + (8825296925*I)*x^4 + 482890100*Sqrt[31]*x^4)])/(3125*S
qrt[(62*(-13 + I*Sqrt[31]))/11]) - (((121*I)/3125)*(-247*I + 119*Sqrt[31])*ArcTa
n[(31*(26809468*I + 6019464*Sqrt[31] - (39236196*I)*x + 21712592*Sqrt[31]*x - (1
96135933*I)*x^2 + 19472266*Sqrt[31]*x^2 - (200932460*I)*x^3 + 8024170*Sqrt[31]*x
^3 - (185896675*I)*x^4 + 15577100*Sqrt[31]*x^4))/(-910772808 - (46000516*I)*Sqrt
[31] - 727715824*x + (277778652*I)*Sqrt[31]*x + 1240038998*x^2 - (326488029*I)*S
qrt[31]*x^2 - 1188688490*x^3 - (285779980*I)*Sqrt[31]*x^3 + 1002301300*x^4 - (21
4634275*I)*Sqrt[31]*x^4 - (2500000*I)*Sqrt[682*(13 + I*Sqrt[31])]*Sqrt[3 - x + 2
*x^2] + (6250000*I)*Sqrt[682*(13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] + (8750000
*I)*Sqrt[682*(13 + I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2] + (25000000*I)*Sqrt[682*
(13 + I*Sqrt[31])]*x^3*Sqrt[3 - x + 2*x^2])])/Sqrt[(62*(13 + I*Sqrt[31]))/11] -
(121*(-247*I + 119*Sqrt[31])*Log[(-3*I + Sqrt[31] - (10*I)*x)^2*(3*I + Sqrt[31]
+ (10*I)*x)^2])/(6250*Sqrt[(62*(13 + I*Sqrt[31]))/11]) + (((121*I)/6250)*(247*I
+ 119*Sqrt[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] - (((121*I)/6250)*(247*I + 119*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*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2])])/Sqrt[(62*(-13 + I*Sqrt[3
1]))/11] + (121*(-247*I + 119*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])])/(6250*Sqrt[(62*(13 + I*Sqrt[31]))/11])
_______________________________________________________________________________________
Maple [B] time = 0.064, size = 4860, normalized size = 21.9 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x)
[Out]
7216203/1600000*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-267449/80000*(2*x^2-x+3)^
(1/2)+20603/12000*x*(2*x^2-x+3)^(1/2)-133/300*x^2*(2*x^2-x+3)^(1/2)+1/5*x^3*(2*x
^2-x+3)^(1/2)+4/33034375*(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)*(75195*2^(1/2)*arctan(1/1169248
7*(-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+103
68*(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)+106294*a
rctan(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)+108099046*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
)-158290154*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)+6/6606875*(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)*(10915*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)+14918*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)*(-775
687+549362*2^(1/2))^(1/2)-5052938*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)-51565338*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)-21/1321375*(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)*(4245*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)*(-7
75687+549362*2^(1/2))^(1/2)+6154*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)+12325786*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)-359414*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)-37/528550*(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)*(2365*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)+3338*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))*(-88
66+6820*2^(1/2))^(1/2)*(-775687+549362*2^(1/2))^(1/2)+3192442*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)-5264358*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)+63
/105710*(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))*(-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))*(-88
66+6820*2^(1/2))^(1/2)*(-775687+549362*2^(1/2))^(1/2)-386*arctan(1/11692487*(-77
5687+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)+27/21142*(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*(-77568
7+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)+218*arctan(1/1169
2487*(-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)+40169
8*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)+27/21142*(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+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))/(-8
866+6820*2^(1/2))^(1/2)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 - x + 3)^(5/2)/(5*x^2 + 3*x + 2),x, algorithm="maxima")
[Out]
integrate((2*x^2 - x + 3)^(5/2)/(5*x^2 + 3*x + 2), x)
_______________________________________________________________________________________
Fricas [A] time = 0.417159, size = 1519, normalized size = 6.84 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 - x + 3)^(5/2)/(5*x^2 + 3*x + 2),x, algorithm="fricas")
[Out]
1/115320000000*sqrt(155)*sqrt(31)*sqrt(5)*(21648609*sqrt(155)*sqrt(31)*sqrt(5)*(
25000*sqrt(2) + 15457)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 1488918
849))*log(-sqrt(2)*(32*x^2 - 16*x + 25) - 8*sqrt(2*x^2 - x + 3)*(4*x - 1)) + 20*
sqrt(155)*sqrt(31)*sqrt(5)*(2400000000*x^3 - 5320000000*x^2 + 15457*sqrt(2)*(480
00*x^3 - 106400*x^2 + 412060*x - 802347) + 20603000000*x - 40117350000)*sqrt(2*x
^2 - x + 3)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 1488918849)) - 232
320*242^(1/4)*sqrt(31)*(15457*sqrt(2) + 50000)*log(468512/5*(2*242^(1/4)*sqrt(15
5)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(25340070869392033*x - 63023045494527892
) + 37682974625135859*x - 88363116363919925)*sqrt((15457*sqrt(2) + 50000)/(77285
0000*sqrt(2) + 1488918849)) + 24092767700750000*x^2 + 220*sqrt(2)*(1966756547000
00*x^2 + 61656718648993*sqrt(2)*(2*x^2 - x + 3) - 98337827350000*x + 29501348205
0000) + 308283593244965*sqrt(2)*(49*x^2 - 151*x + 200) - 74245059649250000*x + 9
8337827350000000)/(61656718648993*sqrt(2)*x^2 + 98337827350000*x^2)) + 232320*24
2^(1/4)*sqrt(31)*(15457*sqrt(2) + 50000)*log(-468512/5*(2*242^(1/4)*sqrt(155)*sq
rt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(25340070869392033*x - 63023045494527892) + 3
7682974625135859*x - 88363116363919925)*sqrt((15457*sqrt(2) + 50000)/(772850000*
sqrt(2) + 1488918849)) - 24092767700750000*x^2 - 220*sqrt(2)*(196675654700000*x^
2 + 61656718648993*sqrt(2)*(2*x^2 - x + 3) - 98337827350000*x + 295013482050000)
- 308283593244965*sqrt(2)*(49*x^2 - 151*x + 200) + 74245059649250000*x - 983378
27350000000)/(61656718648993*sqrt(2)*x^2 + 98337827350000*x^2)) + 164520660480*2
42^(1/4)*sqrt(2)*arctan(31*(sqrt(155)*sqrt(5)*(25000*sqrt(2)*(x - 6) + 15457*x -
92742)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 1488918849)) + 10*242^
(1/4)*sqrt(2*x^2 - x + 3)*(98*sqrt(2) + 443))/(2*sqrt(155)*sqrt(31)*sqrt(5)*sqrt
(2/5)*(25000*sqrt(2)*x + 15457*x)*sqrt((2*242^(1/4)*sqrt(155)*sqrt(5)*sqrt(2*x^2
- x + 3)*(sqrt(2)*(25340070869392033*x - 63023045494527892) + 37682974625135859
*x - 88363116363919925)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 148891
8849)) + 24092767700750000*x^2 + 220*sqrt(2)*(196675654700000*x^2 + 616567186489
93*sqrt(2)*(2*x^2 - x + 3) - 98337827350000*x + 295013482050000) + 3082835932449
65*sqrt(2)*(49*x^2 - 151*x + 200) - 74245059649250000*x + 98337827350000000)/(61
656718648993*sqrt(2)*x^2 + 98337827350000*x^2))*sqrt((15457*sqrt(2) + 50000)/(77
2850000*sqrt(2) + 1488918849)) + sqrt(155)*sqrt(31)*sqrt(5)*(25000*sqrt(2)*(19*x
- 22) + 293683*x - 340054)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 14
88918849)) + 310*242^(1/4)*sqrt(31)*sqrt(2*x^2 - x + 3)*(54*sqrt(2) + 11))) + 16
4520660480*242^(1/4)*sqrt(2)*arctan(-31*(sqrt(155)*sqrt(5)*(25000*sqrt(2)*(x - 6
) + 15457*x - 92742)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 148891884
9)) - 10*242^(1/4)*sqrt(2*x^2 - x + 3)*(98*sqrt(2) + 443))/(2*sqrt(155)*sqrt(31)
*sqrt(5)*sqrt(2/5)*(25000*sqrt(2)*x + 15457*x)*sqrt(-(2*242^(1/4)*sqrt(155)*sqrt
(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(25340070869392033*x - 63023045494527892) + 376
82974625135859*x - 88363116363919925)*sqrt((15457*sqrt(2) + 50000)/(772850000*sq
rt(2) + 1488918849)) - 24092767700750000*x^2 - 220*sqrt(2)*(196675654700000*x^2
+ 61656718648993*sqrt(2)*(2*x^2 - x + 3) - 98337827350000*x + 295013482050000) -
308283593244965*sqrt(2)*(49*x^2 - 151*x + 200) + 74245059649250000*x - 98337827
350000000)/(61656718648993*sqrt(2)*x^2 + 98337827350000*x^2))*sqrt((15457*sqrt(2
) + 50000)/(772850000*sqrt(2) + 1488918849)) + sqrt(155)*sqrt(31)*sqrt(5)*(25000
*sqrt(2)*(19*x - 22) + 293683*x - 340054)*sqrt((15457*sqrt(2) + 50000)/(77285000
0*sqrt(2) + 1488918849)) - 310*242^(1/4)*sqrt(31)*sqrt(2*x^2 - x + 3)*(54*sqrt(2
) + 11))))/((15457*sqrt(2) + 50000)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt
(2) + 1488918849)))
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (2 x^{2} - x + 3\right )^{\frac{5}{2}}}{5 x^{2} + 3 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x**2-x+3)**(5/2)/(5*x**2+3*x+2),x)
[Out]
Integral((2*x**2 - x + 3)**(5/2)/(5*x**2 + 3*x + 2), x)
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 - x + 3)^(5/2)/(5*x^2 + 3*x + 2),x, algorithm="giac")
[Out]
Exception raised: RuntimeError