3.83 \(\int \frac{1}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx\)
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))/Sqrt[3 - x + 2*x^2]] - Sqrt[(-13 + 10*Sqrt[2])/682]*A
rcTanh[(Sqrt[11/(31*(-13 + 10*Sqrt[2]))]*(7 - 3*Sqrt[2] + (13 - 10*Sqrt[2])*x))/
Sqrt[3 - x + 2*x^2]]
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Rubi [A] time = 0.627494, 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
\[ \sqrt{\frac{1}{682} \left (13+10 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (13+10 \sqrt{2}\right )}} \left (\left (13+10 \sqrt{2}\right ) x+3 \sqrt{2}+7\right )}{\sqrt{2 x^2-x+3}}\right )-\sqrt{\frac{1}{682} \left (10 \sqrt{2}-13\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (10 \sqrt{2}-13\right )}} \left (\left (13-10 \sqrt{2}\right ) x-3 \sqrt{2}+7\right )}{\sqrt{2 x^2-x+3}}\right ) \]
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))/Sqrt[3 - x + 2*x^2]] - Sqrt[(-13 + 10*Sqrt[2])/682]*A
rcTanh[(Sqrt[11/(31*(-13 + 10*Sqrt[2]))]*(7 - 3*Sqrt[2] + (13 - 10*Sqrt[2])*x))/
Sqrt[3 - x + 2*x^2]]
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Rubi in Sympy [A] time = 55.0243, size = 172, normalized size = 1.16 \[ \frac{\sqrt{682} \left (22 + 22 \sqrt{2}\right ) \left (33 \sqrt{2} + 77\right ) \operatorname{atan}{\left (\frac{\sqrt{341} \left (x \left (143 + 110 \sqrt{2}\right ) + 33 \sqrt{2} + 77\right )}{341 \sqrt{13 + 10 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{165044 \sqrt{13 + 10 \sqrt{2}}} + \frac{\sqrt{682} \left (- 33 \sqrt{2} + 77\right ) \left (- 22 \sqrt{2} + 22\right ) \operatorname{atanh}{\left (\frac{\sqrt{341} \left (x \left (- 110 \sqrt{2} + 143\right ) - 33 \sqrt{2} + 77\right )}{341 \sqrt{-13 + 10 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{165044 \sqrt{-13 + 10 \sqrt{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(5*x**2+3*x+2)/(2*x**2-x+3)**(1/2),x)
[Out]
sqrt(682)*(22 + 22*sqrt(2))*(33*sqrt(2) + 77)*atan(sqrt(341)*(x*(143 + 110*sqrt(
2)) + 33*sqrt(2) + 77)/(341*sqrt(13 + 10*sqrt(2))*sqrt(2*x**2 - x + 3)))/(165044
*sqrt(13 + 10*sqrt(2))) + sqrt(682)*(-33*sqrt(2) + 77)*(-22*sqrt(2) + 22)*atanh(
sqrt(341)*(x*(-110*sqrt(2) + 143) - 33*sqrt(2) + 77)/(341*sqrt(-13 + 10*sqrt(2))
*sqrt(2*x**2 - x + 3)))/(165044*sqrt(-13 + 10*sqrt(2)))
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Mathematica [C] time = 6.41329, size = 959, normalized size = 6.48 \[ -\frac{5 i \tan ^{-1}\left (\frac{11 \left (550 x^4-2200 x^3-62 i \sqrt{31} x^2+1906 x^2+31 i \sqrt{31} x+1797 x-93 i \sqrt{31}+1759\right )}{-1100 \sqrt{31} x^4-550 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3-110 \sqrt{31} x^3+6820 i x^3+1245 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2-1078 \sqrt{31} x^2-1364 i x^2+725 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-1111 \sqrt{31} x+9207 i x+630 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+363 \sqrt{31}+3069 i}\right )}{\sqrt{\frac{341}{2} \left (-13+i \sqrt{31}\right )}}-\frac{5 \tan ^{-1}\left (\frac{31 \left (100 i x^2-50 i x+11 \sqrt{31}+7 i\right ) \left (2 x^2-x+3\right )}{-1100 i \sqrt{31} x^4+100 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3-110 i \sqrt{31} x^3+6820 x^3+35 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2-1078 i \sqrt{31} x^2-1364 x^2+25 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-1111 i \sqrt{31} x+9207 x-10 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+363 i \sqrt{31}+3069}\right )}{\sqrt{\frac{341}{2} \left (13+i \sqrt{31}\right )}}+\frac{5 i \log \left (\left (-10 i x+\sqrt{31}-3 i\right )^2 \left (10 i x+\sqrt{31}+3 i\right )^2\right )}{\sqrt{682 \left (13+i \sqrt{31}\right )}}-\frac{5 \log \left (\left (-10 i x+\sqrt{31}-3 i\right )^2 \left (10 i x+\sqrt{31}+3 i\right )^2\right )}{\sqrt{682 \left (-13+i \sqrt{31}\right )}}+\frac{5 \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 )}{\sqrt{682 \left (-13+i \sqrt{31}\right )}}-\frac{5 i \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 )}{\sqrt{682 \left (13+i \sqrt{31}\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)),x]
[Out]
((-5*I)*ArcTan[(11*(1759 - (93*I)*Sqrt[31] + 1797*x + (31*I)*Sqrt[31]*x + 1906*x
^2 - (62*I)*Sqrt[31]*x^2 - 2200*x^3 + 550*x^4))/(3069*I + 363*Sqrt[31] + (9207*I
)*x - 1111*Sqrt[31]*x - (1364*I)*x^2 - 1078*Sqrt[31]*x^2 + (6820*I)*x^3 - 110*Sq
rt[31]*x^3 - 1100*Sqrt[31]*x^4 + 630*Sqrt[22*(-13 + I*Sqrt[31])]*Sqrt[3 - x + 2*
x^2] + 725*Sqrt[22*(-13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] + 1245*Sqrt[22*(-13
+ I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2] - 550*Sqrt[22*(-13 + I*Sqrt[31])]*x^3*Sq
rt[3 - x + 2*x^2])])/Sqrt[(341*(-13 + I*Sqrt[31]))/2] - (5*ArcTan[(31*(7*I + 11*
Sqrt[31] - (50*I)*x + (100*I)*x^2)*(3 - x + 2*x^2))/(3069 + (363*I)*Sqrt[31] + 9
207*x - (1111*I)*Sqrt[31]*x - 1364*x^2 - (1078*I)*Sqrt[31]*x^2 + 6820*x^3 - (110
*I)*Sqrt[31]*x^3 - (1100*I)*Sqrt[31]*x^4 - (10*I)*Sqrt[682*(13 + I*Sqrt[31])]*Sq
rt[3 - x + 2*x^2] + (25*I)*Sqrt[682*(13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] + (
35*I)*Sqrt[682*(13 + I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2] + (100*I)*Sqrt[682*(13
+ I*Sqrt[31])]*x^3*Sqrt[3 - x + 2*x^2])])/Sqrt[(341*(13 + I*Sqrt[31]))/2] - (5*
Log[(-3*I + Sqrt[31] - (10*I)*x)^2*(3*I + Sqrt[31] + (10*I)*x)^2])/Sqrt[682*(-13
+ I*Sqrt[31])] + ((5*I)*Log[(-3*I + Sqrt[31] - (10*I)*x)^2*(3*I + Sqrt[31] + (1
0*I)*x)^2])/Sqrt[682*(13 + I*Sqrt[31])] + (5*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[68
2*(-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[682*(-13 + I*Sqrt[31])] - ((5*I)*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])])/Sqrt[682*(13 + I*Sqrt[31])]
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Maple [B] time = 0.006, size = 684, normalized size = 4.6 \[ \text{result too large to display} \]
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)*(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))*(-886
6+6820*2^(1/2))^(1/2)*(-775687+549362*2^(1/2))^(1/2)+520*arctan(1/11692487*(-775
687+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))/(-8866+6820*2^(1/2))
^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt{2 \, x^{2} - x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)*sqrt(2*x^2 - x + 3)),x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)*sqrt(2*x^2 - x + 3)), x)
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Fricas [A] time = 0.330157, size = 1299, normalized size = 8.78 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)*sqrt(2*x^2 - x + 3)),x, algorithm="fricas")
[Out]
1/422840*sqrt(341)*sqrt(31)*sqrt(5)*(200^(1/4)*sqrt(31)*(10*sqrt(2) - 13)*log(-4
0*(sqrt(341)*200^(1/4)*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(86551*x + 35845) -
122396*x - 50706)*sqrt((13*sqrt(2) - 20)/(260*sqrt(2) - 369)) + 3464300*x^2 + 22
0*sqrt(2)*(28280*x^2 - 9997*sqrt(2)*(2*x^2 - x + 3) - 14140*x + 42420) - 49985*s
qrt(2)*(49*x^2 - 151*x + 200) - 10675700*x + 14140000)/(9997*sqrt(2)*x^2 - 14140
*x^2)) - 200^(1/4)*sqrt(31)*(10*sqrt(2) - 13)*log(40*(sqrt(341)*200^(1/4)*sqrt(5
)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(86551*x + 35845) - 122396*x - 50706)*sqrt((13*sq
rt(2) - 20)/(260*sqrt(2) - 369)) - 3464300*x^2 - 220*sqrt(2)*(28280*x^2 - 9997*s
qrt(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)) + 124*200^(1/4)*arctan
(155*(sqrt(341)*sqrt(5)*(10*sqrt(2)*(x - 6) - 13*x + 78)*sqrt((13*sqrt(2) - 20)/
(260*sqrt(2) - 369)) + 11*200^(1/4)*sqrt(2*x^2 - x + 3)*(7*sqrt(2) - 6))/(2*sqrt
(341)*sqrt(31)*sqrt(10)*sqrt(5)*(10*sqrt(2)*x - 13*x)*sqrt(-(sqrt(341)*200^(1/4)
*sqrt(5)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(86551*x + 35845) - 122396*x - 50706)*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
+ 200) - 10675700*x + 14140000)/(9997*sqrt(2)*x^2 - 14140*x^2))*sqrt((13*sqrt(2
) - 20)/(260*sqrt(2) - 369)) + 5*sqrt(341)*sqrt(31)*sqrt(5)*(10*sqrt(2)*(19*x -
22) - 247*x + 286)*sqrt((13*sqrt(2) - 20)/(260*sqrt(2) - 369)) - 1705*200^(1/4)*
sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2) - 2))) + 124*200^(1/4)*arctan(-155*(sqrt(3
41)*sqrt(5)*(10*sqrt(2)*(x - 6) - 13*x + 78)*sqrt((13*sqrt(2) - 20)/(260*sqrt(2)
- 369)) - 11*200^(1/4)*sqrt(2*x^2 - x + 3)*(7*sqrt(2) - 6))/(2*sqrt(341)*sqrt(3
1)*sqrt(10)*sqrt(5)*(10*sqrt(2)*x - 13*x)*sqrt((sqrt(341)*200^(1/4)*sqrt(5)*sqrt
(2*x^2 - x + 3)*(sqrt(2)*(86551*x + 35845) - 122396*x - 50706)*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 + 200) + 106
75700*x - 14140000)/(9997*sqrt(2)*x^2 - 14140*x^2))*sqrt((13*sqrt(2) - 20)/(260*
sqrt(2) - 369)) + 5*sqrt(341)*sqrt(31)*sqrt(5)*(10*sqrt(2)*(19*x - 22) - 247*x +
286)*sqrt((13*sqrt(2) - 20)/(260*sqrt(2) - 369)) + 1705*200^(1/4)*sqrt(31)*sqrt
(2*x^2 - x + 3)*(sqrt(2) - 2))))/((10*sqrt(2) - 13)*sqrt((13*sqrt(2) - 20)/(260*
sqrt(2) - 369)))
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )}\, dx \]
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)
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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(1/((5*x^2 + 3*x + 2)*sqrt(2*x^2 - x + 3)),x, algorithm="giac")
[Out]
Exception raised: RuntimeError