∖int 1________________________________ ∖left(3−x+2x2∖right)5∕2∖left(2+3x+5x2∖right)∖,dx

3.97 \(\int \frac{1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx\)

Optimal. Leaf size=199 \[ \frac{3603-658 x}{128018 \sqrt{2 x^2-x+3}}+\frac{13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}+\frac{1}{484} \sqrt{\frac{1}{682} \left (25000 \sqrt{2}-15457\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (25000 \sqrt{2}-15457\right )}} \left (\left (247+345 \sqrt{2}\right ) x-98 \sqrt{2}+443\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{484} \sqrt{\frac{1}{682} \left (15457+25000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (15457+25000 \sqrt{2}\right )}} \left (\left (247-345 \sqrt{2}\right ) x+98 \sqrt{2}+443\right )}{\sqrt{2 x^2-x+3}}\right ) \]

[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*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(-15457 + 25000*S

qrt[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

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Rubi [A] time = 0.942163, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{3603-658 x}{128018 \sqrt{2 x^2-x+3}}+\frac{13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}+\frac{1}{484} \sqrt{\frac{1}{682} \left (25000 \sqrt{2}-15457\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (25000 \sqrt{2}-15457\right )}} \left (\left (247+345 \sqrt{2}\right ) x-98 \sqrt{2}+443\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{484} \sqrt{\frac{1}{682} \left (15457+25000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (15457+25000 \sqrt{2}\right )}} \left (\left (247-345 \sqrt{2}\right ) x+98 \sqrt{2}+443\right )}{\sqrt{2 x^2-x+3}}\right ) \]

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*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(-15457 + 25000*S

qrt[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

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Rubi in Sympy [A] time = 105.101, size = 238, normalized size = 1.2 \[ \frac{- 119427 x + \frac{1307889}{2}}{23235267 \sqrt{2 x^{2} - x + 3}} + \frac{- 66 x + 143}{8349 \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}} - \frac{\sqrt{682} \left (- 57032019 \sqrt{2} + \frac{23235267}{2}\right ) \left (- \frac{103502553 \sqrt{2}}{2} + \frac{935747571}{4}\right ) \operatorname{atan}{\left (\frac{4 \sqrt{341} \left (x \left (\frac{521737359}{4} + \frac{728742465 \sqrt{2}}{4}\right ) - \frac{103502553 \sqrt{2}}{2} + \frac{935747571}{4}\right )}{65481207 \sqrt{-15457 + 25000 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{184098272703399549 \sqrt{-15457 + 25000 \sqrt{2}}} - \frac{\sqrt{682} \left (\frac{23235267}{2} + 57032019 \sqrt{2}\right ) \left (\frac{103502553 \sqrt{2}}{2} + \frac{935747571}{4}\right ) \operatorname{atanh}{\left (\frac{4 \sqrt{341} \left (x \left (- \frac{728742465 \sqrt{2}}{4} + \frac{521737359}{4}\right ) + \frac{103502553 \sqrt{2}}{2} + \frac{935747571}{4}\right )}{65481207 \sqrt{15457 + 25000 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{184098272703399549 \sqrt{15457 + 25000 \sqrt{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate(1/(2*x**2-x+3)**(5/2)/(5*x**2+3*x+2),x)

[Out]

(-119427*x + 1307889/2)/(23235267*sqrt(2*x**2 - x + 3)) + (-66*x + 143)/(8349*(2

*x**2 - x + 3)**(3/2)) - sqrt(682)*(-57032019*sqrt(2) + 23235267/2)*(-103502553*

sqrt(2)/2 + 935747571/4)*atan(4*sqrt(341)*(x*(521737359/4 + 728742465*sqrt(2)/4)

- 103502553*sqrt(2)/2 + 935747571/4)/(65481207*sqrt(-15457 + 25000*sqrt(2))*sqr

t(2*x**2 - x + 3)))/(184098272703399549*sqrt(-15457 + 25000*sqrt(2))) - sqrt(682

)*(23235267/2 + 57032019*sqrt(2))*(103502553*sqrt(2)/2 + 935747571/4)*atanh(4*sq

rt(341)*(x*(-728742465*sqrt(2)/4 + 521737359/4) + 103502553*sqrt(2)/2 + 93574757

1/4)/(65481207*sqrt(15457 + 25000*sqrt(2))*sqrt(2*x**2 - x + 3)))/(1840982727033

99549*sqrt(15457 + 25000*sqrt(2)))

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Mathematica [C] time = 6.4295, size = 1176, normalized size = 5.91 \[ \sqrt{2 x^2-x+3} \left (\frac{3603-658 x}{128018 \left (2 x^2-x+3\right )}+\frac{13-6 x}{759 \left (2 x^2-x+3\right )^2}\right )-\frac{5 \left (-69 i+13 \sqrt{31}\right ) \tan ^{-1}\left (\frac{526291 i \sqrt{31} x^4+1223508 x^4-110000 i \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3+375280 i \sqrt{31} x^3+187550 x^3+249000 i \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2+657545 i \sqrt{31} x^2+1185998 x^2+145000 i \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-72154 i \sqrt{31} x+1323762 x+126000 i \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}-31062 i \sqrt{31}-374418}{230516 \sqrt{31} x^4+2998917 i x^4-221650 \sqrt{31} x^3-2776640 i x^3+411246 \sqrt{31} x^2+6774415 i x^2-165726 \sqrt{31} x+2234202 i x+18414 \sqrt{31}+4112406 i}\right )}{484 \sqrt{682 \left (-13+i \sqrt{31}\right )}}+\frac{5 i \left (69 i+13 \sqrt{31}\right ) \tan ^{-1}\left (\frac{31 \left (7436 \sqrt{31} x^4-17707 i x^4-7150 \sqrt{31} x^3-106560 i x^3+13266 \sqrt{31} x^2-10465 i x^2-5346 \sqrt{31} x+24058 i x+594 \sqrt{31}-3626 i\right )}{526291 i \sqrt{31} x^4-1223508 x^4+20000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3+375280 i \sqrt{31} x^3-187550 x^3+7000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2+657545 i \sqrt{31} x^2-1185998 x^2+5000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-72154 i \sqrt{31} x-1323762 x-2000 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}-31062 i \sqrt{31}+374418}\right )}{484 \sqrt{682 \left (13+i \sqrt{31}\right )}}+\frac{5 \left (69 i+13 \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 )}{968 \sqrt{682 \left (13+i \sqrt{31}\right )}}-\frac{5 i \left (-69 i+13 \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 )}{968 \sqrt{682 \left (-13+i \sqrt{31}\right )}}+\frac{5 i \left (-69 i+13 \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 )}{968 \sqrt{682 \left (-13+i \sqrt{31}\right )}}-\frac{5 \left (69 i+13 \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 )}{968 \sqrt{682 \left (13+i \sqrt{31}\right )}} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)),x]

[Out]

Sqrt[3 - x + 2*x^2]*((13 - 6*x)/(759*(3 - x + 2*x^2)^2) + (3603 - 658*x)/(128018

*(3 - x + 2*x^2))) - (5*(-69*I + 13*Sqrt[31])*ArcTan[(-374418 - (31062*I)*Sqrt[3

1] + 1323762*x - (72154*I)*Sqrt[31]*x + 1185998*x^2 + (657545*I)*Sqrt[31]*x^2 +

187550*x^3 + (375280*I)*Sqrt[31]*x^3 + 1223508*x^4 + (526291*I)*Sqrt[31]*x^4 + (

126000*I)*Sqrt[22*(-13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2] + (145000*I)*Sqrt[22*(

-13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] + (249000*I)*Sqrt[22*(-13 + I*Sqrt[31])

]*x^2*Sqrt[3 - x + 2*x^2] - (110000*I)*Sqrt[22*(-13 + I*Sqrt[31])]*x^3*Sqrt[3 -

x + 2*x^2])/(4112406*I + 18414*Sqrt[31] + (2234202*I)*x - 165726*Sqrt[31]*x + (6

774415*I)*x^2 + 411246*Sqrt[31]*x^2 - (2776640*I)*x^3 - 221650*Sqrt[31]*x^3 + (2

998917*I)*x^4 + 230516*Sqrt[31]*x^4)])/(484*Sqrt[682*(-13 + I*Sqrt[31])]) + (((5

*I)/484)*(69*I + 13*Sqrt[31])*ArcTan[(31*(-3626*I + 594*Sqrt[31] + (24058*I)*x -

5346*Sqrt[31]*x - (10465*I)*x^2 + 13266*Sqrt[31]*x^2 - (106560*I)*x^3 - 7150*Sq

rt[31]*x^3 - (17707*I)*x^4 + 7436*Sqrt[31]*x^4))/(374418 - (31062*I)*Sqrt[31] -

1323762*x - (72154*I)*Sqrt[31]*x - 1185998*x^2 + (657545*I)*Sqrt[31]*x^2 - 18755

0*x^3 + (375280*I)*Sqrt[31]*x^3 - 1223508*x^4 + (526291*I)*Sqrt[31]*x^4 - (2000*

I)*Sqrt[682*(13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2] + (5000*I)*Sqrt[682*(13 + I*S

qrt[31])]*x*Sqrt[3 - x + 2*x^2] + (7000*I)*Sqrt[682*(13 + I*Sqrt[31])]*x^2*Sqrt[

3 - x + 2*x^2] + (20000*I)*Sqrt[682*(13 + I*Sqrt[31])]*x^3*Sqrt[3 - x + 2*x^2])]

)/Sqrt[682*(13 + I*Sqrt[31])] - (((5*I)/968)*(-69*I + 13*Sqrt[31])*Log[(-3*I + S

qrt[31] - (10*I)*x)^2*(3*I + Sqrt[31] + (10*I)*x)^2])/Sqrt[682*(-13 + I*Sqrt[31]

)] + (5*(69*I + 13*Sqrt[31])*Log[(-3*I + Sqrt[31] - (10*I)*x)^2*(3*I + Sqrt[31]

+ (10*I)*x)^2])/(968*Sqrt[682*(13 + I*Sqrt[31])]) + (((5*I)/968)*(-69*I + 13*Sqr

t[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[682*(-13

+ I*Sqrt[31])] - (5*(69*I + 13*Sqrt[31])*Log[(2 + 3*x + 5*x^2)*(-1858*I + 66*Sqr

t[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])])/(968*Sqrt[682*(13 + I*Sqrt[31])])

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Maple [B] time = 0.04, size = 751, normalized size = 3.8 \[ \text{result too large to display} \]

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*(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)*(10111*2^(1/2)*arctan(1/11692487*(-775687+549

362*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)+13910*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)-993674*a

rctanh(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)-42685698*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)-1/506*(4*x-1)/(2*x^2-x+3)^(3/2)-329/256036*(4*x-1)/(2*x^2-x+3)^(

1/2)+1/66/(2*x^2-x+3)^(3/2)+13/484/(2*x^2-x+3)^(1/2)

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((5*x^2 + 3*x + 2)*(2*x^2 - x + 3)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((5*x^2 + 3*x + 2)*(2*x^2 - x + 3)^(5/2)), x)

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Fricas [A] time = 0.359485, size = 1601, normalized size = 8.05 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((5*x^2 + 3*x + 2)*(2*x^2 - x + 3)^(5/2)),x, algorithm="fricas")

[Out]

-1/110752294271520*232562^(3/4)*sqrt(31)*sqrt(5)*(8*232562^(1/4)*sqrt(31)*sqrt(5

)*(197400000*x^3 - 1179600000*x^2 + 15457*sqrt(2)*(3948*x^3 - 23592*x^2 + 19767*

x - 39005) + 988350000*x - 1950250000)*sqrt(2*x^2 - x + 3)*sqrt((15457*sqrt(2) +

50000)/(772850000*sqrt(2) + 1488918849)) - 1123856268*sqrt(5)*sqrt(2)*(4*x^4 -

4*x^3 + 13*x^2 - 6*x + 9)*arctan(31*(232562^(1/4)*sqrt(5)*(15457*sqrt(2)*(x - 6)

+ 50000*x - 300000)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 148891884

9)) + 44*sqrt(5)*sqrt(2*x^2 - x + 3)*(98*sqrt(2) + 443))/(2*232562^(1/4)*sqrt(31

)*sqrt(5)*(15457*sqrt(2)*x + 50000*x)*sqrt(sqrt(2)*(2*232562^(1/4)*sqrt(2*x^2 -

x + 3)*(sqrt(2)*(37682974625135859*x - 88363116363919925) + 50680141738784066*x

- 126046090989055784)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 14889188

49)) + 17307457613600000*x^2 + sqrt(2)*(4818553540150000*x^2 + 61656718648993*sq

rt(2)*(49*x^2 - 151*x + 200) - 14849011929850000*x + 19667565470000000) + 542579

1241111384*sqrt(2)*(2*x^2 - x + 3) - 8653728806800000*x + 25961186420400000)/(61

656718648993*sqrt(2)*x^2 + 98337827350000*x^2))*sqrt((15457*sqrt(2) + 50000)/(77

2850000*sqrt(2) + 1488918849)) + 232562^(1/4)*sqrt(31)*sqrt(5)*(15457*sqrt(2)*(1

9*x - 22) + 950000*x - 1100000)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2)

+ 1488918849)) + 1364*sqrt(31)*sqrt(5)*sqrt(2*x^2 - x + 3)*(54*sqrt(2) + 11))) -

1123856268*sqrt(5)*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*arctan(-31*(23256

2^(1/4)*sqrt(5)*(15457*sqrt(2)*(x - 6) + 50000*x - 300000)*sqrt((15457*sqrt(2) +

50000)/(772850000*sqrt(2) + 1488918849)) - 44*sqrt(5)*sqrt(2*x^2 - x + 3)*(98*s

qrt(2) + 443))/(2*232562^(1/4)*sqrt(31)*sqrt(5)*(15457*sqrt(2)*x + 50000*x)*sqrt

(-sqrt(2)*(2*232562^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(37682974625135859*x - 88

363116363919925) + 50680141738784066*x - 126046090989055784)*sqrt((15457*sqrt(2)

+ 50000)/(772850000*sqrt(2) + 1488918849)) - 17307457613600000*x^2 - sqrt(2)*(4

818553540150000*x^2 + 61656718648993*sqrt(2)*(49*x^2 - 151*x + 200) - 1484901192

9850000*x + 19667565470000000) - 5425791241111384*sqrt(2)*(2*x^2 - x + 3) + 8653

728806800000*x - 25961186420400000)/(61656718648993*sqrt(2)*x^2 + 98337827350000

*x^2))*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 1488918849)) + 232562^(

1/4)*sqrt(31)*sqrt(5)*(15457*sqrt(2)*(19*x - 22) + 950000*x - 1100000)*sqrt((154

57*sqrt(2) + 50000)/(772850000*sqrt(2) + 1488918849)) - 1364*sqrt(31)*sqrt(5)*sq

rt(2*x^2 - x + 3)*(54*sqrt(2) + 11))) - 1587*sqrt(31)*sqrt(5)*(200000*x^4 - 2000

00*x^3 + 650000*x^2 + 15457*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9) - 300000*

x + 450000)*log(39062500*sqrt(2)*(2*232562^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(3

7682974625135859*x - 88363116363919925) + 50680141738784066*x - 1260460909890557

84)*sqrt((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 1488918849)) + 17307457613

600000*x^2 + sqrt(2)*(4818553540150000*x^2 + 61656718648993*sqrt(2)*(49*x^2 - 15

1*x + 200) - 14849011929850000*x + 19667565470000000) + 5425791241111384*sqrt(2)

*(2*x^2 - x + 3) - 8653728806800000*x + 25961186420400000)/(61656718648993*sqrt(

2)*x^2 + 98337827350000*x^2)) + 1587*sqrt(31)*sqrt(5)*(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 + 4500

00)*log(-39062500*sqrt(2)*(2*232562^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(37682974

625135859*x - 88363116363919925) + 50680141738784066*x - 126046090989055784)*sqr

t((15457*sqrt(2) + 50000)/(772850000*sqrt(2) + 1488918849)) - 17307457613600000*

x^2 - sqrt(2)*(4818553540150000*x^2 + 61656718648993*sqrt(2)*(49*x^2 - 151*x + 2

00) - 14849011929850000*x + 19667565470000000) - 5425791241111384*sqrt(2)*(2*x^2

- x + 3) + 8653728806800000*x - 25961186420400000)/(61656718648993*sqrt(2)*x^2

+ 98337827350000*x^2)))/((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((15457*sqrt(2) + 500

00)/(772850000*sqrt(2) + 1488918849)))

_______________________________________________________________________________________

Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )}\, dx \]

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)

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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)*(2*x^2 - x + 3)^(5/2)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError

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