3.2305 \(\int \frac{(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx\)

Optimal. Leaf size=296 \[ -\frac{(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac{8}{155} (2 x+1)^{3/2}+\frac{604}{775} \sqrt{2 x+1}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

(604*Sqrt[1 + 2*x])/775 - (8*(1 + 2*x)^(3/2))/155 - ((5 - 4*x)*(1 + 2*x)^(5/2))/
(31*(2 + 3*x + 5*x^2)) + (Sqrt[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqr
t[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/775 - (Sqrt[
(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[
1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/775 + (Sqrt[(5682718 + 968975*Sqrt[35])/310
]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775 - (Sq
rt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/775

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Rubi [A]  time = 1.35451, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac{8}{155} (2 x+1)^{3/2}+\frac{604}{775} \sqrt{2 x+1}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(604*Sqrt[1 + 2*x])/775 - (8*(1 + 2*x)^(3/2))/155 - ((5 - 4*x)*(1 + 2*x)^(5/2))/
(31*(2 + 3*x + 5*x^2)) + (Sqrt[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqr
t[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/775 - (Sqrt[
(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[
1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/775 + (Sqrt[(5682718 + 968975*Sqrt[35])/310
]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775 - (Sq
rt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/775

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Rubi in Sympy [A]  time = 86.8752, size = 403, normalized size = 1.36 \[ - \frac{\left (- 4 x + 5\right ) \left (2 x + 1\right )^{\frac{5}{2}}}{31 \left (5 x^{2} + 3 x + 2\right )} - \frac{8 \left (2 x + 1\right )^{\frac{3}{2}}}{155} + \frac{604 \sqrt{2 x + 1}}{775} + \frac{\sqrt{14} \left (\frac{814 \sqrt{35}}{5} + 1057\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{10850 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{14} \left (\frac{814 \sqrt{35}}{5} + 1057\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{10850 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (\frac{1628 \sqrt{35}}{5} + 2114\right )}{10} + \frac{2114 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{5425 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (\frac{1628 \sqrt{35}}{5} + 2114\right )}{10} + \frac{2114 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{5425 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x)**(7/2)/(5*x**2+3*x+2)**2,x)

[Out]

-(-4*x + 5)*(2*x + 1)**(5/2)/(31*(5*x**2 + 3*x + 2)) - 8*(2*x + 1)**(3/2)/155 +
604*sqrt(2*x + 1)/775 + sqrt(14)*(814*sqrt(35)/5 + 1057)*log(2*x - sqrt(10)*sqrt
(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(10850*sqrt(2 + sqrt(35))) - sq
rt(14)*(814*sqrt(35)/5 + 1057)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x +
1)/5 + 1 + sqrt(35)/5)/(10850*sqrt(2 + sqrt(35))) - sqrt(35)*(-sqrt(10)*sqrt(2 +
sqrt(35))*(1628*sqrt(35)/5 + 2114)/10 + 2114*sqrt(10)*sqrt(2 + sqrt(35))/5)*ata
n(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(542
5*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) - sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35
))*(1628*sqrt(35)/5 + 2114)/10 + 2114*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(1
0)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(5425*sqrt(-
2 + sqrt(35))*sqrt(2 + sqrt(35)))

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Mathematica [C]  time = 0.917613, size = 150, normalized size = 0.51 \[ \frac{\sqrt{2 x+1} \left (2480 x^2+1132 x+1003\right )}{775 \left (5 x^2+3 x+2\right )}+\frac{2 \left (25234+3657 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{24025 \sqrt{-10-5 i \sqrt{31}}}+\frac{2 \left (25234-3657 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{24025 \sqrt{5 i \left (\sqrt{31}+2 i\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(Sqrt[1 + 2*x]*(1003 + 1132*x + 2480*x^2))/(775*(2 + 3*x + 5*x^2)) + (2*(25234 +
(3657*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/(24025*Sqrt[-1
0 - (5*I)*Sqrt[31]]) + (2*(25234 - (3657*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt
[-2 + I*Sqrt[31]]])/(24025*Sqrt[(5*I)*(2*I + Sqrt[31])])

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Maple [B]  time = 0.136, size = 494, normalized size = 1.7 \[{\frac{16}{25}\sqrt{1+2\,x}}+{\frac{16}{25} \left ( -{\frac{89}{310} \left ( 1+2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{189}{620}\sqrt{1+2\,x}} \right ) \left ( \left ( 1+2\,x \right ) ^{2}-{\frac{8\,x}{5}}+{\frac{3}{5}} \right ) ^{-1}}+{\frac{256\,\sqrt{20+10\,\sqrt{35}}\sqrt{35}}{120125}\ln \left ( 5+10\,x+\sqrt{35}-\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{3657\,\sqrt{20+10\,\sqrt{35}}}{240250}\ln \left ( 5+10\,x+\sqrt{35}-\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{ \left ( 10240+5120\,\sqrt{35} \right ) \sqrt{35}}{120125\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }+{\frac{73140+36570\,\sqrt{35}}{120125\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{604\,\sqrt{35}}{775\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{256\,\sqrt{20+10\,\sqrt{35}}\sqrt{35}}{120125}\ln \left ( 5+10\,x+\sqrt{35}+\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }-{\frac{3657\,\sqrt{20+10\,\sqrt{35}}}{240250}\ln \left ( 5+10\,x+\sqrt{35}+\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{ \left ( 10240+5120\,\sqrt{35} \right ) \sqrt{35}}{120125\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }+{\frac{73140+36570\,\sqrt{35}}{120125\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{604\,\sqrt{35}}{775\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x)

[Out]

16/25*(1+2*x)^(1/2)+16/25*(-89/310*(1+2*x)^(3/2)+189/620*(1+2*x)^(1/2))/((1+2*x)
^2-8/5*x+3/5)+256/120125*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2)
)*(20+10*35^(1/2))^(1/2)*35^(1/2)+3657/240250*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(
20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)+512/120125/(-20+10*35^(1/2))^(1/2)
*arctan((10*(1+2*x)^(1/2)-(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+1
0*35^(1/2))*35^(1/2)+3657/120125/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2
)-(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))-604/775/(-20
+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)-(20+10*35^(1/2))^(1/2))/(-20+10*35^
(1/2))^(1/2))*35^(1/2)-256/120125*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/
2))^(1/2))*(20+10*35^(1/2))^(1/2)*35^(1/2)-3657/240250*ln(5+10*x+35^(1/2)+(1+2*x
)^(1/2)*(20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)+512/120125/(-20+10*35^(1/
2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/
2))*(20+10*35^(1/2))*35^(1/2)+3657/120125/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+
2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))-604
/775/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-
20+10*35^(1/2))^(1/2))*35^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{7}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2, x)

_______________________________________________________________________________________

Fricas [A]  time = 0.281502, size = 1311, normalized size = 4.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2,x, algorithm="fricas")

[Out]

1/182626277750*4805^(3/4)*sqrt(1582)*sqrt(31)*(4805^(1/4)*sqrt(1582)*sqrt(31)*(4
844875*sqrt(7)*(2480*x^2 + 1132*x + 1003) + 5682718*sqrt(5)*(2480*x^2 + 1132*x +
1003))*sqrt(2*x + 1)*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*
sqrt(7)*sqrt(5) + 65155223139399)) + 16794436*4379767^(1/4)*sqrt(5)*(5*x^2 + 3*x
+ 2)*arctan(122605*4379767^(1/4)*sqrt(31)*(2560*sqrt(7) + 3657*sqrt(5))/(4805^(
1/4)*sqrt(3503)*sqrt(1582)*sqrt(31)*(4844875*sqrt(7) + 5682718*sqrt(5))*sqrt(sqr
t(5)*(4379767^(1/4)*4805^(1/4)*sqrt(1582)*(3207316266100669246015564763351367186
8488934691072016679539595765063499686*sqrt(7)*sqrt(5) + 189747389198847177228133
091180719008457375909791895406491061056330399984715)*sqrt(2*x + 1)*sqrt((5682718
*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(5) + 65155223139399))
+ 27685*sqrt(5)*(317534406223123480749314905540279766234457246893704330102071544
3940500*sqrt(7)*sqrt(5)*(2*x + 1) + 37571177621903234650010699940319130769036533
643095191237457284137193998*x + 187855888109516173250053499701595653845182668215
47595618728642068596999) + 27685*sqrt(7)*(31753440622312348074931490554027976623
44572468937043301020715443940500*sqrt(7)*sqrt(5) + 18785588810951617325005349970
159565384518266821547595618728642068596999))/(3175344062231234807493149055402797
662344572468937043301020715443940500*sqrt(7)*sqrt(5) + 1878558881095161732500534
9970159565384518266821547595618728642068596999))*sqrt((5682718*sqrt(7)*sqrt(5) +
33914125)/(11012823348100*sqrt(7)*sqrt(5) + 65155223139399)) + 122605*4805^(1/4
)*sqrt(1582)*sqrt(2*x + 1)*(4844875*sqrt(7) + 5682718*sqrt(5))*sqrt((5682718*sqr
t(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(5) + 65155223139399)) + 38
00755*4379767^(1/4)*(755*sqrt(7) + 814*sqrt(5)))) + 16794436*4379767^(1/4)*sqrt(
5)*(5*x^2 + 3*x + 2)*arctan(122605*4379767^(1/4)*sqrt(31)*(2560*sqrt(7) + 3657*s
qrt(5))/(4805^(1/4)*sqrt(3503)*sqrt(1582)*sqrt(31)*(4844875*sqrt(7) + 5682718*sq
rt(5))*sqrt(-sqrt(5)*(4379767^(1/4)*4805^(1/4)*sqrt(1582)*(320731626610066924601
55647633513671868488934691072016679539595765063499686*sqrt(7)*sqrt(5) + 18974738
9198847177228133091180719008457375909791895406491061056330399984715)*sqrt(2*x +
1)*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(5) + 6
5155223139399)) - 27685*sqrt(5)*(31753440622312348074931490554027976623445724689
37043301020715443940500*sqrt(7)*sqrt(5)*(2*x + 1) + 3757117762190323465001069994
0319130769036533643095191237457284137193998*x + 18785588810951617325005349970159
565384518266821547595618728642068596999) - 27685*sqrt(7)*(3175344062231234807493
149055402797662344572468937043301020715443940500*sqrt(7)*sqrt(5) + 1878558881095
1617325005349970159565384518266821547595618728642068596999))/(317534406223123480
7493149055402797662344572468937043301020715443940500*sqrt(7)*sqrt(5) + 187855888
10951617325005349970159565384518266821547595618728642068596999))*sqrt((5682718*s
qrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(5) + 65155223139399)) +
122605*4805^(1/4)*sqrt(1582)*sqrt(2*x + 1)*(4844875*sqrt(7) + 5682718*sqrt(5))*s
qrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(5) + 65155
223139399)) - 3800755*4379767^(1/4)*(755*sqrt(7) + 814*sqrt(5)))) - 4379767^(1/4
)*sqrt(31)*(4844875*sqrt(7)*(5*x^2 + 3*x + 2) + 5682718*sqrt(5)*(5*x^2 + 3*x + 2
))*log(33642812/25*sqrt(5)*(4379767^(1/4)*4805^(1/4)*sqrt(1582)*(320731626610066
92460155647633513671868488934691072016679539595765063499686*sqrt(7)*sqrt(5) + 18
9747389198847177228133091180719008457375909791895406491061056330399984715)*sqrt(
2*x + 1)*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(
5) + 65155223139399)) + 27685*sqrt(5)*(31753440622312348074931490554027976623445
72468937043301020715443940500*sqrt(7)*sqrt(5)*(2*x + 1) + 3757117762190323465001
0699940319130769036533643095191237457284137193998*x + 18785588810951617325005349
970159565384518266821547595618728642068596999) + 27685*sqrt(7)*(3175344062231234
807493149055402797662344572468937043301020715443940500*sqrt(7)*sqrt(5) + 1878558
8810951617325005349970159565384518266821547595618728642068596999))/(317534406223
1234807493149055402797662344572468937043301020715443940500*sqrt(7)*sqrt(5) + 187
85588810951617325005349970159565384518266821547595618728642068596999)) + 4379767
^(1/4)*sqrt(31)*(4844875*sqrt(7)*(5*x^2 + 3*x + 2) + 5682718*sqrt(5)*(5*x^2 + 3*
x + 2))*log(-33642812/25*sqrt(5)*(4379767^(1/4)*4805^(1/4)*sqrt(1582)*(320731626
61006692460155647633513671868488934691072016679539595765063499686*sqrt(7)*sqrt(5
) + 189747389198847177228133091180719008457375909791895406491061056330399984715)
*sqrt(2*x + 1)*sqrt((5682718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)
*sqrt(5) + 65155223139399)) - 27685*sqrt(5)*(31753440622312348074931490554027976
62344572468937043301020715443940500*sqrt(7)*sqrt(5)*(2*x + 1) + 3757117762190323
4650010699940319130769036533643095191237457284137193998*x + 18785588810951617325
005349970159565384518266821547595618728642068596999) - 27685*sqrt(7)*(3175344062
231234807493149055402797662344572468937043301020715443940500*sqrt(7)*sqrt(5) + 1
8785588810951617325005349970159565384518266821547595618728642068596999))/(317534
4062231234807493149055402797662344572468937043301020715443940500*sqrt(7)*sqrt(5)
+ 18785588810951617325005349970159565384518266821547595618728642068596999)))/((
4844875*sqrt(7)*(5*x^2 + 3*x + 2) + 5682718*sqrt(5)*(5*x^2 + 3*x + 2))*sqrt((568
2718*sqrt(7)*sqrt(5) + 33914125)/(11012823348100*sqrt(7)*sqrt(5) + 6515522313939
9)))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*x)**(7/2)/(5*x**2+3*x+2)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{7}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2,x, algorithm="giac")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2, x)