3.2312 \(\int \frac{(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx\)

Optimal. Leaf size=313 \[ -\frac{(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac{(1143-1088 x) (2 x+1)^{3/2}}{9610 \left (5 x^2+3 x+2\right )}-\frac{1584 \sqrt{2 x+1}}{24025}+\frac{3 \sqrt{\frac{1}{310} \left (64681225 \sqrt{35}-250141922\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{48050}-\frac{3 \sqrt{\frac{1}{310} \left (64681225 \sqrt{35}-250141922\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{48050}-\frac{3 \sqrt{\frac{1}{310} \left (250141922+64681225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{24025}+\frac{3 \sqrt{\frac{1}{310} \left (250141922+64681225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{24025} \]

[Out]

(-1584*Sqrt[1 + 2*x])/24025 - ((5 - 4*x)*(1 + 2*x)^(7/2))/(62*(2 + 3*x + 5*x^2)^
2) - ((1143 - 1088*x)*(1 + 2*x)^(3/2))/(9610*(2 + 3*x + 5*x^2)) - (3*Sqrt[(25014
1922 + 64681225*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x
])/Sqrt[10*(-2 + Sqrt[35])]])/24025 + (3*Sqrt[(250141922 + 64681225*Sqrt[35])/31
0]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]
)/24025 + (3*Sqrt[(-250141922 + 64681225*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(
2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/48050 - (3*Sqrt[(-250141922 + 64681
225*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 +
2*x)])/48050

_______________________________________________________________________________________

Rubi [A]  time = 1.50656, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac{(1143-1088 x) (2 x+1)^{3/2}}{9610 \left (5 x^2+3 x+2\right )}-\frac{1584 \sqrt{2 x+1}}{24025}+\frac{3 \sqrt{\frac{1}{310} \left (64681225 \sqrt{35}-250141922\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{48050}-\frac{3 \sqrt{\frac{1}{310} \left (64681225 \sqrt{35}-250141922\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{48050}-\frac{3 \sqrt{\frac{1}{310} \left (250141922+64681225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{24025}+\frac{3 \sqrt{\frac{1}{310} \left (250141922+64681225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{24025} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x)^(9/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

(-1584*Sqrt[1 + 2*x])/24025 - ((5 - 4*x)*(1 + 2*x)^(7/2))/(62*(2 + 3*x + 5*x^2)^
2) - ((1143 - 1088*x)*(1 + 2*x)^(3/2))/(9610*(2 + 3*x + 5*x^2)) - (3*Sqrt[(25014
1922 + 64681225*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x
])/Sqrt[10*(-2 + Sqrt[35])]])/24025 + (3*Sqrt[(250141922 + 64681225*Sqrt[35])/31
0]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]
)/24025 + (3*Sqrt[(-250141922 + 64681225*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(
2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/48050 - (3*Sqrt[(-250141922 + 64681
225*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 +
2*x)])/48050

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 86.9251, size = 418, normalized size = 1.34 \[ - \frac{\left (- 1088 x + 1143\right ) \left (2 x + 1\right )^{\frac{3}{2}}}{9610 \left (5 x^{2} + 3 x + 2\right )} - \frac{\left (- 4 x + 5\right ) \left (2 x + 1\right )^{\frac{7}{2}}}{62 \left (5 x^{2} + 3 x + 2\right )^{2}} - \frac{1584 \sqrt{2 x + 1}}{24025} - \frac{\sqrt{14} \left (- \frac{22137 \sqrt{35}}{5} + 5544\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{672700 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- \frac{22137 \sqrt{35}}{5} + 5544\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{672700 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (\frac{11088 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5} - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{44274 \sqrt{35}}{5} + 11088\right )}{10}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{336350 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (\frac{11088 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5} - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{44274 \sqrt{35}}{5} + 11088\right )}{10}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{336350 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x)**(9/2)/(5*x**2+3*x+2)**3,x)

[Out]

-(-1088*x + 1143)*(2*x + 1)**(3/2)/(9610*(5*x**2 + 3*x + 2)) - (-4*x + 5)*(2*x +
1)**(7/2)/(62*(5*x**2 + 3*x + 2)**2) - 1584*sqrt(2*x + 1)/24025 - sqrt(14)*(-22
137*sqrt(35)/5 + 5544)*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1
+ sqrt(35)/5)/(672700*sqrt(2 + sqrt(35))) + sqrt(14)*(-22137*sqrt(35)/5 + 5544)
*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(672700
*sqrt(2 + sqrt(35))) + sqrt(35)*(11088*sqrt(10)*sqrt(2 + sqrt(35))/5 - sqrt(10)*
sqrt(2 + sqrt(35))*(-44274*sqrt(35)/5 + 11088)/10)*atan(sqrt(10)*(sqrt(2*x + 1)
- sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(336350*sqrt(-2 + sqrt(35))*sq
rt(2 + sqrt(35))) + sqrt(35)*(11088*sqrt(10)*sqrt(2 + sqrt(35))/5 - sqrt(10)*sqr
t(2 + sqrt(35))*(-44274*sqrt(35)/5 + 11088)/10)*atan(sqrt(10)*(sqrt(2*x + 1) + s
qrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(336350*sqrt(-2 + sqrt(35))*sqrt(
2 + sqrt(35)))

_______________________________________________________________________________________

Mathematica [C]  time = 1.11442, size = 161, normalized size = 0.51 \[ \frac{-\frac{155 \sqrt{2 x+1} \left (86150 x^3+144557 x^2+87291 x+27977\right )}{2 \left (5 x^2+3 x+2\right )^2}+\frac{3 \left (228749-23998 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}+\frac{3 \left (228749+23998 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}}{3723875} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x)^(9/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

((-155*Sqrt[1 + 2*x]*(27977 + 87291*x + 144557*x^2 + 86150*x^3))/(2*(2 + 3*x + 5
*x^2)^2) + (3*(228749 - (23998*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sq
rt[31]]])/Sqrt[(-I/5)*(-2*I + Sqrt[31])] + (3*(228749 + (23998*I)*Sqrt[31])*ArcT
an[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[(I/5)*(2*I + Sqrt[31])])/3723875

_______________________________________________________________________________________

Maple [B]  time = 0.056, size = 662, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x)

[Out]

1600*(-1723/768800*(1+2*x)^(7/2)-3833/4805000*(1+2*x)^(5/2)-14693/19220000*(1+2*
x)^(3/2)-4851/2402500*(1+2*x)^(1/2))/(5*(1+2*x)^2-8*x+3)^2-35997/7447750*ln(-(2*
5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)*(2*5^(1/2
)*7^(1/2)+4)^(1/2)*5^(1/2)+23721/2979100*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)
*(1+2*x)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-35997
/744775/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/
2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)+23721/
1489550/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/
2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2
)*7^(1/2)+1584/24025/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4
)^(1/2)*5^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*7^(1/2)
+35997/7447750*ln(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+
2*x)^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-23721/2979100*ln(5^(1/2)*7^(1/2)
+10*x+5+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))*7^(1/2)*(2*5^(1/2)*7^
(1/2)+4)^(1/2)-35997/744775/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/
2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2
)*7^(1/2)+4)+23721/1489550/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2
)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)
*7^(1/2)+4)*5^(1/2)*7^(1/2)+1584/24025/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*
(1+2*x)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)
)*5^(1/2)*7^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{9}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(9/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(9/2)/(5*x^2 + 3*x + 2)^3, x)

_______________________________________________________________________________________

Fricas [A]  time = 0.286905, size = 1434, normalized size = 4.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(9/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="fricas")

[Out]

-1/755823900975500*sqrt(105602)*4805^(3/4)*sqrt(31)*(sqrt(105602)*4805^(1/4)*sqr
t(31)*(323406125*sqrt(7)*(86150*x^3 + 144557*x^2 + 87291*x + 27977) - 250141922*
sqrt(5)*(86150*x^3 + 144557*x^2 + 87291*x + 27977))*sqrt(2*x + 1)*sqrt((25014192
2*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt(7)*sqrt(5) - 20899911150
4375959)) + 19347824532*19515619207^(1/4)*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12
*x + 4)*arctan(8184155*19515619207^(1/4)*sqrt(31)*(39535*sqrt(7) - 23998*sqrt(5)
)/(sqrt(233833)*sqrt(105602)*4805^(1/4)*sqrt(31)*(323406125*sqrt(7) - 250141922*
sqrt(5))*sqrt(sqrt(5)*(19515619207^(1/4)*sqrt(105602)*4805^(1/4)*(46027949230603
829019315248495939203970818237008559845609636245010906105219935562476310292821*s
qrt(7)*sqrt(5) - 272304962426556613682844277648619745181082072977995537167958516
243683729674761969752175995260)*sqrt(2*x + 1)*sqrt((250141922*sqrt(7)*sqrt(5) -
2263842875)/(32358971877628900*sqrt(7)*sqrt(5) - 208999111504375959)) + 1848035*
sqrt(5)*(87017964052957797337208059672811164557519700853295090161628785193525753
0573818652464500*sqrt(7)*sqrt(5)*(2*x + 1) - 10296107698092025625520320851652379
012251908564769041680092626442342953147427423262647598*x - 514805384904601281276
0160425826189506125954282384520840046313221171476573713711631323799) + 1848035*s
qrt(7)*(870179640529577973372080596728111645575197008532950901616287851935257530
573818652464500*sqrt(7)*sqrt(5) - 5148053849046012812760160425826189506125954282
384520840046313221171476573713711631323799))/(8701796405295779733720805967281116
45575197008532950901616287851935257530573818652464500*sqrt(7)*sqrt(5) - 51480538
49046012812760160425826189506125954282384520840046313221171476573713711631323799
))*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt(7)*sqrt
(5) - 208999111504375959)) + 8184155*sqrt(105602)*4805^(1/4)*sqrt(2*x + 1)*(3234
06125*sqrt(7) - 250141922*sqrt(5))*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)
/(32358971877628900*sqrt(7)*sqrt(5) - 208999111504375959)) + 253708805*195156192
07^(1/4)*(1320*sqrt(7) - 7379*sqrt(5)))) + 19347824532*19515619207^(1/4)*sqrt(5)
*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(8184155*19515619207^(1/4)*sqrt(31)
*(39535*sqrt(7) - 23998*sqrt(5))/(sqrt(233833)*sqrt(105602)*4805^(1/4)*sqrt(31)*
(323406125*sqrt(7) - 250141922*sqrt(5))*sqrt(-sqrt(5)*(19515619207^(1/4)*sqrt(10
5602)*4805^(1/4)*(46027949230603829019315248495939203970818237008559845609636245
010906105219935562476310292821*sqrt(7)*sqrt(5) - 2723049624265566136828442776486
19745181082072977995537167958516243683729674761969752175995260)*sqrt(2*x + 1)*sq
rt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt(7)*sqrt(5) -
208999111504375959)) - 1848035*sqrt(5)*(870179640529577973372080596728111645575
197008532950901616287851935257530573818652464500*sqrt(7)*sqrt(5)*(2*x + 1) - 102
96107698092025625520320851652379012251908564769041680092626442342953147427423262
647598*x - 514805384904601281276016042582618950612595428238452084004631322117147
6573713711631323799) - 1848035*sqrt(7)*(8701796405295779733720805967281116455751
97008532950901616287851935257530573818652464500*sqrt(7)*sqrt(5) - 51480538490460
12812760160425826189506125954282384520840046313221171476573713711631323799))/(87
01796405295779733720805967281116455751970085329509016162878519352575305738186524
64500*sqrt(7)*sqrt(5) - 51480538490460128127601604258261895061259542823845208400
46313221171476573713711631323799))*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)
/(32358971877628900*sqrt(7)*sqrt(5) - 208999111504375959)) + 8184155*sqrt(105602
)*4805^(1/4)*sqrt(2*x + 1)*(323406125*sqrt(7) - 250141922*sqrt(5))*sqrt((2501419
22*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt(7)*sqrt(5) - 2089991115
04375959)) - 253708805*19515619207^(1/4)*(1320*sqrt(7) - 7379*sqrt(5)))) - 3*195
15619207^(1/4)*sqrt(31)*(323406125*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
- 250141922*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*log(5052897297/25*sq
rt(5)*(19515619207^(1/4)*sqrt(105602)*4805^(1/4)*(460279492306038290193152484959
39203970818237008559845609636245010906105219935562476310292821*sqrt(7)*sqrt(5) -
2723049624265566136828442776486197451810820729779955371679585162436837296747619
69752175995260)*sqrt(2*x + 1)*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(323
58971877628900*sqrt(7)*sqrt(5) - 208999111504375959)) + 1848035*sqrt(5)*(8701796
40529577973372080596728111645575197008532950901616287851935257530573818652464500
*sqrt(7)*sqrt(5)*(2*x + 1) - 102961076980920256255203208516523790122519085647690
41680092626442342953147427423262647598*x - 5148053849046012812760160425826189506
125954282384520840046313221171476573713711631323799) + 1848035*sqrt(7)*(87017964
0529577973372080596728111645575197008532950901616287851935257530573818652464500*
sqrt(7)*sqrt(5) - 51480538490460128127601604258261895061259542823845208400463132
21171476573713711631323799))/(87017964052957797337208059672811164557519700853295
0901616287851935257530573818652464500*sqrt(7)*sqrt(5) - 514805384904601281276016
0425826189506125954282384520840046313221171476573713711631323799)) + 3*195156192
07^(1/4)*sqrt(31)*(323406125*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 250
141922*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*log(-5052897297/25*sqrt(5)
*(19515619207^(1/4)*sqrt(105602)*4805^(1/4)*(46027949230603829019315248495939203
970818237008559845609636245010906105219935562476310292821*sqrt(7)*sqrt(5) - 2723
04962426556613682844277648619745181082072977995537167958516243683729674761969752
175995260)*sqrt(2*x + 1)*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(32358971
877628900*sqrt(7)*sqrt(5) - 208999111504375959)) - 1848035*sqrt(5)*(870179640529
577973372080596728111645575197008532950901616287851935257530573818652464500*sqrt
(7)*sqrt(5)*(2*x + 1) - 10296107698092025625520320851652379012251908564769041680
092626442342953147427423262647598*x - 514805384904601281276016042582618950612595
4282384520840046313221171476573713711631323799) - 1848035*sqrt(7)*(8701796405295
77973372080596728111645575197008532950901616287851935257530573818652464500*sqrt(
7)*sqrt(5) - 5148053849046012812760160425826189506125954282384520840046313221171
476573713711631323799))/(8701796405295779733720805967281116455751970085329509016
16287851935257530573818652464500*sqrt(7)*sqrt(5) - 51480538490460128127601604258
26189506125954282384520840046313221171476573713711631323799)))/((323406125*sqrt(
7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 250141922*sqrt(5)*(25*x^4 + 30*x^3 +
29*x^2 + 12*x + 4))*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(3235897187762
8900*sqrt(7)*sqrt(5) - 208999111504375959)))

_______________________________________________________________________________________

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)**(9/2)/(5*x**2+3*x+2)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{9}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(9/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="giac")

[Out]

integrate((2*x + 1)^(9/2)/(5*x^2 + 3*x + 2)^3, x)