3.2315 \(\int \frac{(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx\)
Optimal. Leaf size=300 \[ -\frac{\sqrt{2 x+1} (5-4 x)}{62 \left (5 x^2+3 x+2\right )^2}+\frac{\sqrt{2 x+1} (120 x+67)}{1922 \left (5 x^2+3 x+2\right )}-\frac{3 \sqrt{\frac{1}{434} \left (2705 \sqrt{35}-15082\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1922}+\frac{3 \sqrt{\frac{1}{434} \left (2705 \sqrt{35}-15082\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1922}-\frac{3}{961} \sqrt{\frac{1}{434} \left (15082+2705 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{3}{961} \sqrt{\frac{1}{434} \left (15082+2705 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
[Out]
-((5 - 4*x)*Sqrt[1 + 2*x])/(62*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(67 + 120*x
))/(1922*(2 + 3*x + 5*x^2)) - (3*Sqrt[(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[
10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/961 + (3*Sqrt[
(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])
/Sqrt[10*(-2 + Sqrt[35])]])/961 - (3*Sqrt[(-15082 + 2705*Sqrt[35])/434]*Log[Sqrt
[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/1922 + (3*Sqrt[(-15
082 + 2705*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] +
5*(1 + 2*x)])/1922
_______________________________________________________________________________________
Rubi [A] time = 1.32141, antiderivative size = 300, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364
\[ -\frac{\sqrt{2 x+1} (5-4 x)}{62 \left (5 x^2+3 x+2\right )^2}+\frac{\sqrt{2 x+1} (120 x+67)}{1922 \left (5 x^2+3 x+2\right )}-\frac{3 \sqrt{\frac{1}{434} \left (2705 \sqrt{35}-15082\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1922}+\frac{3 \sqrt{\frac{1}{434} \left (2705 \sqrt{35}-15082\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1922}-\frac{3}{961} \sqrt{\frac{1}{434} \left (15082+2705 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{3}{961} \sqrt{\frac{1}{434} \left (15082+2705 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*x)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
-((5 - 4*x)*Sqrt[1 + 2*x])/(62*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(67 + 120*x
))/(1922*(2 + 3*x + 5*x^2)) - (3*Sqrt[(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[
10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/961 + (3*Sqrt[
(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])
/Sqrt[10*(-2 + Sqrt[35])]])/961 - (3*Sqrt[(-15082 + 2705*Sqrt[35])/434]*Log[Sqrt
[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/1922 + (3*Sqrt[(-15
082 + 2705*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] +
5*(1 + 2*x)])/1922
_______________________________________________________________________________________
Rubi in Sympy [A] time = 79.3769, size = 400, normalized size = 1.33 \[ - \frac{\left (- 4 x + 5\right ) \sqrt{2 x + 1}}{62 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{\sqrt{2 x + 1} \left (840 x + 469\right )}{13454 \left (5 x^{2} + 3 x + 2\right )} - \frac{\sqrt{14} \left (- 84 \sqrt{35} + 819\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{188356 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- 84 \sqrt{35} + 819\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{188356 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 168 \sqrt{35} + 1638\right )}{10} + \frac{1638 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{94178 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 168 \sqrt{35} + 1638\right )}{10} + \frac{1638 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{94178 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+2*x)**(3/2)/(5*x**2+3*x+2)**3,x)
[Out]
-(-4*x + 5)*sqrt(2*x + 1)/(62*(5*x**2 + 3*x + 2)**2) + sqrt(2*x + 1)*(840*x + 46
9)/(13454*(5*x**2 + 3*x + 2)) - sqrt(14)*(-84*sqrt(35) + 819)*log(2*x - sqrt(10)
*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(188356*sqrt(2 + sqrt(35))
) + sqrt(14)*(-84*sqrt(35) + 819)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x
+ 1)/5 + 1 + sqrt(35)/5)/(188356*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt
(2 + sqrt(35))*(-168*sqrt(35) + 1638)/10 + 1638*sqrt(10)*sqrt(2 + sqrt(35))/5)*a
tan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(9
4178*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt
(35))*(-168*sqrt(35) + 1638)/10 + 1638*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(
10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(94178*sqrt
(-2 + sqrt(35))*sqrt(2 + sqrt(35)))
_______________________________________________________________________________________
Mathematica [C] time = 1.19848, size = 161, normalized size = 0.54 \[ \frac{\frac{31 \sqrt{2 x+1} \left (600 x^3+695 x^2+565 x-21\right )}{2 \left (5 x^2+3 x+2\right )^2}+\frac{3 \left (124-47 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}+\frac{3 \left (124+47 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}}{29791} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*x)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
((31*Sqrt[1 + 2*x]*(-21 + 565*x + 695*x^2 + 600*x^3))/(2*(2 + 3*x + 5*x^2)^2) +
(3*(124 - (47*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/Sqrt[(-
I/5)*(-2*I + Sqrt[31])] + (3*(124 + (47*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[
-2 + I*Sqrt[31]]])/Sqrt[(I/5)*(2*I + Sqrt[31])])/29791
_______________________________________________________________________________________
Maple [B] time = 0.054, size = 662, normalized size = 2.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x)
[Out]
1600*(3/7688*(1+2*x)^(7/2)-41/153760*(1+2*x)^(5/2)+4/4805*(1+2*x)^(3/2)-819/7688
00*(1+2*x)^(1/2))/(5*(1+2*x)^2-8*x+3)^2-141/119164*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)+327/417074*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)-705/59582/(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)+327/208537/(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)+234/6727/(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)+141/119164*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)-327/417074*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)-705/59582/(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))/(1
0*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)+327/208537/(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)+234/6727/(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. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)^(3/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="maxima")
[Out]
integrate((2*x + 1)^(3/2)/(5*x^2 + 3*x + 2)^3, x)
_______________________________________________________________________________________
Fricas [A] time = 0.274041, size = 1434, normalized size = 4.78 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)^(3/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="fricas")
[Out]
1/2168371896740*6727^(3/4)*sqrt(5410)*sqrt(31)*(6727^(1/4)*sqrt(5410)*sqrt(31)*(
15082*sqrt(7)*(600*x^3 + 695*x^2 + 565*x - 21) - 18935*sqrt(5)*(600*x^3 + 695*x^
2 + 565*x - 21))*sqrt(2*x + 1)*sqrt((15082*sqrt(7)*sqrt(5) - 94675)/(81593620*sq
rt(7)*sqrt(5) - 483562599)) - 357492*36585125^(1/4)*sqrt(7)*(25*x^4 + 30*x^3 + 2
9*x^2 + 12*x + 4)*arctan(117397*36585125^(1/4)*sqrt(31)*(235*sqrt(7) - 218*sqrt(
5))/(6727^(1/4)*sqrt(5410)*sqrt(16771/7)*sqrt(31)*(15082*sqrt(7) - 18935*sqrt(5)
)*sqrt(sqrt(7)*(36585125^(1/4)*6727^(1/4)*sqrt(5410)*(44613309328142145955797180
85172449276997287896*sqrt(7)*sqrt(5) - 26393589737339133130143791869529533761565
180961)*sqrt(2*x + 1)*sqrt((15082*sqrt(7)*sqrt(5) - 94675)/(81593620*sqrt(7)*sqr
t(5) - 483562599)) + 13525*sqrt(7)*(71194118033712629247693899941318820766528100
*sqrt(7)*sqrt(5)*(2*x + 1) - 842380164749711027459827993730507633551345998*x - 4
21190082374855513729913996865253816775672999) + 18935*sqrt(5)*(71194118033712629
247693899941318820766528100*sqrt(7)*sqrt(5) - 4211900823748555137299139968652538
16775672999))/(71194118033712629247693899941318820766528100*sqrt(7)*sqrt(5) - 42
1190082374855513729913996865253816775672999))*sqrt((15082*sqrt(7)*sqrt(5) - 9467
5)/(81593620*sqrt(7)*sqrt(5) - 483562599)) + 83855*6727^(1/4)*sqrt(5410)*sqrt(2*
x + 1)*(15082*sqrt(7) - 18935*sqrt(5))*sqrt((15082*sqrt(7)*sqrt(5) - 94675)/(815
93620*sqrt(7)*sqrt(5) - 483562599)) + 3639307*36585125^(1/4)*(20*sqrt(7) - 39*sq
rt(5)))) - 357492*36585125^(1/4)*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*a
rctan(117397*36585125^(1/4)*sqrt(31)*(235*sqrt(7) - 218*sqrt(5))/(6727^(1/4)*sqr
t(5410)*sqrt(16771/7)*sqrt(31)*(15082*sqrt(7) - 18935*sqrt(5))*sqrt(-sqrt(7)*(36
585125^(1/4)*6727^(1/4)*sqrt(5410)*(44613309328142145955797180851724492769972878
96*sqrt(7)*sqrt(5) - 26393589737339133130143791869529533761565180961)*sqrt(2*x +
1)*sqrt((15082*sqrt(7)*sqrt(5) - 94675)/(81593620*sqrt(7)*sqrt(5) - 483562599))
- 13525*sqrt(7)*(71194118033712629247693899941318820766528100*sqrt(7)*sqrt(5)*(
2*x + 1) - 842380164749711027459827993730507633551345998*x - 4211900823748555137
29913996865253816775672999) - 18935*sqrt(5)*(71194118033712629247693899941318820
766528100*sqrt(7)*sqrt(5) - 421190082374855513729913996865253816775672999))/(711
94118033712629247693899941318820766528100*sqrt(7)*sqrt(5) - 42119008237485551372
9913996865253816775672999))*sqrt((15082*sqrt(7)*sqrt(5) - 94675)/(81593620*sqrt(
7)*sqrt(5) - 483562599)) + 83855*6727^(1/4)*sqrt(5410)*sqrt(2*x + 1)*(15082*sqrt
(7) - 18935*sqrt(5))*sqrt((15082*sqrt(7)*sqrt(5) - 94675)/(81593620*sqrt(7)*sqrt
(5) - 483562599)) - 3639307*36585125^(1/4)*(20*sqrt(7) - 39*sqrt(5)))) + 3*36585
125^(1/4)*sqrt(31)*(15082*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 18935*
sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*log(3773475/7*sqrt(7)*(36585125^(
1/4)*6727^(1/4)*sqrt(5410)*(4461330932814214595579718085172449276997287896*sqrt(
7)*sqrt(5) - 26393589737339133130143791869529533761565180961)*sqrt(2*x + 1)*sqrt
((15082*sqrt(7)*sqrt(5) - 94675)/(81593620*sqrt(7)*sqrt(5) - 483562599)) + 13525
*sqrt(7)*(71194118033712629247693899941318820766528100*sqrt(7)*sqrt(5)*(2*x + 1)
- 842380164749711027459827993730507633551345998*x - 421190082374855513729913996
865253816775672999) + 18935*sqrt(5)*(7119411803371262924769389994131882076652810
0*sqrt(7)*sqrt(5) - 421190082374855513729913996865253816775672999))/(71194118033
712629247693899941318820766528100*sqrt(7)*sqrt(5) - 4211900823748555137299139968
65253816775672999)) - 3*36585125^(1/4)*sqrt(31)*(15082*sqrt(7)*(25*x^4 + 30*x^3
+ 29*x^2 + 12*x + 4) - 18935*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*log(
-3773475/7*sqrt(7)*(36585125^(1/4)*6727^(1/4)*sqrt(5410)*(4461330932814214595579
718085172449276997287896*sqrt(7)*sqrt(5) - 2639358973733913313014379186952953376
1565180961)*sqrt(2*x + 1)*sqrt((15082*sqrt(7)*sqrt(5) - 94675)/(81593620*sqrt(7)
*sqrt(5) - 483562599)) - 13525*sqrt(7)*(7119411803371262924769389994131882076652
8100*sqrt(7)*sqrt(5)*(2*x + 1) - 842380164749711027459827993730507633551345998*x
- 421190082374855513729913996865253816775672999) - 18935*sqrt(5)*(7119411803371
2629247693899941318820766528100*sqrt(7)*sqrt(5) - 421190082374855513729913996865
253816775672999))/(71194118033712629247693899941318820766528100*sqrt(7)*sqrt(5)
- 421190082374855513729913996865253816775672999)))/((15082*sqrt(7)*(25*x^4 + 30*
x^3 + 29*x^2 + 12*x + 4) - 18935*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*
sqrt((15082*sqrt(7)*sqrt(5) - 94675)/(81593620*sqrt(7)*sqrt(5) - 483562599)))
_______________________________________________________________________________________
Sympy [A] time = 140.295, size = 490, normalized size = 1.63 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+2*x)**(3/2)/(5*x**2+3*x+2)**3,x)
[Out]
229120*(2*x + 1)**(7/2)/(-24109568*x + 5381600*(2*x + 1)**4 - 8610560*(2*x + 1)*
*3 + 18512704*(2*x + 1)**2 - 1506848) - 1774080*(2*x + 1)**(7/2)/(-168766976*x +
37671200*(2*x + 1)**4 - 60273920*(2*x + 1)**3 + 129588928*(2*x + 1)**2 - 105479
36) - 259072*(2*x + 1)**(5/2)/(-24109568*x + 5381600*(2*x + 1)**4 - 8610560*(2*x
+ 1)**3 + 18512704*(2*x + 1)**2 - 1506848) - 940352*(2*x + 1)**(5/2)/(-16876697
6*x + 37671200*(2*x + 1)**4 - 60273920*(2*x + 1)**3 + 129588928*(2*x + 1)**2 - 1
0547936) + 3017984*(2*x + 1)**(3/2)/(5*(-24109568*x + 5381600*(2*x + 1)**4 - 861
0560*(2*x + 1)**3 + 18512704*(2*x + 1)**2 - 1506848)) - 6868736*(2*x + 1)**(3/2)
/(5*(-168766976*x + 37671200*(2*x + 1)**4 - 60273920*(2*x + 1)**3 + 129588928*(2
*x + 1)**2 - 10547936)) + 128*(2*x + 1)**(3/2)/(-6944*x + 4340*(2*x + 1)**2 + 26
04) - 974848*sqrt(2*x + 1)/(5*(-24109568*x + 5381600*(2*x + 1)**4 - 8610560*(2*x
+ 1)**3 + 18512704*(2*x + 1)**2 - 1506848)) - 5403328*sqrt(2*x + 1)/(-168766976
*x + 37671200*(2*x + 1)**4 - 60273920*(2*x + 1)**3 + 129588928*(2*x + 1)**2 - 10
547936) + 1728*sqrt(2*x + 1)/(5*(-6944*x + 4340*(2*x + 1)**2 + 2604)) + 256*Root
Sum(75465931487403231630327808*_t**4 + 9053854476152406016*_t**2 + 333142578125,
Lambda(_t, _t*log(21632117045402271744*_t**3/158378125 + 10865340674816*_t/1108
646875 + sqrt(2*x + 1))))/5 - 448*RootSum(3697830642882758349886062592*_t**4 + 2
111968303753265086464*_t**2 + 705698730253125, Lambda(_t, _t*log(-34594382834112
09322496*_t**3/1377792122625 + 251494140770688*_t/357205365125 + sqrt(2*x + 1)))
)/5 + 64*RootSum(19950060344639488*_t**4 + 498437272576*_t**2 + 10878125, Lambda
(_t, _t*log(-11049511452672*_t**3/2205125 + 307918256*_t/2205125 + sqrt(2*x + 1)
)))/5
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)^(3/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="giac")
[Out]
integrate((2*x + 1)^(3/2)/(5*x^2 + 3*x + 2)^3, x)