3.2299 \(\int \frac{(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx\)

Optimal. Leaf size=266 \[ \frac{4}{15} (2 x+1)^{3/2}+\frac{16}{25} \sqrt{2 x+1}+\frac{1}{25} \sqrt{\frac{1}{310} \left (1225 \sqrt{35}-7162\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{25} \sqrt{\frac{1}{310} \left (1225 \sqrt{35}-7162\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{25} \sqrt{\frac{2}{155} \left (7162+1225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{25} \sqrt{\frac{2}{155} \left (7162+1225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

(16*Sqrt[1 + 2*x])/25 + (4*(1 + 2*x)^(3/2))/15 + (Sqrt[(2*(7162 + 1225*Sqrt[35])
)/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35
])]])/25 - (Sqrt[(2*(7162 + 1225*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])]
+ 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/25 + (Sqrt[(-7162 + 1225*Sqrt[35
])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/25
- (Sqrt[(-7162 + 1225*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt
[1 + 2*x] + 5*(1 + 2*x)])/25

_______________________________________________________________________________________

Rubi [A]  time = 1.29594, antiderivative size = 266, 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 \[ \frac{4}{15} (2 x+1)^{3/2}+\frac{16}{25} \sqrt{2 x+1}+\frac{1}{25} \sqrt{\frac{1}{310} \left (1225 \sqrt{35}-7162\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{25} \sqrt{\frac{1}{310} \left (1225 \sqrt{35}-7162\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{25} \sqrt{\frac{2}{155} \left (7162+1225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{25} \sqrt{\frac{2}{155} \left (7162+1225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(16*Sqrt[1 + 2*x])/25 + (4*(1 + 2*x)^(3/2))/15 + (Sqrt[(2*(7162 + 1225*Sqrt[35])
)/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35
])]])/25 - (Sqrt[(2*(7162 + 1225*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])]
+ 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/25 + (Sqrt[(-7162 + 1225*Sqrt[35
])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/25
- (Sqrt[(-7162 + 1225*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt
[1 + 2*x] + 5*(1 + 2*x)])/25

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 77.2329, size = 379, normalized size = 1.42 \[ \frac{4 \left (2 x + 1\right )^{\frac{3}{2}}}{15} + \frac{16 \sqrt{2 x + 1}}{25} + \frac{\sqrt{14} \left (- \frac{19 \sqrt{35}}{5} + 28\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{350 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{14} \left (- \frac{19 \sqrt{35}}{5} + 28\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{350 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{38 \sqrt{35}}{5} + 56\right )}{10} + \frac{56 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{175 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{38 \sqrt{35}}{5} + 56\right )}{10} + \frac{56 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{175 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(2*x + 1)**(3/2)/15 + 16*sqrt(2*x + 1)/25 + sqrt(14)*(-19*sqrt(35)/5 + 28)*log
(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(350*sqrt(2
+ sqrt(35))) - sqrt(14)*(-19*sqrt(35)/5 + 28)*log(2*x + sqrt(10)*sqrt(2 + sqrt(
35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(350*sqrt(2 + sqrt(35))) - sqrt(35)*(-sqr
t(10)*sqrt(2 + sqrt(35))*(-38*sqrt(35)/5 + 56)/10 + 56*sqrt(10)*sqrt(2 + sqrt(35
))/5)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(3
5)))/(175*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) - sqrt(35)*(-sqrt(10)*sqrt(2 +
sqrt(35))*(-38*sqrt(35)/5 + 56)/10 + 56*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqr
t(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(175*sqrt
(-2 + sqrt(35))*sqrt(2 + sqrt(35)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.537306, size = 137, normalized size = 0.52 \[ \frac{4}{75} \sqrt{2 x+1} (10 x+17)+\frac{2 i \left (178 \sqrt{31}+589 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{775 \sqrt{-10-5 i \sqrt{31}}}-\frac{2 i \left (178 \sqrt{31}-589 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{775 \sqrt{5 i \left (\sqrt{31}+2 i\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[1 + 2*x]*(17 + 10*x))/75 + (((2*I)/775)*(589*I + 178*Sqrt[31])*ArcTan[Sq
rt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/Sqrt[-10 - (5*I)*Sqrt[31]] - (((2*I)/775)*(
-589*I + 178*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[(5*I)*
(2*I + Sqrt[31])]

_______________________________________________________________________________________

Maple [B]  time = 0.047, size = 625, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/15*(1+2*x)^(3/2)+16/25*(1+2*x)^(1/2)+89/3875*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)
-27/1550*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)+1
0*x+5)*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+178/775/(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)-27/775/(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)-16/25/(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)-89/3875*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)+27/1550
*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)+178/775/(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)-27/775/(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)-16/25/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((1
0*(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. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{5}{2}}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.294033, size = 1192, normalized size = 4.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/22343250*4805^(3/4)*sqrt(31)*sqrt(2)*(4*4805^(1/4)*sqrt(31)*sqrt(2)*(6125*sqrt
(7)*(10*x + 17) - 7162*sqrt(5)*(10*x + 17))*sqrt(2*x + 1)*sqrt((7162*sqrt(7)*sqr
t(5) - 42875)/(17546900*sqrt(7)*sqrt(5) - 103816119)) - 3*sqrt(31)*7^(1/4)*(6125
*sqrt(7) - 7162*sqrt(5))*log(42532/25*sqrt(5)*(4805^(1/4)*7^(1/4)*sqrt(2)*(82306
25376258393076926858088337463298045381*sqrt(7)*sqrt(5) - 48693036390748949794311
793462698151573366360)*sqrt(2*x + 1)*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(175469
00*sqrt(7)*sqrt(5) - 103816119)) + 35*sqrt(5)*(326087381553402705926541359046633
13974500*sqrt(7)*sqrt(5)*(2*x + 1) - 385831793106395283574239901229957825433198*
x - 192915896553197641787119950614978912716599) + 35*sqrt(7)*(326087381553402705
92654135904663313974500*sqrt(7)*sqrt(5) - 19291589655319764178711995061497891271
6599))/(32608738155340270592654135904663313974500*sqrt(7)*sqrt(5) - 192915896553
197641787119950614978912716599)) + 3*sqrt(31)*7^(1/4)*(6125*sqrt(7) - 7162*sqrt(
5))*log(-42532/25*sqrt(5)*(4805^(1/4)*7^(1/4)*sqrt(2)*(8230625376258393076926858
088337463298045381*sqrt(7)*sqrt(5) - 4869303639074894979431179346269815157336636
0)*sqrt(2*x + 1)*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) -
103816119)) - 35*sqrt(5)*(32608738155340270592654135904663313974500*sqrt(7)*sqr
t(5)*(2*x + 1) - 385831793106395283574239901229957825433198*x - 1929158965531976
41787119950614978912716599) - 35*sqrt(7)*(32608738155340270592654135904663313974
500*sqrt(7)*sqrt(5) - 192915896553197641787119950614978912716599))/(326087381553
40270592654135904663313974500*sqrt(7)*sqrt(5) - 19291589655319764178711995061497
8912716599)) - 74028*7^(1/4)*sqrt(5)*arctan(1085*sqrt(31)*7^(1/4)*(135*sqrt(7) -
178*sqrt(5))/(4805^(1/4)*sqrt(217)*sqrt(31)*sqrt(2)*(6125*sqrt(7) - 7162*sqrt(5
))*sqrt(sqrt(5)*(4805^(1/4)*7^(1/4)*sqrt(2)*(82306253762583930769268580883374632
98045381*sqrt(7)*sqrt(5) - 48693036390748949794311793462698151573366360)*sqrt(2*
x + 1)*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) - 103816119
)) + 35*sqrt(5)*(32608738155340270592654135904663313974500*sqrt(7)*sqrt(5)*(2*x
+ 1) - 385831793106395283574239901229957825433198*x - 19291589655319764178711995
0614978912716599) + 35*sqrt(7)*(32608738155340270592654135904663313974500*sqrt(7
)*sqrt(5) - 192915896553197641787119950614978912716599))/(3260873815534027059265
4135904663313974500*sqrt(7)*sqrt(5) - 192915896553197641787119950614978912716599
))*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) - 103816119)) +
1085*4805^(1/4)*sqrt(2)*sqrt(2*x + 1)*(6125*sqrt(7) - 7162*sqrt(5))*sqrt((7162*
sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) - 103816119)) + 33635*7^(1/4)
*(20*sqrt(7) - 19*sqrt(5)))) - 74028*7^(1/4)*sqrt(5)*arctan(1085*sqrt(31)*7^(1/4
)*(135*sqrt(7) - 178*sqrt(5))/(4805^(1/4)*sqrt(217)*sqrt(31)*sqrt(2)*(6125*sqrt(
7) - 7162*sqrt(5))*sqrt(-sqrt(5)*(4805^(1/4)*7^(1/4)*sqrt(2)*(823062537625839307
6926858088337463298045381*sqrt(7)*sqrt(5) - 486930363907489497943117934626981515
73366360)*sqrt(2*x + 1)*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sq
rt(5) - 103816119)) - 35*sqrt(5)*(32608738155340270592654135904663313974500*sqrt
(7)*sqrt(5)*(2*x + 1) - 385831793106395283574239901229957825433198*x - 192915896
553197641787119950614978912716599) - 35*sqrt(7)*(3260873815534027059265413590466
3313974500*sqrt(7)*sqrt(5) - 192915896553197641787119950614978912716599))/(32608
738155340270592654135904663313974500*sqrt(7)*sqrt(5) - 1929158965531976417871199
50614978912716599))*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5
) - 103816119)) + 1085*4805^(1/4)*sqrt(2)*sqrt(2*x + 1)*(6125*sqrt(7) - 7162*sqr
t(5))*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) - 103816119)
) - 33635*7^(1/4)*(20*sqrt(7) - 19*sqrt(5)))))/((6125*sqrt(7) - 7162*sqrt(5))*sq
rt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) - 103816119)))

_______________________________________________________________________________________

Sympy [A]  time = 44.193, size = 97, normalized size = 0.36 \[ \frac{4 \left (2 x + 1\right )^{\frac{3}{2}}}{15} + \frac{16 \sqrt{2 x + 1}}{25} - \frac{76 \operatorname{RootSum}{\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log{\left (9920 t^{3} + 8 t + \sqrt{2 x + 1} \right )} \right )\right )}}{25} - \frac{112 \operatorname{RootSum}{\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log{\left (- \frac{27776 t^{3}}{5} + \frac{108 t}{5} + \sqrt{2 x + 1} \right )} \right )\right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(2*x + 1)**(3/2)/15 + 16*sqrt(2*x + 1)/25 - 76*RootSum(1230080*_t**4 + 1984*_t
**2 + 7, Lambda(_t, _t*log(9920*_t**3 + 8*_t + sqrt(2*x + 1))))/25 - 112*RootSum
(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/5 + s
qrt(2*x + 1))))/25

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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