3.2304 \(\int \frac{1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx\)

Optimal. Leaf size=266 \[ -\frac{16}{49 \sqrt{2 x+1}}-\frac{4}{21 (2 x+1)^{3/2}}-\frac{1}{49} \sqrt{\frac{1}{434} \left (1225 \sqrt{35}-7162\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{49} \sqrt{\frac{1}{434} \left (1225 \sqrt{35}-7162\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{49} \sqrt{\frac{2}{217} \left (7162+1225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{49} \sqrt{\frac{2}{217} \left (7162+1225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

-4/(21*(1 + 2*x)^(3/2)) - 16/(49*Sqrt[1 + 2*x]) + (Sqrt[(2*(7162 + 1225*Sqrt[35]
))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[3
5])]])/49 - (Sqrt[(2*(7162 + 1225*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])
] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/49 - (Sqrt[(-7162 + 1225*Sqrt[3
5])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/49
+ (Sqrt[(-7162 + 1225*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqr
t[1 + 2*x] + 5*(1 + 2*x)])/49

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Rubi [A]  time = 1.17813, 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 \[ -\frac{16}{49 \sqrt{2 x+1}}-\frac{4}{21 (2 x+1)^{3/2}}-\frac{1}{49} \sqrt{\frac{1}{434} \left (1225 \sqrt{35}-7162\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{49} \sqrt{\frac{1}{434} \left (1225 \sqrt{35}-7162\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{49} \sqrt{\frac{2}{217} \left (7162+1225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{49} \sqrt{\frac{2}{217} \left (7162+1225 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-4/(21*(1 + 2*x)^(3/2)) - 16/(49*Sqrt[1 + 2*x]) + (Sqrt[(2*(7162 + 1225*Sqrt[35]
))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[3
5])]])/49 - (Sqrt[(2*(7162 + 1225*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])
] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/49 - (Sqrt[(-7162 + 1225*Sqrt[3
5])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/49
+ (Sqrt[(-7162 + 1225*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqr
t[1 + 2*x] + 5*(1 + 2*x)])/49

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Rubi in Sympy [A]  time = 77.9081, size = 372, normalized size = 1.4 \[ \frac{\sqrt{14} \left (- 4 \sqrt{35} + 19\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{686 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{14} \left (- 4 \sqrt{35} + 19\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{686 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 8 \sqrt{35} + 38\right )}{10} + \frac{38 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{343 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 8 \sqrt{35} + 38\right )}{10} + \frac{38 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{343 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} - \frac{16}{49 \sqrt{2 x + 1}} - \frac{4}{21 \left (2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(14)*(-4*sqrt(35) + 19)*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/
5 + 1 + sqrt(35)/5)/(686*sqrt(2 + sqrt(35))) - sqrt(14)*(-4*sqrt(35) + 19)*log(2
*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(686*sqrt(2 +
sqrt(35))) - sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(-8*sqrt(35) + 38)/10 + 38*
sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(
35))/10)/sqrt(-2 + sqrt(35)))/(343*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) - sqr
t(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(-8*sqrt(35) + 38)/10 + 38*sqrt(10)*sqrt(2 +
sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2
+ sqrt(35)))/(343*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) - 16/(49*sqrt(2*x + 1)
) - 4/(21*(2*x + 1)**(3/2))

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Mathematica [C]  time = 1.02346, size = 139, normalized size = 0.52 \[ \frac{2 \left (-\frac{62 (24 x+19)}{(2 x+1)^{3/2}}+\frac{3 i \left (27 \sqrt{31}+124 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}-\frac{3 \left (124+27 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}\right )}{4557} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-62*(19 + 24*x))/(1 + 2*x)^(3/2) + ((3*I)*(124*I + 27*Sqrt[31])*ArcTan[Sqrt
[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/Sqrt[(-I/5)*(-2*I + Sqrt[31])] - (3*(124 + (2
7*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[(I/5)*(2*I + S
qrt[31])]))/4557

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Maple [B]  time = 0.05, 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/(1+2*x)^(5/2)/(5*x^2+3*x+2),x)

[Out]

-4/21/(1+2*x)^(3/2)-16/49/(1+2*x)^(1/2)+27/3038*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
)-89/10633*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)+135/1519/(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)-178/10633/(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)-76/343/(10*5^(1/2)*7^(1/2)-2
0)^(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)-27/3038*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)+
89/10633*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)+135/1519/(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)-178/10633/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arct
an((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)-76/343/(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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.283635, size = 1241, normalized size = 4.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/61309878*6727^(3/4)*sqrt(31)*sqrt(2)*(74028*sqrt(7)*5^(1/4)*(2*x + 1)^(3/2)*a
rctan(217*sqrt(31)*(135*sqrt(7)*5^(1/4) - 178*5^(3/4))/(6727^(1/4)*sqrt(31)*sqrt
(155/7)*sqrt(2)*(7162*sqrt(7) - 8575*sqrt(5))*sqrt(sqrt(7)*(6727^(1/4)*sqrt(2)*(
1391229611164255708408908384648518616381896*sqrt(7)*5^(3/4) - 823062537625839307
6926858088337463298045381*5^(1/4))*sqrt(2*x + 1)*sqrt((7162*sqrt(7)*sqrt(5) - 42
875)/(17546900*sqrt(7)*sqrt(5) - 103816119)) + 5*sqrt(7)*(3260873815534027059265
4135904663313974500*sqrt(7)*sqrt(5)*(2*x + 1) - 38583179310639528357423990122995
7825433198*x - 192915896553197641787119950614978912716599) + 1141305835436909470
742894756663215989107500*sqrt(7) - 1350411275872383492509839654304852389016193*s
qrt(5))/(32608738155340270592654135904663313974500*sqrt(7)*sqrt(5) - 19291589655
3197641787119950614978912716599))*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*
sqrt(7)*sqrt(5) - 103816119)) + 155*6727^(1/4)*sqrt(2)*sqrt(2*x + 1)*(7162*sqrt(
7) - 8575*sqrt(5))*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5)
- 103816119)) + 134540*sqrt(7)*5^(1/4) - 127813*5^(3/4))) + 74028*sqrt(7)*5^(1/
4)*(2*x + 1)^(3/2)*arctan(217*sqrt(31)*(135*sqrt(7)*5^(1/4) - 178*5^(3/4))/(6727
^(1/4)*sqrt(31)*sqrt(155/7)*sqrt(2)*(7162*sqrt(7) - 8575*sqrt(5))*sqrt(-sqrt(7)*
(6727^(1/4)*sqrt(2)*(1391229611164255708408908384648518616381896*sqrt(7)*5^(3/4)
- 8230625376258393076926858088337463298045381*5^(1/4))*sqrt(2*x + 1)*sqrt((7162
*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) - 103816119)) - 5*sqrt(7)*(3
2608738155340270592654135904663313974500*sqrt(7)*sqrt(5)*(2*x + 1) - 38583179310
6395283574239901229957825433198*x - 192915896553197641787119950614978912716599)
- 1141305835436909470742894756663215989107500*sqrt(7) + 135041127587238349250983
9654304852389016193*sqrt(5))/(32608738155340270592654135904663313974500*sqrt(7)*
sqrt(5) - 192915896553197641787119950614978912716599))*sqrt((7162*sqrt(7)*sqrt(5
) - 42875)/(17546900*sqrt(7)*sqrt(5) - 103816119)) + 155*6727^(1/4)*sqrt(2)*sqrt
(2*x + 1)*(7162*sqrt(7) - 8575*sqrt(5))*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(175
46900*sqrt(7)*sqrt(5) - 103816119)) - 134540*sqrt(7)*5^(1/4) + 127813*5^(3/4)))
+ 3*sqrt(31)*(7162*sqrt(7)*5^(1/4)*(2*x + 1) - 8575*5^(3/4)*(2*x + 1))*sqrt(2*x
+ 1)*log(387500/7*sqrt(7)*(6727^(1/4)*sqrt(2)*(139122961116425570840890838464851
8616381896*sqrt(7)*5^(3/4) - 8230625376258393076926858088337463298045381*5^(1/4)
)*sqrt(2*x + 1)*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) -
103816119)) + 5*sqrt(7)*(32608738155340270592654135904663313974500*sqrt(7)*sqrt(
5)*(2*x + 1) - 385831793106395283574239901229957825433198*x - 192915896553197641
787119950614978912716599) + 1141305835436909470742894756663215989107500*sqrt(7)
- 1350411275872383492509839654304852389016193*sqrt(5))/(326087381553402705926541
35904663313974500*sqrt(7)*sqrt(5) - 192915896553197641787119950614978912716599))
- 3*sqrt(31)*(7162*sqrt(7)*5^(1/4)*(2*x + 1) - 8575*5^(3/4)*(2*x + 1))*sqrt(2*x
+ 1)*log(-387500/7*sqrt(7)*(6727^(1/4)*sqrt(2)*(1391229611164255708408908384648
518616381896*sqrt(7)*5^(3/4) - 8230625376258393076926858088337463298045381*5^(1/
4))*sqrt(2*x + 1)*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5)
- 103816119)) - 5*sqrt(7)*(32608738155340270592654135904663313974500*sqrt(7)*sqr
t(5)*(2*x + 1) - 385831793106395283574239901229957825433198*x - 1929158965531976
41787119950614978912716599) - 1141305835436909470742894756663215989107500*sqrt(7
) + 1350411275872383492509839654304852389016193*sqrt(5))/(3260873815534027059265
4135904663313974500*sqrt(7)*sqrt(5) - 192915896553197641787119950614978912716599
)) + 4*6727^(1/4)*sqrt(31)*sqrt(2)*(7162*sqrt(7)*(24*x + 19) - 8575*sqrt(5)*(24*
x + 19))*sqrt((7162*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) - 1038161
19)))/((7162*sqrt(7)*(2*x + 1) - 8575*sqrt(5)*(2*x + 1))*sqrt(2*x + 1)*sqrt((716
2*sqrt(7)*sqrt(5) - 42875)/(17546900*sqrt(7)*sqrt(5) - 103816119)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((2*x + 1)**(5/2)*(5*x**2 + 3*x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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