3.279 \(\int \frac{(1+2 y) \sqrt{1-5 y-5 y^2}}{y (1+y) (2+y) \sqrt{1-y-y^2}} \, dy\)

Optimal. Leaf size=142 \[ -\frac{1}{4} \tanh ^{-1}\left (\frac{(1-3 y) \sqrt{-5 y^2-5 y+1}}{(1-5 y) \sqrt{-y^2-y+1}}\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{(3 y+4) \sqrt{-5 y^2-5 y+1}}{(5 y+6) \sqrt{-y^2-y+1}}\right )+\frac{9}{4} \tanh ^{-1}\left (\frac{(7 y+11) \sqrt{-5 y^2-5 y+1}}{3 (5 y+7) \sqrt{-y^2-y+1}}\right ) \]

[Out]

-ArcTanh[((1 - 3*y)*Sqrt[1 - 5*y - 5*y^2])/((1 - 5*y)*Sqrt[1 - y - y^2])]/4 - Ar
cTanh[((4 + 3*y)*Sqrt[1 - 5*y - 5*y^2])/((6 + 5*y)*Sqrt[1 - y - y^2])]/2 + (9*Ar
cTanh[((11 + 7*y)*Sqrt[1 - 5*y - 5*y^2])/(3*(7 + 5*y)*Sqrt[1 - y - y^2])])/4

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Rubi [F]  time = 5.56654, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{(1+2 y) \sqrt{1-5 y-5 y^2}}{y (1+y) (2+y) \sqrt{1-y-y^2}},y\right ) \]

Verification is Not applicable to the result.

[In]  Int[((1 + 2*y)*Sqrt[1 - 5*y - 5*y^2])/(y*(1 + y)*(2 + y)*Sqrt[1 - y - y^2]),y]

[Out]

Defer[Int][Sqrt[1 - 5*y - 5*y^2]/(y*Sqrt[1 - y - y^2]), y]/2 + Defer[Int][Sqrt[1
- 5*y - 5*y^2]/((1 + y)*Sqrt[1 - y - y^2]), y] - (3*Defer[Int][Sqrt[1 - 5*y - 5
*y^2]/((2 + y)*Sqrt[1 - y - y^2]), y])/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*y)*(-5*y**2-5*y+1)**(1/2)/y/(1+y)/(2+y)/(-y**2-y+1)**(1/2),y)

[Out]

Timed out

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Mathematica [C]  time = 2.3852, size = 630, normalized size = 4.44 \[ \frac{\left (-1-\frac{2}{\sqrt{5}}\right ) \left (2 y+\sqrt{5}+1\right )^2 \sqrt{\frac{10 y+3 \sqrt{5}+5}{10 y+5 \sqrt{5}+5}} \left (20 \left (\sqrt{5} \sqrt{\frac{-10 y+3 \sqrt{5}-5}{2 y+\sqrt{5}+1}} \sqrt{\frac{-2 y+\sqrt{5}-1}{2 y+\sqrt{5}+1}}-4 \sqrt{\frac{-10 y+3 \sqrt{5}-5}{2 y+\sqrt{5}+1}} \sqrt{\frac{-2 y+\sqrt{5}-1}{2 y+\sqrt{5}+1}}-2 \sqrt{5} \sqrt{-\frac{2 \sqrt{5} y+\sqrt{5}-5}{2 y+\sqrt{5}+1}} \sqrt{-\frac{2 \sqrt{5} y+\sqrt{5}-3}{2 y+\sqrt{5}+1}}+5 \sqrt{-\frac{2 \sqrt{5} y+\sqrt{5}-5}{2 y+\sqrt{5}+1}} \sqrt{-\frac{2 \sqrt{5} y+\sqrt{5}-3}{2 y+\sqrt{5}+1}}\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{\frac{10 y+3 \sqrt{5}+5}{2 y+\sqrt{5}+1}}}{\sqrt{15}}\right ),\frac{15}{16}\right )+\sqrt{\frac{-10 y+3 \sqrt{5}-5}{2 y+\sqrt{5}+1}} \sqrt{\frac{-2 y+\sqrt{5}-1}{2 y+\sqrt{5}+1}} \left (9 \sqrt{5} \Pi \left (\frac{5}{8}-\frac{\sqrt{5}}{8};\sin ^{-1}\left (\frac{2 \sqrt{\frac{10 y+3 \sqrt{5}+5}{2 y+\sqrt{5}+1}}}{\sqrt{15}}\right )|\frac{15}{16}\right )+\left (9 \sqrt{5}-20\right ) \Pi \left (-\frac{3}{8} \left (-5+\sqrt{5}\right );\sin ^{-1}\left (\frac{2 \sqrt{\frac{10 y+3 \sqrt{5}+5}{2 y+\sqrt{5}+1}}}{\sqrt{15}}\right )|\frac{15}{16}\right )+2 \sqrt{5} \Pi \left (\frac{3}{8} \left (5+\sqrt{5}\right );\sin ^{-1}\left (\frac{2 \sqrt{\frac{10 y+3 \sqrt{5}+5}{2 y+\sqrt{5}+1}}}{\sqrt{15}}\right )|\frac{15}{16}\right )\right )\right )}{16 \sqrt{-5 y^2-5 y+1} \sqrt{-y^2-y+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((1 + 2*y)*Sqrt[1 - 5*y - 5*y^2])/(y*(1 + y)*(2 + y)*Sqrt[1 - y - y^2]),y]

[Out]

((-1 - 2/Sqrt[5])*(1 + Sqrt[5] + 2*y)^2*Sqrt[(5 + 3*Sqrt[5] + 10*y)/(5 + 5*Sqrt[
5] + 10*y)]*(20*(-4*Sqrt[(-5 + 3*Sqrt[5] - 10*y)/(1 + Sqrt[5] + 2*y)]*Sqrt[(-1 +
Sqrt[5] - 2*y)/(1 + Sqrt[5] + 2*y)] + Sqrt[5]*Sqrt[(-5 + 3*Sqrt[5] - 10*y)/(1 +
Sqrt[5] + 2*y)]*Sqrt[(-1 + Sqrt[5] - 2*y)/(1 + Sqrt[5] + 2*y)] + 5*Sqrt[-((-5 +
Sqrt[5] + 2*Sqrt[5]*y)/(1 + Sqrt[5] + 2*y))]*Sqrt[-((-3 + Sqrt[5] + 2*Sqrt[5]*y
)/(1 + Sqrt[5] + 2*y))] - 2*Sqrt[5]*Sqrt[-((-5 + Sqrt[5] + 2*Sqrt[5]*y)/(1 + Sqr
t[5] + 2*y))]*Sqrt[-((-3 + Sqrt[5] + 2*Sqrt[5]*y)/(1 + Sqrt[5] + 2*y))])*Ellipti
cF[ArcSin[(2*Sqrt[(5 + 3*Sqrt[5] + 10*y)/(1 + Sqrt[5] + 2*y)])/Sqrt[15]], 15/16]
+ Sqrt[(-5 + 3*Sqrt[5] - 10*y)/(1 + Sqrt[5] + 2*y)]*Sqrt[(-1 + Sqrt[5] - 2*y)/(
1 + Sqrt[5] + 2*y)]*(9*Sqrt[5]*EllipticPi[5/8 - Sqrt[5]/8, ArcSin[(2*Sqrt[(5 + 3
*Sqrt[5] + 10*y)/(1 + Sqrt[5] + 2*y)])/Sqrt[15]], 15/16] + (-20 + 9*Sqrt[5])*Ell
ipticPi[(-3*(-5 + Sqrt[5]))/8, ArcSin[(2*Sqrt[(5 + 3*Sqrt[5] + 10*y)/(1 + Sqrt[5
] + 2*y)])/Sqrt[15]], 15/16] + 2*Sqrt[5]*EllipticPi[(3*(5 + Sqrt[5]))/8, ArcSin[
(2*Sqrt[(5 + 3*Sqrt[5] + 10*y)/(1 + Sqrt[5] + 2*y)])/Sqrt[15]], 15/16])))/(16*Sq
rt[1 - 5*y - 5*y^2]*Sqrt[1 - y - y^2])

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Maple [C]  time = 0.177, size = 352, normalized size = 2.5 \[ -1200\,{\frac{\sqrt{-5\,{y}^{2}-5\,y+1}\sqrt{-{y}^{2}-y+1} \left ( 10\,y+5+3\,\sqrt{5} \right ) ^{2}\sqrt{5}}{\sqrt{5\,{y}^{4}+10\,{y}^{3}-{y}^{2}-6\,y+1}\sqrt{ \left ( -10\,y+3\,\sqrt{5}-5 \right ) \left ( 10\,y+5+3\,\sqrt{5} \right ) \left ( -2\,y+\sqrt{5}-1 \right ) \left ( 2\,y+1+\sqrt{5} \right ) } \left ( 5+3\,\sqrt{5} \right ) \left ( 3\,\sqrt{5}-5 \right ) \left ( -5+\sqrt{5} \right ) \left ( 5+\sqrt{5} \right ) }\sqrt{-{\frac{-10\,y+3\,\sqrt{5}-5}{10\,y+5+3\,\sqrt{5}}}}\sqrt{{\frac{-2\,y+\sqrt{5}-1}{10\,y+5+3\,\sqrt{5}}}}\sqrt{{\frac{2\,y+1+\sqrt{5}}{10\,y+5+3\,\sqrt{5}}}} \left ({\it EllipticPi} \left ( 1/2\,\sqrt{-{\frac{-10\,y+3\,\sqrt{5}-5}{10\,y+5+3\,\sqrt{5}}}},-4\,{\frac{5+3\,\sqrt{5}}{3\,\sqrt{5}-5}},4 \right ) +2\,{\it EllipticPi} \left ( 1/2\,\sqrt{-{\frac{-10\,y+3\,\sqrt{5}-5}{10\,y+5+3\,\sqrt{5}}}},-4\,{\frac{3\,\sqrt{5}-5}{5+3\,\sqrt{5}}},4 \right ) -3\,{\it EllipticPi} \left ( 1/2\,\sqrt{-{\frac{-10\,y+3\,\sqrt{5}-5}{10\,y+5+3\,\sqrt{5}}}},-4\,{\frac{-5+\sqrt{5}}{5+\sqrt{5}}},4 \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*y)*(-5*y^2-5*y+1)^(1/2)/y/(1+y)/(2+y)/(-y^2-y+1)^(1/2),y)

[Out]

-1200*(-5*y^2-5*y+1)^(1/2)*(-y^2-y+1)^(1/2)*(-(-10*y+3*5^(1/2)-5)/(10*y+5+3*5^(1
/2)))^(1/2)*(10*y+5+3*5^(1/2))^2*5^(1/2)*((-2*y+5^(1/2)-1)/(10*y+5+3*5^(1/2)))^(
1/2)*((2*y+1+5^(1/2))/(10*y+5+3*5^(1/2)))^(1/2)*(EllipticPi(1/2*(-(-10*y+3*5^(1/
2)-5)/(10*y+5+3*5^(1/2)))^(1/2),-4*(5+3*5^(1/2))/(3*5^(1/2)-5),4)+2*EllipticPi(1
/2*(-(-10*y+3*5^(1/2)-5)/(10*y+5+3*5^(1/2)))^(1/2),-4*(3*5^(1/2)-5)/(5+3*5^(1/2)
),4)-3*EllipticPi(1/2*(-(-10*y+3*5^(1/2)-5)/(10*y+5+3*5^(1/2)))^(1/2),-4*(-5+5^(
1/2))/(5+5^(1/2)),4))/(5*y^4+10*y^3-y^2-6*y+1)^(1/2)/((-10*y+3*5^(1/2)-5)*(10*y+
5+3*5^(1/2))*(-2*y+5^(1/2)-1)*(2*y+1+5^(1/2)))^(1/2)/(5+3*5^(1/2))/(3*5^(1/2)-5)
/(-5+5^(1/2))/(5+5^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-5*y^2 - 5*y + 1)*(2*y + 1)/(sqrt(-y^2 - y + 1)*(y + 2)*(y + 1)*y),y, algorithm="maxima")

[Out]

integrate(sqrt(-5*y^2 - 5*y + 1)*(2*y + 1)/(sqrt(-y^2 - y + 1)*(y + 2)*(y + 1)*y
), y)

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Fricas [A]  time = 0.281836, size = 301, normalized size = 2.12 \[ \frac{9}{8} \, \log \left (-\frac{235 \, y^{4} + 935 \, y^{3} - 3 \,{\left (35 \, y^{2} + 104 \, y + 77\right )} \sqrt{-y^{2} - y + 1} \sqrt{-5 \, y^{2} - 5 \, y + 1} + 1086 \, y^{2} + 131 \, y - 281}{y^{4} + 8 \, y^{3} + 24 \, y^{2} + 32 \, y + 16}\right ) + \frac{1}{4} \, \log \left (\frac{35 \, y^{4} + 125 \, y^{3} +{\left (15 \, y^{2} + 38 \, y + 24\right )} \sqrt{-y^{2} - y + 1} \sqrt{-5 \, y^{2} - 5 \, y + 1} + 131 \, y^{2} + 16 \, y - 26}{y^{4} + 4 \, y^{3} + 6 \, y^{2} + 4 \, y + 1}\right ) + \frac{1}{8} \, \log \left (\frac{35 \, y^{4} + 15 \, y^{3} +{\left (15 \, y^{2} - 8 \, y + 1\right )} \sqrt{-y^{2} - y + 1} \sqrt{-5 \, y^{2} - 5 \, y + 1} - 34 \, y^{2} + 11 \, y - 1}{y^{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-5*y^2 - 5*y + 1)*(2*y + 1)/(sqrt(-y^2 - y + 1)*(y + 2)*(y + 1)*y),y, algorithm="fricas")

[Out]

9/8*log(-(235*y^4 + 935*y^3 - 3*(35*y^2 + 104*y + 77)*sqrt(-y^2 - y + 1)*sqrt(-5
*y^2 - 5*y + 1) + 1086*y^2 + 131*y - 281)/(y^4 + 8*y^3 + 24*y^2 + 32*y + 16)) +
1/4*log((35*y^4 + 125*y^3 + (15*y^2 + 38*y + 24)*sqrt(-y^2 - y + 1)*sqrt(-5*y^2
- 5*y + 1) + 131*y^2 + 16*y - 26)/(y^4 + 4*y^3 + 6*y^2 + 4*y + 1)) + 1/8*log((35
*y^4 + 15*y^3 + (15*y^2 - 8*y + 1)*sqrt(-y^2 - y + 1)*sqrt(-5*y^2 - 5*y + 1) - 3
4*y^2 + 11*y - 1)/y^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (2 y + 1\right ) \sqrt{- 5 y^{2} - 5 y + 1}}{y \left (y + 1\right ) \left (y + 2\right ) \sqrt{- y^{2} - y + 1}}\, dy \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*y)*(-5*y**2-5*y+1)**(1/2)/y/(1+y)/(2+y)/(-y**2-y+1)**(1/2),y)

[Out]

Integral((2*y + 1)*sqrt(-5*y**2 - 5*y + 1)/(y*(y + 1)*(y + 2)*sqrt(-y**2 - y + 1
)), y)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-5*y^2 - 5*y + 1)*(2*y + 1)/(sqrt(-y^2 - y + 1)*(y + 2)*(y + 1)*y),y, algorithm="giac")

[Out]

integrate(sqrt(-5*y^2 - 5*y + 1)*(2*y + 1)/(sqrt(-y^2 - y + 1)*(y + 2)*(y + 1)*y
), y)