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

________________________________________________________________________________________

Rubi [F]  time = 3.3236, 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., Rules used = {} \[ \int \frac{(1+2 y) \sqrt{1-5 y-5 y^2}}{y (1+y) (2+y) \sqrt{1-y-y^2}} \, dy \]

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

Rubi steps

\begin{align*} \int \frac{(1+2 y) \sqrt{1-5 y-5 y^2}}{y (1+y) (2+y) \sqrt{1-y-y^2}} \, dy &=\int \left (\frac{\sqrt{1-5 y-5 y^2}}{2 y \sqrt{1-y-y^2}}+\frac{\sqrt{1-5 y-5 y^2}}{(1+y) \sqrt{1-y-y^2}}-\frac{3 \sqrt{1-5 y-5 y^2}}{2 (2+y) \sqrt{1-y-y^2}}\right ) \, dy\\ &=\frac{1}{2} \int \frac{\sqrt{1-5 y-5 y^2}}{y \sqrt{1-y-y^2}} \, dy-\frac{3}{2} \int \frac{\sqrt{1-5 y-5 y^2}}{(2+y) \sqrt{1-y-y^2}} \, dy+\int \frac{\sqrt{1-5 y-5 y^2}}{(1+y) \sqrt{1-y-y^2}} \, dy\\ \end{align*}

Mathematica [C]  time = 1.52404, 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 + Sqrt[5] + 2*y))]*Sqrt[-((-3 + Sqrt[5] + 2*Sqrt[5]*y)/(1 + Sqrt[5] + 2*y))
])*EllipticF[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 - Sq
rt[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*Sqrt[1 - 5*y - 5*y^2]*Sqrt[1 - y - y^2])

________________________________________________________________________________________

Maple [C]  time = 0.213, size = 354, normalized size = 2.5 \begin{align*} 300\,{\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+5+3\,\sqrt{5} \right ) \left ( -10\,y-5+3\,\sqrt{5} \right ) \left ( \sqrt{5}-1-2\,y \right ) \left ( 2\,y+1+\sqrt{5} \right ) } \left ( 5+\sqrt{5} \right ) \left ( \sqrt{5}-5 \right ) \left ( 3\,\sqrt{5}-5 \right ) \left ( 5+3\,\sqrt{5} \right ) }\sqrt{-{\frac{10\,y+5+3\,\sqrt{5}}{-10\,y-5+3\,\sqrt{5}}}}\sqrt{{\frac{\sqrt{5}-1-2\,y}{-10\,y-5+3\,\sqrt{5}}}}\sqrt{{\frac{2\,y+1+\sqrt{5}}{-10\,y-5+3\,\sqrt{5}}}} \left ( 3\,{\it EllipticPi} \left ( 2\,\sqrt{-{\frac{10\,y+5+3\,\sqrt{5}}{-10\,y-5+3\,\sqrt{5}}}},-1/4\,{\frac{5+\sqrt{5}}{\sqrt{5}-5}},1/4 \right ) -{\it EllipticPi} \left ( 2\,\sqrt{-{\frac{10\,y+5+3\,\sqrt{5}}{-10\,y-5+3\,\sqrt{5}}}},-1/4\,{\frac{3\,\sqrt{5}-5}{5+3\,\sqrt{5}}},1/4 \right ) -2\,{\it EllipticPi} \left ( 2\,\sqrt{-{\frac{10\,y+5+3\,\sqrt{5}}{-10\,y-5+3\,\sqrt{5}}}},-1/4\,{\frac{5+3\,\sqrt{5}}{3\,\sqrt{5}-5}},1/4 \right ) \right ) } \end{align*}

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]

300*(-5*y^2-5*y+1)^(1/2)*(-y^2-y+1)^(1/2)*(-(10*y+5+3*5^(1/2))/(-10*y-5+3*5^(1/2)))^(1/2)*(-10*y-5+3*5^(1/2))^
2*((5^(1/2)-1-2*y)/(-10*y-5+3*5^(1/2)))^(1/2)*5^(1/2)*((2*y+1+5^(1/2))/(-10*y-5+3*5^(1/2)))^(1/2)*(3*EllipticP
i(2*(-(10*y+5+3*5^(1/2))/(-10*y-5+3*5^(1/2)))^(1/2),-1/4*(5+5^(1/2))/(5^(1/2)-5),1/4)-EllipticPi(2*(-(10*y+5+3
*5^(1/2))/(-10*y-5+3*5^(1/2)))^(1/2),-1/4*(3*5^(1/2)-5)/(5+3*5^(1/2)),1/4)-2*EllipticPi(2*(-(10*y+5+3*5^(1/2))
/(-10*y-5+3*5^(1/2)))^(1/2),-1/4*(5+3*5^(1/2))/(3*5^(1/2)-5),1/4))/(5*y^4+10*y^3-y^2-6*y+1)^(1/2)/((10*y+5+3*5
^(1/2))*(-10*y-5+3*5^(1/2))*(5^(1/2)-1-2*y)*(2*y+1+5^(1/2)))^(1/2)/(5+5^(1/2))/(5^(1/2)-5)/(3*5^(1/2)-5)/(5+3*
5^(1/2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, 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)

________________________________________________________________________________________

Fricas [A]  time = 2.8141, size = 572, normalized size = 4.03 \begin{align*} \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 ) \end{align*}

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, 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 + 1
31*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) - 34*y^2 + 11*y - 1)/y^4)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, 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)