3.3.79 \(\int \frac {(1+2 y) \sqrt {1-5 y-5 y^2}}{y (1+y) (2+y) \sqrt {1-y-y^2}} \, dy\) [279]

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

[Out]

-1/4*arctanh((1-3*y)*(-5*y^2-5*y+1)^(1/2)/(1-5*y)/(-y^2-y+1)^(1/2))-1/2*arctanh((4+3*y)*(-5*y^2-5*y+1)^(1/2)/(
6+5*y)/(-y^2-y+1)^(1/2))+9/4*arctanh(1/3*(11+7*y)*(-5*y^2-5*y+1)^(1/2)/(7+5*y)/(-y^2-y+1)^(1/2))

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Rubi [F]
time = 1.94, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(1+2 y) \sqrt {1-5 y-5 y^2}}{y (1+y) (2+y) \sqrt {1-y-y^2}} \, dy \end {gather*}

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 3.48, size = 630, normalized size = 4.44 \begin {gather*} \frac {\left (-1-\frac {2}{\sqrt {5}}\right ) \left (1+\sqrt {5}+2 y\right )^2 \sqrt {\frac {5+3 \sqrt {5}+10 y}{5+5 \sqrt {5}+10 y}} \left (20 \left (-4 \sqrt {\frac {-5+3 \sqrt {5}-10 y}{1+\sqrt {5}+2 y}} \sqrt {\frac {-1+\sqrt {5}-2 y}{1+\sqrt {5}+2 y}}+\sqrt {5} \sqrt {\frac {-5+3 \sqrt {5}-10 y}{1+\sqrt {5}+2 y}} \sqrt {\frac {-1+\sqrt {5}-2 y}{1+\sqrt {5}+2 y}}+5 \sqrt {-\frac {-5+\sqrt {5}+2 \sqrt {5} y}{1+\sqrt {5}+2 y}} \sqrt {-\frac {-3+\sqrt {5}+2 \sqrt {5} y}{1+\sqrt {5}+2 y}}-2 \sqrt {5} \sqrt {-\frac {-5+\sqrt {5}+2 \sqrt {5} y}{1+\sqrt {5}+2 y}} \sqrt {-\frac {-3+\sqrt {5}+2 \sqrt {5} y}{1+\sqrt {5}+2 y}}\right ) F\left (\sin ^{-1}\left (\frac {2 \sqrt {\frac {5+3 \sqrt {5}+10 y}{1+\sqrt {5}+2 y}}}{\sqrt {15}}\right )|\frac {15}{16}\right )+\sqrt {\frac {-5+3 \sqrt {5}-10 y}{1+\sqrt {5}+2 y}} \sqrt {\frac {-1+\sqrt {5}-2 y}{1+\sqrt {5}+2 y}} \left (9 \sqrt {5} \Pi \left (\frac {5}{8}-\frac {\sqrt {5}}{8};\sin ^{-1}\left (\frac {2 \sqrt {\frac {5+3 \sqrt {5}+10 y}{1+\sqrt {5}+2 y}}}{\sqrt {15}}\right )|\frac {15}{16}\right )+\left (-20+9 \sqrt {5}\right ) \Pi \left (-\frac {3}{8} \left (-5+\sqrt {5}\right );\sin ^{-1}\left (\frac {2 \sqrt {\frac {5+3 \sqrt {5}+10 y}{1+\sqrt {5}+2 y}}}{\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 {5+3 \sqrt {5}+10 y}{1+\sqrt {5}+2 y}}}{\sqrt {15}}\right )|\frac {15}{16}\right )\right )\right )}{16 \sqrt {1-5 y-5 y^2} \sqrt {1-y-y^2}} \end {gather*}

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])

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.34, size = 354, normalized size = 2.49

method	result	size
default	\(\frac {300 \sqrt {-5 y^{2}-5 y +1}\, \sqrt {-y^{2}-y +1}\, \sqrt {-\frac {10 y +5+3 \sqrt {5}}{-10 y -5+3 \sqrt {5}}}\, \left (-10 y -5+3 \sqrt {5}\right )^{2} \sqrt {\frac {-2 y +\sqrt {5}-1}{-10 y -5+3 \sqrt {5}}}\, \sqrt {5}\, \sqrt {\frac {2 y +1+\sqrt {5}}{-10 y -5+3 \sqrt {5}}}\, \left (3 \EllipticPi \left (2 \sqrt {-\frac {10 y +5+3 \sqrt {5}}{-10 y -5+3 \sqrt {5}}}, -\frac {5+\sqrt {5}}{4 \left (\sqrt {5}-5\right )}, \frac {1}{4}\right )-\EllipticPi \left (2 \sqrt {-\frac {10 y +5+3 \sqrt {5}}{-10 y -5+3 \sqrt {5}}}, -\frac {3 \sqrt {5}-5}{4 \left (5+3 \sqrt {5}\right )}, \frac {1}{4}\right )-2 \EllipticPi \left (2 \sqrt {-\frac {10 y +5+3 \sqrt {5}}{-10 y -5+3 \sqrt {5}}}, -\frac {5+3 \sqrt {5}}{4 \left (3 \sqrt {5}-5\right )}, \frac {1}{4}\right )\right )}{\sqrt {5 y^{4}+10 y^{3}-y^{2}-6 y +1}\, \sqrt {\left (10 y +5+3 \sqrt {5}\right ) \left (-10 y -5+3 \sqrt {5}\right ) \left (-2 y +\sqrt {5}-1\right ) \left (2 y +1+\sqrt {5}\right )}\, \left (5+\sqrt {5}\right ) \left (\sqrt {5}-5\right ) \left (3 \sqrt {5}-5\right ) \left (5+3 \sqrt {5}\right )}\)	\(354\)
elliptic	\(\text {Expression too large to display}\)	\(7002\)

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,method=_RETURNVERBOSE)

[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*((-2*y+5^(1/2)-1)/(-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*Elliptic
Pi(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))*(-2*y+5^(1/2)-1)*(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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [A]
time = 0.37, size = 223, normalized size = 1.57 \begin {gather*} \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 {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

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]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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}} \,d y \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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