3.4.88 \(\int \frac {-1+2 x+2 x^2}{(1-x+3 x^2) \sqrt {-x+x^4}} \, dx\)

Optimal. Leaf size=32 \[ \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x^4-x}}{x^2+x+1}\right ) \]

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Rubi [F]  time = 2.41, 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 x+2 x^2}{\left (1-x+3 x^2\right ) \sqrt {-x+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + 2*x + 2*x^2)/((1 - x + 3*x^2)*Sqrt[-x + x^4]),x]

[Out]

(2*(1 - x)*x*Sqrt[(1 + x + x^2)/(1 - (1 + Sqrt[3])*x)^2]*EllipticF[ArcCos[(1 - (1 - Sqrt[3])*x)/(1 - (1 + Sqrt
[3])*x)], (2 + Sqrt[3])/4])/(3*3^(1/4)*Sqrt[-(((1 - x)*x)/(1 - (1 + Sqrt[3])*x)^2)]*Sqrt[-x + x^4]) - (2*(4 -
I*Sqrt[11])*Sqrt[x]*Sqrt[-1 + x^3]*Defer[Subst][Defer[Int][1/((Sqrt[1 - I*Sqrt[11]] - Sqrt[6]*x)*Sqrt[-1 + x^6
]), x], x, Sqrt[x]])/(3*Sqrt[1 - I*Sqrt[11]]*Sqrt[-x + x^4]) - (2*(4 + I*Sqrt[11])*Sqrt[x]*Sqrt[-1 + x^3]*Defe
r[Subst][Defer[Int][1/((Sqrt[1 + I*Sqrt[11]] - Sqrt[6]*x)*Sqrt[-1 + x^6]), x], x, Sqrt[x]])/(3*Sqrt[1 + I*Sqrt
[11]]*Sqrt[-x + x^4]) - (2*(4 - I*Sqrt[11])*Sqrt[x]*Sqrt[-1 + x^3]*Defer[Subst][Defer[Int][1/((Sqrt[1 - I*Sqrt
[11]] + Sqrt[6]*x)*Sqrt[-1 + x^6]), x], x, Sqrt[x]])/(3*Sqrt[1 - I*Sqrt[11]]*Sqrt[-x + x^4]) - (2*(4 + I*Sqrt[
11])*Sqrt[x]*Sqrt[-1 + x^3]*Defer[Subst][Defer[Int][1/((Sqrt[1 + I*Sqrt[11]] + Sqrt[6]*x)*Sqrt[-1 + x^6]), x],
x, Sqrt[x]])/(3*Sqrt[1 + I*Sqrt[11]]*Sqrt[-x + x^4])

Rubi steps

\begin {align*} \int \frac {-1+2 x+2 x^2}{\left (1-x+3 x^2\right ) \sqrt {-x+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {-1+2 x+2 x^2}{\sqrt {x} \left (1-x+3 x^2\right ) \sqrt {-1+x^3}} \, dx}{\sqrt {-x+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \left (\frac {2}{3 \sqrt {x} \sqrt {-1+x^3}}-\frac {5-8 x}{3 \sqrt {x} \left (1-x+3 x^2\right ) \sqrt {-1+x^3}}\right ) \, dx}{\sqrt {-x+x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {5-8 x}{\sqrt {x} \left (1-x+3 x^2\right ) \sqrt {-1+x^3}} \, dx}{3 \sqrt {-x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1+x^3}} \, dx}{3 \sqrt {-x+x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \left (\frac {-8-2 i \sqrt {11}}{\sqrt {x} \left (-1-i \sqrt {11}+6 x\right ) \sqrt {-1+x^3}}+\frac {-8+2 i \sqrt {11}}{\sqrt {x} \left (-1+i \sqrt {11}+6 x\right ) \sqrt {-1+x^3}}\right ) \, dx}{3 \sqrt {-x+x^4}}+\frac {\left (4 \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}\\ &=\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (2 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (-1+i \sqrt {11}+6 x\right ) \sqrt {-1+x^3}} \, dx}{3 \sqrt {-x+x^4}}+\frac {\left (2 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (-1-i \sqrt {11}+6 x\right ) \sqrt {-1+x^3}} \, dx}{3 \sqrt {-x+x^4}}\\ &=\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (4 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {11}+6 x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}+\frac {\left (4 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {11}+6 x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}\\ &=\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (4 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {1-i \sqrt {11}}}{2 \left (-1+i \sqrt {11}\right ) \left (\sqrt {1-i \sqrt {11}}-\sqrt {6} x\right ) \sqrt {-1+x^6}}+\frac {\sqrt {1-i \sqrt {11}}}{2 \left (-1+i \sqrt {11}\right ) \left (\sqrt {1-i \sqrt {11}}+\sqrt {6} x\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}+\frac {\left (4 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {1+i \sqrt {11}}}{2 \left (-1-i \sqrt {11}\right ) \left (\sqrt {1+i \sqrt {11}}-\sqrt {6} x\right ) \sqrt {-1+x^6}}+\frac {\sqrt {1+i \sqrt {11}}}{2 \left (-1-i \sqrt {11}\right ) \left (\sqrt {1+i \sqrt {11}}+\sqrt {6} x\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^4}}\\ &=\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (2 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {11}}-\sqrt {6} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {1-i \sqrt {11}} \sqrt {-x+x^4}}-\frac {\left (2 \left (4-i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {11}}+\sqrt {6} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {1-i \sqrt {11}} \sqrt {-x+x^4}}-\frac {\left (2 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {11}}-\sqrt {6} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {1+i \sqrt {11}} \sqrt {-x+x^4}}-\frac {\left (2 \left (4+i \sqrt {11}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {11}}+\sqrt {6} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {1+i \sqrt {11}} \sqrt {-x+x^4}}\\ \end {align*}

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Mathematica [C]  time = 2.15, size = 533, normalized size = 16.66 \begin {gather*} -\frac {2 \sqrt {\frac {1-\frac {1}{x}}{1+\sqrt [3]{-1}}} x^2 \left (\frac {i \sqrt {11} \sqrt {\frac {1}{x^2}+\frac {1}{x}+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}-\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-\frac {(-1)^{2/3}}{x}}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{2 i-\sqrt {3}+\sqrt {11}}+\frac {\sqrt {\frac {1}{x^2}+\frac {1}{x}+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}-\sqrt {11}};\sin ^{-1}\left (\sqrt {-\frac {2 i+\frac {i+\sqrt {3}}{x}}{-3 i+\sqrt {3}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )}{2 i-\sqrt {3}+\sqrt {11}}+\frac {(-1)^{5/6} \left (1+\sqrt [3]{-1}\right ) \sqrt {\frac {1}{x^2}+\frac {1}{x}+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}+\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-\frac {(-1)^{2/3}}{x}}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{3-2 i \sqrt {3}+\sqrt {33}}+\frac {i \sqrt {11} \sqrt {\frac {1}{x^2}+\frac {1}{x}+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}+\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-\frac {(-1)^{2/3}}{x}}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{-2 i+\sqrt {3}+\sqrt {11}}-\frac {\left (\frac {1}{x}+\sqrt [3]{-1}\right ) \sqrt {\frac {\frac {(-1)^{2/3}}{x}+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {1-\frac {(-1)^{2/3}}{x}}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {1-\frac {(-1)^{2/3}}{x}}{1+\sqrt [3]{-1}}}}\right )}{\sqrt {x \left (x^3-1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-1 + 2*x + 2*x^2)/((1 - x + 3*x^2)*Sqrt[-x + x^4]),x]

[Out]

(-2*Sqrt[(1 - x^(-1))/(1 + (-1)^(1/3))]*x^2*(-((((-1)^(1/3) + x^(-1))*Sqrt[((-1)^(1/3) + (-1)^(2/3)/x)/(1 + (-
1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)/x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)/x)/(
1 + (-1)^(1/3))]) + (I*Sqrt[11]*Sqrt[1 + x^(-2) + x^(-1)]*EllipticPi[(2*Sqrt[3])/(-2*I + Sqrt[3] - Sqrt[11]),
ArcSin[Sqrt[(1 - (-1)^(2/3)/x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(2*I - Sqrt[3] + Sqrt[11]) + (Sqrt[1 + x^(-2)
+ x^(-1)]*EllipticPi[(2*Sqrt[3])/(-2*I + Sqrt[3] - Sqrt[11]), ArcSin[Sqrt[-((2*I + (I + Sqrt[3])/x)/(-3*I + Sq
rt[3]))]], (1 + I*Sqrt[3])/2])/(2*I - Sqrt[3] + Sqrt[11]) + (I*Sqrt[11]*Sqrt[1 + x^(-2) + x^(-1)]*EllipticPi[(
2*Sqrt[3])/(-2*I + Sqrt[3] + Sqrt[11]), ArcSin[Sqrt[(1 - (-1)^(2/3)/x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-2*I
+ Sqrt[3] + Sqrt[11]) + ((-1)^(5/6)*(1 + (-1)^(1/3))*Sqrt[1 + x^(-2) + x^(-1)]*EllipticPi[(2*Sqrt[3])/(-2*I +
Sqrt[3] + Sqrt[11]), ArcSin[Sqrt[(1 - (-1)^(2/3)/x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(3 - (2*I)*Sqrt[3] + Sqrt
[33])))/Sqrt[x*(-1 + x^3)]

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IntegrateAlgebraic [A]  time = 1.34, size = 32, normalized size = 1.00 \begin {gather*} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-x+x^4}}{1+x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + 2*x + 2*x^2)/((1 - x + 3*x^2)*Sqrt[-x + x^4]),x]

[Out]

Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[-x + x^4])/(1 + x + x^2)]

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fricas [A]  time = 0.54, size = 28, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - 3 \, x - 1\right )}}{4 \, \sqrt {x^{4} - x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+2*x-1)/(3*x^2-x+1)/(x^4-x)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*arctan(1/4*sqrt(2)*(x^2 - 3*x - 1)/sqrt(x^4 - x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{2} + 2 \, x - 1}{\sqrt {x^{4} - x} {\left (3 \, x^{2} - x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+2*x-1)/(3*x^2-x+1)/(x^4-x)^(1/2),x, algorithm="giac")

[Out]

integrate((2*x^2 + 2*x - 1)/(sqrt(x^4 - x)*(3*x^2 - x + 1)), x)

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maple [C]  time = 0.52, size = 62, normalized size = 1.94

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {x^{4}-x}-\RootOf \left (\textit {\_Z}^{2}+2\right )}{3 x^{2}-x +1}\right )}{2}\)	\(62\)
default	\(\frac {4 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{3 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {2 \left (\frac {4}{3}+\frac {i \sqrt {11}}{3}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\frac {5}{6}+\frac {i \sqrt {11}}{6}\right ) \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, -\frac {i \sqrt {3}\, \left (\frac {1}{6}+\frac {i \sqrt {11}}{6}\right )}{2}+\frac {1}{4}+\frac {i \sqrt {11}}{4}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{3 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {2 \left (\frac {4}{3}-\frac {i \sqrt {11}}{3}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\frac {5}{6}-\frac {i \sqrt {11}}{6}\right ) \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, -\frac {i \sqrt {3}\, \left (\frac {1}{6}-\frac {i \sqrt {11}}{6}\right )}{2}+\frac {1}{4}-\frac {i \sqrt {11}}{4}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{3 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(828\)
elliptic	\(\frac {4 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{3 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {2 \left (\frac {4}{9}+\frac {i \sqrt {11}}{9}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\frac {5}{6}+\frac {i \sqrt {11}}{6}\right ) \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, -\frac {i \sqrt {3}\, \left (\frac {1}{6}+\frac {i \sqrt {11}}{6}\right )}{2}+\frac {1}{4}+\frac {i \sqrt {11}}{4}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {2 \left (\frac {4}{9}-\frac {i \sqrt {11}}{9}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\frac {5}{6}-\frac {i \sqrt {11}}{6}\right ) \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, -\frac {i \sqrt {3}\, \left (\frac {1}{6}-\frac {i \sqrt {11}}{6}\right )}{2}+\frac {1}{4}-\frac {i \sqrt {11}}{4}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(828\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2+2*x-1)/(3*x^2-x+1)/(x^4-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*RootOf(_Z^2+2)*ln((RootOf(_Z^2+2)*x^2-3*RootOf(_Z^2+2)*x+4*(x^4-x)^(1/2)-RootOf(_Z^2+2))/(3*x^2-x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{2} + 2 \, x - 1}{\sqrt {x^{4} - x} {\left (3 \, x^{2} - x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+2*x-1)/(3*x^2-x+1)/(x^4-x)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x^2 + 2*x - 1)/(sqrt(x^4 - x)*(3*x^2 - x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {2\,x^2+2\,x-1}{\sqrt {x^4-x}\,\left (3\,x^2-x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 2*x^2 - 1)/((x^4 - x)^(1/2)*(3*x^2 - x + 1)),x)

[Out]

int((2*x + 2*x^2 - 1)/((x^4 - x)^(1/2)*(3*x^2 - x + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x^{2} + 2 x - 1}{\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (3 x^{2} - x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2+2*x-1)/(3*x**2-x+1)/(x**4-x)**(1/2),x)

[Out]

Integral((2*x**2 + 2*x - 1)/(sqrt(x*(x - 1)*(x**2 + x + 1))*(3*x**2 - x + 1)), x)

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