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

Optimal. Leaf size=33 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {x^4+x}}{x^2-x+1}\right )}{\sqrt {3}} \]

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Rubi [F]  time = 2.49, 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+2 x+4 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 + 2*x + 4*x^2)*Sqrt[x + x^4]),x]

[Out]

(x*(1 + 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])/(2*3^(1/4)*Sqrt[(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[x + x^4]) + (Sqrt[3]*(I +
Sqrt[3])*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((Sqrt[-1 - I*Sqrt[3]] - 2*x)*Sqrt[1 + x^6]), x], x,
Sqrt[x]])/(2*Sqrt[-1 - I*Sqrt[3]]*Sqrt[x + x^4]) - (Sqrt[3]*(I - Sqrt[3])*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][D
efer[Int][1/((Sqrt[-1 + I*Sqrt[3]] - 2*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/(2*Sqrt[I*(I + Sqrt[3])]*Sqrt[x + x
^4]) + (Sqrt[3]*(I + Sqrt[3])*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((Sqrt[-1 - I*Sqrt[3]] + 2*x)*Sq
rt[1 + x^6]), x], x, Sqrt[x]])/(2*Sqrt[-1 - I*Sqrt[3]]*Sqrt[x + x^4]) - (Sqrt[3]*(I - Sqrt[3])*Sqrt[x]*Sqrt[1
+ x^3]*Defer[Subst][Defer[Int][1/((Sqrt[-1 + I*Sqrt[3]] + 2*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/(2*Sqrt[I*(I +
Sqrt[3])]*Sqrt[x + x^4])

Rubi steps

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

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Mathematica [C]  time = 1.90, size = 287, normalized size = 8.70 \begin {gather*} -\frac {2 \sqrt {\frac {\frac {1}{x}+1}{1+\sqrt [3]{-1}}} x^2 \left (\frac {2 \sqrt {\frac {1}{x^2}-\frac {1}{x}+1} \Pi \left (\frac {2 i}{3 i+\sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {x+(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}\right )|\sqrt [3]{-1}\right )}{\sqrt {3}+3 i}-\frac {2 i \sqrt {3} \sqrt {\frac {1}{x^2}-\frac {1}{x}+1} \Pi \left (-\frac {2 \sqrt {3}}{3 i+\sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {x+(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}\right )|\sqrt [3]{-1}\right )}{\sqrt {3}+3 i}+\frac {\left (\sqrt [3]{-1}-\frac {1}{x}\right ) \sqrt {\frac {\sqrt [3]{-1}-\frac {(-1)^{2/3}}{x}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {x+(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {x+(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}}\right )}{\sqrt {x^4+x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

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IntegrateAlgebraic [A]  time = 1.13, size = 33, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {x^4+x}}{x^2-x+1}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 28, normalized size = 0.85 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + 4 \, x - 1\right )}}{6 \, \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)/(4*x^2+2*x+1)/(x^4+x)^(1/2),x, algorithm="fricas")

[Out]

-1/3*sqrt(3)*arctan(1/6*sqrt(3)*(2*x^2 + 4*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 (4 \, x^{2} + 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)/(4*x^2+2*x+1)/(x^4+x)^(1/2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.18, size = 830, normalized size = 25.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1/2-1/2*I*3^(1/2))*((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)*(1+x)^2*(-(x-1/2+1/2*I*3^(1/2))/
(1/2-1/2*I*3^(1/2))/(1+x))^(1/2)*(-(x-1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)/(3/2+1/2*I*3^(1/2))/
(x*(1+x)*(x-1/2+1/2*I*3^(1/2))*(x-1/2-1/2*I*3^(1/2)))^(1/2)*EllipticF(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2
))/(1+x))^(1/2),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+3
*(1/4-1/12*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))*((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)*(1+x)^2*(-(
x-1/2+1/2*I*3^(1/2))/(1/2-1/2*I*3^(1/2))/(1+x))^(1/2)*(-(x-1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)
/(3/2+1/2*I*3^(1/2))/(x*(1+x)*(x-1/2+1/2*I*3^(1/2))*(x-1/2-1/2*I*3^(1/2)))^(1/2)*(-1+1/3*I*3^(1/2))*(EllipticF
(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*
I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+(-1-I*3^(1/2))*EllipticPi(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(
1+x))^(1/2),-2/3*I*3^(1/2)*(-1/4+1/4*I*3^(1/2))-2/3*I*3^(1/2),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2
+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)))+3*(1/4+1/12*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))*((3/2+1/2*I*3^(1/2))
*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)*(1+x)^2*(-(x-1/2+1/2*I*3^(1/2))/(1/2-1/2*I*3^(1/2))/(1+x))^(1/2)*(-(x-1/2-
1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)/(3/2+1/2*I*3^(1/2))/(x*(1+x)*(x-1/2+1/2*I*3^(1/2))*(x-1/2-1/2*
I*3^(1/2)))^(1/2)*(-1-1/3*I*3^(1/2))*(EllipticF(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),((-3/2
+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+(I*3^(1/2)-1)*EllipticP
i(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),-2/3*I*3^(1/2)*(-1/4-1/4*I*3^(1/2))+1/3*I*3^(1/2),((
-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)))

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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 (4 \, x^{2} + 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)/(4*x^2+2*x+1)/(x^4+x)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

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