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

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

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Rubi [F]  time = 2.08, 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+3 x+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 + 3*x + x^2)*Sqrt[x + x^4]),x]

[Out]

(2*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])/(3^(1/4)*Sqrt[(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[x + x^4]) + (Sqrt[-17 + 5*
Sqrt[13]]*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((Sqrt[-3 + Sqrt[13]] - Sqrt[2]*x)*Sqrt[1 + x^6]), x
], x, Sqrt[x]])/Sqrt[x + x^4] - (I*Sqrt[17 + 5*Sqrt[13]]*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((I*S
qrt[3 + Sqrt[13]] - Sqrt[2]*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/Sqrt[x + x^4] + (Sqrt[-17 + 5*Sqrt[13]]*Sqrt[x
]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((Sqrt[-3 + Sqrt[13]] + Sqrt[2]*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/
Sqrt[x + x^4] - (I*Sqrt[17 + 5*Sqrt[13]]*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((I*Sqrt[3 + Sqrt[13]
] + Sqrt[2]*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/Sqrt[x + x^4]

Rubi steps

\begin {align*} \int \frac {-1-2 x+2 x^2}{\left (-1+3 x+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+3 x+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}{\sqrt {x} \sqrt {1+x^3}}+\frac {1-8 x}{\sqrt {x} \left (-1+3 x+x^2\right ) \sqrt {1+x^3}}\right ) \, dx}{\sqrt {x+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1-8 x}{\sqrt {x} \left (-1+3 x+x^2\right ) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \left (\frac {-8+2 \sqrt {13}}{\sqrt {x} \left (3-\sqrt {13}+2 x\right ) \sqrt {1+x^3}}+\frac {-8-2 \sqrt {13}}{\sqrt {x} \left (3+\sqrt {13}+2 x\right ) \sqrt {1+x^3}}\right ) \, dx}{\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 )}{\sqrt {x+x^4}}\\ &=\frac {2 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 )}{\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 (4-\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (3-\sqrt {13}+2 x\right ) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}}-\frac {\left (2 \left (4+\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (3+\sqrt {13}+2 x\right ) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}}\\ &=\frac {2 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 )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}-\frac {\left (4 \left (4-\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3-\sqrt {13}+2 x^2\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}-\frac {\left (4 \left (4+\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3+\sqrt {13}+2 x^2\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}\\ &=\frac {2 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 )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}-\frac {\left (4 \left (4-\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {-3+\sqrt {13}}}{2 \left (3-\sqrt {13}\right ) \left (\sqrt {-3+\sqrt {13}}-\sqrt {2} x\right ) \sqrt {1+x^6}}+\frac {\sqrt {-3+\sqrt {13}}}{2 \left (3-\sqrt {13}\right ) \left (\sqrt {-3+\sqrt {13}}+\sqrt {2} x\right ) \sqrt {1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}-\frac {\left (4 \left (4+\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {i}{2 \sqrt {3+\sqrt {13}} \left (i \sqrt {3+\sqrt {13}}-\sqrt {2} x\right ) \sqrt {1+x^6}}+\frac {i}{2 \sqrt {3+\sqrt {13}} \left (i \sqrt {3+\sqrt {13}}+\sqrt {2} x\right ) \sqrt {1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}\\ &=\frac {2 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 )}{\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 (4-\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-3+\sqrt {13}}-\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-3+\sqrt {13}} \sqrt {x+x^4}}+\frac {\left (2 \left (4-\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-3+\sqrt {13}}+\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-3+\sqrt {13}} \sqrt {x+x^4}}-\frac {\left (2 i \left (4+\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i \sqrt {3+\sqrt {13}}-\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {3+\sqrt {13}} \sqrt {x+x^4}}-\frac {\left (2 i \left (4+\sqrt {13}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i \sqrt {3+\sqrt {13}}+\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {3+\sqrt {13}} \sqrt {x+x^4}}\\ \end {align*}

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Mathematica [C]  time = 1.24, size = 298, normalized size = 9.31 \begin {gather*} -\frac {2 \sqrt {\frac {1}{x^2}-\frac {1}{x}+1} \sqrt {\frac {\frac {1}{x}+1}{1+\sqrt [3]{-1}}} x^2 \left (\frac {\sqrt {3} \left (-i \sqrt {3} x+\sqrt [3]{-1}+1\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {x+(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}\right )|\sqrt [3]{-1}\right )}{x+(-1)^{2/3}}-\frac {6 i \left (\left (-1+5 \sqrt [3]{-1}+\sqrt {13}+\sqrt [3]{-1} \sqrt {13}\right ) \Pi \left (\frac {2 \sqrt {3}}{2 i+\sqrt {3}+i \sqrt {13}};\sin ^{-1}\left (\sqrt {\frac {x+(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}\right )|\sqrt [3]{-1}\right )-\left (1-5 \sqrt [3]{-1}+\sqrt {13}+\sqrt [3]{-1} \sqrt {13}\right ) \Pi \left (\frac {2 i \sqrt {3}}{-3+2 \sqrt [3]{-1}+\sqrt {13}};\sin ^{-1}\left (\sqrt {\frac {x+(-1)^{2/3}}{\left (1+\sqrt [3]{-1}\right ) x}}\right )|\sqrt [3]{-1}\right )\right )}{-12-4 i \sqrt {3}}\right )}{3 \sqrt {x^4+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 + x^(-2) - x^(-1)]*Sqrt[(1 + x^(-1))/(1 + (-1)^(1/3))]*x^2*((Sqrt[3]*(1 + (-1)^(1/3) - I*Sqrt[3]*x)
*EllipticF[ArcSin[Sqrt[((-1)^(2/3) + x)/((1 + (-1)^(1/3))*x)]], (-1)^(1/3)])/((-1)^(2/3) + x) - ((6*I)*((-1 +
5*(-1)^(1/3) + Sqrt[13] + (-1)^(1/3)*Sqrt[13])*EllipticPi[(2*Sqrt[3])/(2*I + Sqrt[3] + I*Sqrt[13]), ArcSin[Sqr
t[((-1)^(2/3) + x)/((1 + (-1)^(1/3))*x)]], (-1)^(1/3)] - (1 - 5*(-1)^(1/3) + Sqrt[13] + (-1)^(1/3)*Sqrt[13])*E
llipticPi[((2*I)*Sqrt[3])/(-3 + 2*(-1)^(1/3) + Sqrt[13]), ArcSin[Sqrt[((-1)^(2/3) + x)/((1 + (-1)^(1/3))*x)]],
(-1)^(1/3)]))/(-12 - (4*I)*Sqrt[3])))/(3*Sqrt[x + x^4])

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IntegrateAlgebraic [A]  time = 1.38, size = 32, normalized size = 1.00 \begin {gather*} \sqrt {2} \tanh ^{-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 + 3*x + x^2)*Sqrt[x + x^4]),x]

[Out]

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

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fricas [B]  time = 0.54, size = 68, normalized size = 2.12 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (-\frac {17 \, x^{4} + 6 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{4} + x} {\left (3 \, x^{2} + x + 1\right )} + 7 \, x^{2} + 10 \, x + 1}{x^{4} + 6 \, x^{3} + 7 \, x^{2} - 6 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log(-(17*x^4 + 6*x^3 + 4*sqrt(2)*sqrt(x^4 + x)*(3*x^2 + x + 1) + 7*x^2 + 10*x + 1)/(x^4 + 6*x^3 +
7*x^2 - 6*x + 1))

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

[Out]

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

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maple [C]  time = 0.39, size = 56, normalized size = 1.75

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}-2\right ) x +\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {x^{4}+x}}{x^{2}+3 x -1}\right )}{2}\)	\(56\)
default	\(\text {Expression too large to display}\)	\(13798\)
elliptic	\(\text {Expression too large to display}\)	\(15100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(_Z^2-2)*ln((3*RootOf(_Z^2-2)*x^2+RootOf(_Z^2-2)*x+RootOf(_Z^2-2)+4*(x^4+x)^(1/2))/(x^2+3*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 (x^{2} + 3 \, 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)/(x^2+3*x-1)/(x^4+x)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

int(-(2*x - 2*x^2 + 1)/((x + x^4)^(1/2)*(3*x + 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 (x^{2} + 3 x - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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