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

Optimal. Leaf size=81 \[ \frac {\sqrt {3 x^4+2 x^2-2 x} x}{2 \left (x^3+2 x-2\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {3 x^4+2 x^2-2 x}}{3 x^3+2 x-2}\right )}{2 \sqrt {2}} \]

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Rubi [F]  time = 1.26, 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 {(-3+2 x) \sqrt {-2 x+2 x^2+3 x^4}}{\left (-2+2 x+x^3\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.47, size = 85, normalized size = 1.05 \begin {gather*} -\frac {x^2 \left (-6 x^3+\sqrt {\frac {4}{x^3}-\frac {4}{x^2}-6} \left (x^3+2 x-2\right ) \tan ^{-1}\left (\sqrt {\frac {1}{x^3}-\frac {1}{x^2}-\frac {3}{2}}\right )-4 x+4\right )}{4 \left (x^3+2 x-2\right ) \sqrt {x \left (3 x^3+2 x-2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(x^2*(4 - 4*x - 6*x^3 + Sqrt[-6 + 4/x^3 - 4/x^2]*(-2 + 2*x + x^3)*ArcTan[Sqrt[-3/2 + x^(-3) - x^(-2)]]))/
((-2 + 2*x + x^3)*Sqrt[x*(-2 + 2*x + 3*x^3)])

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IntegrateAlgebraic [A]  time = 0.44, size = 81, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {-2 x+2 x^2+3 x^4}}{2 \left (-2+2 x+x^3\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {-2 x+2 x^2+3 x^4}}{-2+2 x+3 x^3}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 132, normalized size = 1.63 \begin {gather*} \frac {\sqrt {2} {\left (x^{3} + 2 \, x - 2\right )} \log \left (-\frac {49 \, x^{6} + 36 \, x^{4} - 36 \, x^{3} + 4 \, \sqrt {2} {\left (5 \, x^{4} + 2 \, x^{2} - 2 \, x\right )} \sqrt {3 \, x^{4} + 2 \, x^{2} - 2 \, x} + 4 \, x^{2} - 8 \, x + 4}{x^{6} + 4 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} - 8 \, x + 4}\right ) + 8 \, \sqrt {3 \, x^{4} + 2 \, x^{2} - 2 \, x} x}{16 \, {\left (x^{3} + 2 \, x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(sqrt(2)*(x^3 + 2*x - 2)*log(-(49*x^6 + 36*x^4 - 36*x^3 + 4*sqrt(2)*(5*x^4 + 2*x^2 - 2*x)*sqrt(3*x^4 + 2*
x^2 - 2*x) + 4*x^2 - 8*x + 4)/(x^6 + 4*x^4 - 4*x^3 + 4*x^2 - 8*x + 4)) + 8*sqrt(3*x^4 + 2*x^2 - 2*x)*x)/(x^3 +
2*x - 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.62, size = 100, normalized size = 1.23

method	result	size
trager	\(\frac {x \sqrt {3 x^{4}+2 x^{2}-2 x}}{2 x^{3}+4 x -4}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+4 \sqrt {3 x^{4}+2 x^{2}-2 x}\, x +2 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -2 \RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{3}+2 x -2}\right )}{8}\)	\(100\)
default	\(\text {Expression too large to display}\)	\(1689\)
elliptic	\(\text {Expression too large to display}\)	\(1689\)
risch	\(\text {Expression too large to display}\)	\(1699\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*(3*x^4+2*x^2-2*x)^(1/2)/(x^3+2*x-2)+1/8*RootOf(_Z^2-2)*ln(-(5*RootOf(_Z^2-2)*x^3+4*(3*x^4+2*x^2-2*x)^(1/
2)*x+2*RootOf(_Z^2-2)*x-2*RootOf(_Z^2-2))/(x^3+2*x-2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x-3\right )\,\sqrt {3\,x^4+2\,x^2-2\,x}}{{\left (x^3+2\,x-2\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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