3.11.83 \(\int \frac {(-3+2 x) \sqrt {-2 x+2 x^2+3 x^4}}{(-2+2 x+x^3)^2} \, dx\) [1083]

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

[Out]

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

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Rubi [F]
time = 0.77, 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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 3.26, size = 79, normalized size = 0.98 \begin {gather*} \frac {\sqrt {x \left (-2+2 x+3 x^3\right )} \left (\frac {2 x^{3/2}}{-2+2 x+x^3}+\frac {\tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {-1+x+\frac {3 x^3}{2}}}\right )}{\sqrt {-1+x+\frac {3 x^3}{2}}}\right )}{4 \sqrt {x}} \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]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.10, size = 1689, normalized size = 20.85

method	result	size
trager	\(\frac {x \sqrt {3 x^{4}+2 x^{2}-2 x}}{2 x^{3}+4 x -4}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -4 \sqrt {3 x^{4}+2 x^{2}-2 x}\, x -2 \RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{3}+2 x -2}\right )}{8}\)	\(99\)
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)-1/1728*108^(1/2)*sum(_alpha^2*((9+89^(1/2))^(1/3)-2/(9+89^(1/2))^(1/
3)+I*3^(1/2)*((9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)))*(x*(-3*(9+89^(1/2))^(1/3)+6/(9+89^(1/2))^(1/3)-I*3^(1/
2)*((9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)))/(-(9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)-I*3^(1/2)*((9+89^(1/2)
)^(1/3)+2/(9+89^(1/2))^(1/3)))/(3*x-(9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)))^(1/2)*(3*x-(9+89^(1/2))^(1/3)+2/
(9+89^(1/2))^(1/3))^2*(((9+89^(1/2))^(1/3)-2/(9+89^(1/2))^(1/3))*(6*x+(9+89^(1/2))^(1/3)-2/(9+89^(1/2))^(1/3)-
I*3^(1/2)*((9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)))/(-(9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)+I*3^(1/2)*((9+8
9^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)))/(3*x-(9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)))^(1/2)*(((9+89^(1/2))^(1/3
)-2/(9+89^(1/2))^(1/3))*(6*x+(9+89^(1/2))^(1/3)-2/(9+89^(1/2))^(1/3)+I*3^(1/2)*((9+89^(1/2))^(1/3)+2/(9+89^(1/
2))^(1/3)))/(-(9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)-I*3^(1/2)*((9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)))/(3*
x-(9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)))^(1/2)/(-3*(9+89^(1/2))^(1/3)+6/(9+89^(1/2))^(1/3)-I*3^(1/2)*((9+89
^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3)))/((9+89^(1/2))^(1/3)-2/(9+89^(1/2))^(1/3))/(x*(3*x-(9+89^(1/2))^(1/3)+2/(9
+89^(1/2))^(1/3))*(6*x+(9+89^(1/2))^(1/3)-2/(9+89^(1/2))^(1/3)-I*3^(1/2)*((9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1
/3)))*(6*x+(9+89^(1/2))^(1/3)-2/(9+89^(1/2))^(1/3)+I*3^(1/2)*((9+89^(1/2))^(1/3)+2/(9+89^(1/2))^(1/3))))^(1/2)
*(-44*_alpha^2-8*_alpha-96-31*_alpha^2*(9+89^(1/2))^(2/3)+3*_alpha^2*(9+89^(1/2))^(2/3)*89^(1/2)+6*_alpha^2*(9
+89^(1/2))^(1/3)-2*_alpha^2*(9+89^(1/2))^(1/3)*89^(1/2)-31*_alpha*(9+89^(1/2))^(2/3)+3*_alpha*(9+89^(1/2))^(2/
3)*89^(1/2)-2*_alpha*(9+89^(1/2))^(1/3)*89^(1/2)+6*_alpha*(9+89^(1/2))^(1/3)-66*(9+89^(1/2))^(2/3)+6*(9+89^(1/
2))^(2/3)*89^(1/2)+30*(9+89^(1/2))^(1/3)-6*(9+89^(1/2))^(1/3)*89^(1/2))*(6*EllipticF(((-1/2*(9+89^(1/2))^(1/3)
+1/(9+89^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(9+89^(1/2))^(1/3)+2/3/(9+89^(1/2))^(1/3)))*x/(-1/6*(9+89^(1/2))^(1/3
)+1/3/(9+89^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(9+89^(1/2))^(1/3)+2/3/(9+89^(1/2))^(1/3)))/(x-1/3*(9+89^(1/2))^(1
/3)+2/3/(9+89^(1/2))^(1/3)))^(1/2),((1/2*(9+89^(1/2))^(1/3)-1/(9+89^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(9+89^(1/2
))^(1/3)+2/3/(9+89^(1/2))^(1/3)))*(1/6*(9+89^(1/2))^(1/3)-1/3/(9+89^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(9+89^(1/2
))^(1/3)+2/3/(9+89^(1/2))^(1/3)))/(1/6*(9+89^(1/2))^(1/3)-1/3/(9+89^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(9+89^(1/2
))^(1/3)+2/3/(9+89^(1/2))^(1/3)))/(1/2*(9+89^(1/2))^(1/3)-1/(9+89^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(9+89^(1/2))
^(1/3)+2/3/(9+89^(1/2))^(1/3))))^(1/2))-(_alpha^2+2)*((9+89^(1/2))^(1/3)-2/(9+89^(1/2))^(1/3))*EllipticPi(((-1
/2*(9+89^(1/2))^(1/3)+1/(9+89^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(9+89^(1/2))^(1/3)+2/3/(9+89^(1/2))^(1/3)))*x/(-
1/6*(9+89^(1/2))^(1/3)+1/3/(9+89^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(9+89^(1/2))^(1/3)+2/3/(9+89^(1/2))^(1/3)))/(
x-1/3*(9+89^(1/2))^(1/3)+2/3/(9+89^(1/2))^(1/3)))^(1/2),1/2+1/16*I*3^(1/2)*(9+89^(1/2))^(2/3)*_alpha^2-1/712*I
*3^(1/2)*(9+89^(1/2))^(1/3)*89^(1/2)*_alpha^2-1/36*I*(9+89^(1/2))^(1/3)*3^(1/2)-1/72*I*(9+89^(1/2))^(1/3)*3^(1
/2)*_alpha^2-1/4*(9+89^(1/2))^(2/3)+1/12*I*(9+89^(1/2))^(2/3)*3^(1/2)-85/12816*I*_alpha^2*(9+89^(1/2))^(2/3)*8
9^(1/2)*3^(1/2)-1/356*(9+89^(1/2))^(1/3)*89^(1/2)-29/3204*I*(9+89^(1/2))^(2/3)*89^(1/2)*3^(1/2)+29/1068*(9+89^
(1/2))^(2/3)*89^(1/2)+85/4272*_alpha^2*(9+89^(1/2))^(2/3)*89^(1/2)+1/801*I*3^(1/2)*89^(1/2)*_alpha^2-3/16*_alp
ha^2*(9+89^(1/2))^(2/3)-3/712*_alpha^2*(9+89^(1/2))^(1/3)*89^(1/2)-1/24*_alpha^2*(9+89^(1/2))^(1/3)-1/12*(9+89
^(1/2))^(1/3)-1/1068*I*(9+89^(1/2))^(1/3)*3^(1/2)*89^(1/2)+31/1602*I*3^(1/2)*89^(1/2),((1/2*(9+89^(1/2))^(1/3)
-1/(9+89^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(9+89^(1/2))^(1/3)+2/3/(9+89^(1/2))^(1/3)))*(1/6*(9+89^(1/2))^(1/3)-1
/3/(9+89^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(9+89^(1/2))^(1/3)+2/3/(9+89^(1/2))^(1/3)))/(1/6*(9+89^(1/2))^(1/3)-1
/3/(9+89^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(9+89^(1/2))^(1/3)+2/3/(9+89^(1/2))^(1/3)))/(1/2*(9+89^(1/2))^(1/3)-1
/(9+89^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(9+89^(1/2))^(1/3)+2/3/(9+89^(1/2))^(1/3))))^(1/2))),_alpha=RootOf(_Z^3
+2*_Z-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((-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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Fricas [A]
time = 0.39, 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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Sympy [F(-1)] Timed out
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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Giac [F]
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((-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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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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