\(\int \frac {x^3}{\sqrt [4]{2-3 x^2} (4-3 x^2)} \, dx\) [1033]

Optimal result

Integrand size = 24, antiderivative size = 106 \[ \int \frac {x^3}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2}{27} \left (2-3 x^2\right )^{3/4}+\frac {2}{9} \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {2}{9} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \]

[Out]

2/27*(-3*x^2+2)^(3/4)+2/9*2^(1/4)*arctan(1/2*(2^(1/2)-(-3*x^2+2)^(1/2))*2^(1/4)/(-3*x^2+2)^(1/4))+2/9*2^(1/4)*
arctanh(1/2*(2^(1/2)+(-3*x^2+2)^(1/2))*2^(1/4)/(-3*x^2+2)^(1/4))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {451, 267, 450} \[ \int \frac {x^3}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2}{9} \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {2}{9} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2-3 x^2}+\sqrt {2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {2}{27} \left (2-3 x^2\right )^{3/4} \]

[In]

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

[Out]

(2*(2 - 3*x^2)^(3/4))/27 + (2*2^(1/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))])/9 + (2*
2^(1/4)*ArcTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))])/9

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 450

Int[(x_)/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[(-(Sqrt[2]*Rt[a, 4]*d)^(-1))*A
rcTan[(Rt[a, 4]^2 - Sqrt[a + b*x^2])/(Sqrt[2]*Rt[a, 4]*(a + b*x^2)^(1/4))], x] - Simp[(1/(Sqrt[2]*Rt[a, 4]*d))
*ArcTanh[(Rt[a, 4]^2 + Sqrt[a + b*x^2])/(Sqrt[2]*Rt[a, 4]*(a + b*x^2)^(1/4))], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c - 2*a*d, 0] && PosQ[a]

Rule 451

Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegrand[x^m/((a +
b*x^2)^(1/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && IntegerQ[m] && (PosQ[a]
|| IntegerQ[m/2])

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {x}{3 \sqrt [4]{2-3 x^2}}+\frac {4 x}{3 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {x}{\sqrt [4]{2-3 x^2}} \, dx\right )+\frac {4}{3} \int \frac {x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx \\ & = \frac {2}{27} \left (2-3 x^2\right )^{3/4}+\frac {2}{9} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {2}{9} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2}{27} \left (\left (2-3 x^2\right )^{3/4}+3 \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+3 \sqrt [4]{2} \text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )\right ) \]

[In]

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

[Out]

(2*((2 - 3*x^2)^(3/4) + 3*2^(1/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))] + 3*2^(1/4)*
ArcTanh[(2*(4 - 6*x^2)^(1/4))/(2 + Sqrt[4 - 6*x^2])]))/27

Maple [A] (verified)

Time = 4.64 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(\frac {2 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}{27}-\frac {\ln \left (\frac {-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}{2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}\right ) 2^{\frac {1}{4}}}{9}-\frac {2 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right ) 2^{\frac {1}{4}}}{9}-\frac {2 \arctan \left (-1+2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}\right ) 2^{\frac {1}{4}}}{9}\)	\(117\)
trager	\(\frac {2 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}{27}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{9}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{9}\)	\(198\)
risch	\(-\frac {2 \left (3 x^{2}-2\right )}{27 \left (-3 x^{2}+2\right )^{\frac {1}{4}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{9}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{9}\)	\(205\)

[In]

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

[Out]

2/27*(-3*x^2+2)^(3/4)-1/9*ln((-2^(3/4)*(-3*x^2+2)^(1/4)+2^(1/2)+(-3*x^2+2)^(1/2))/(2^(3/4)*(-3*x^2+2)^(1/4)+2^
(1/2)+(-3*x^2+2)^(1/2)))*2^(1/4)-2/9*arctan(2^(1/4)*(-3*x^2+2)^(1/4)+1)*2^(1/4)-2/9*arctan(-1+2^(1/4)*(-3*x^2+
2)^(1/4))*2^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.96 \[ \int \frac {x^3}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {1}{9} \, \left (-8\right )^{\frac {1}{4}} \log \left (\left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {1}{9} i \, \left (-8\right )^{\frac {1}{4}} \log \left (i \, \left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {1}{9} i \, \left (-8\right )^{\frac {1}{4}} \log \left (-i \, \left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {1}{9} \, \left (-8\right )^{\frac {1}{4}} \log \left (-\left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {2}{27} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

-1/9*(-8)^(1/4)*log((-8)^(3/4) + 4*(-3*x^2 + 2)^(1/4)) + 1/9*I*(-8)^(1/4)*log(I*(-8)^(3/4) + 4*(-3*x^2 + 2)^(1
/4)) - 1/9*I*(-8)^(1/4)*log(-I*(-8)^(3/4) + 4*(-3*x^2 + 2)^(1/4)) + 1/9*(-8)^(1/4)*log(-(-8)^(3/4) + 4*(-3*x^2
+ 2)^(1/4)) + 2/27*(-3*x^2 + 2)^(3/4)

Sympy [F]

\[ \int \frac {x^3}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{3}}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, dx \]

[In]

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

[Out]

-Integral(x**3/(3*x**2*(2 - 3*x**2)**(1/4) - 4*(2 - 3*x**2)**(1/4)), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.22 \[ \int \frac {x^3}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {2}{9} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {2}{9} \cdot 2^{\frac {1}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{9} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {1}{9} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{27} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

-2/9*2^(1/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 2/9*2^(1/4)*arctan(-1/2*2^(1/4)*(2^(3/4) -
2*(-3*x^2 + 2)^(1/4))) + 1/9*2^(1/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 1/9*2^(1/
4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*(-3*x^2 + 2)^(3/4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.22 \[ \int \frac {x^3}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {2}{9} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {2}{9} \cdot 2^{\frac {1}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{9} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {1}{9} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{27} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

-2/9*2^(1/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 2/9*2^(1/4)*arctan(-1/2*2^(1/4)*(2^(3/4) -
2*(-3*x^2 + 2)^(1/4))) + 1/9*2^(1/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 1/9*2^(1/
4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*(-3*x^2 + 2)^(3/4)

Mupad [B] (verification not implemented)

Time = 5.43 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.57 \[ \int \frac {x^3}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2\,{\left (2-3\,x^2\right )}^{3/4}}{27}+2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {2}{9}+\frac {2}{9}{}\mathrm {i}\right )+2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {2}{9}-\frac {2}{9}{}\mathrm {i}\right ) \]

[In]

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

[Out]

(2*(2 - 3*x^2)^(3/4))/27 - 2^(1/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 + 1i/2))*(2/9 + 2i/9) - 2^(1/4)*atan(2^
(1/4)*(2 - 3*x^2)^(1/4)*(1/2 - 1i/2))*(2/9 - 2i/9)