\(\int \frac {x^5}{(2-3 x^2)^{3/4} (4-3 x^2)} \, dx\) [1062]

Optimal result

Integrand size = 24, antiderivative size = 173 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {4}{9} \sqrt [4]{2-3 x^2}-\frac {2}{135} \left (2-3 x^2\right )^{5/4}-\frac {4}{27} 2^{3/4} \arctan \left (1+\sqrt [4]{4-6 x^2}\right )+\frac {4}{27} 2^{3/4} \arctan \left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )+\frac {2}{27} 2^{3/4} \log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )-\frac {2}{27} 2^{3/4} \log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right ) \]

[Out]

4/9*(-3*x^2+2)^(1/4)-2/135*(-3*x^2+2)^(5/4)-4/27*2^(3/4)*arctan(1+(-6*x^2+4)^(1/4))-4/27*2^(3/4)*arctan(-1+2^(
1/4)*(-3*x^2+2)^(1/4))+2/27*2^(3/4)*ln(-2^(3/4)*(-3*x^2+2)^(1/4)+2^(1/2)+(-3*x^2+2)^(1/2))-2/27*2^(3/4)*ln(2^(
3/4)*(-3*x^2+2)^(1/4)+2^(1/2)+(-3*x^2+2)^(1/2))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {454, 267, 272, 45, 455, 65, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {4}{27} 2^{3/4} \arctan \left (\sqrt [4]{4-6 x^2}+1\right )+\frac {4}{27} 2^{3/4} \arctan \left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )-\frac {2}{135} \left (2-3 x^2\right )^{5/4}+\frac {4}{9} \sqrt [4]{2-3 x^2}+\frac {2}{27} 2^{3/4} \log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )-\frac {2}{27} 2^{3/4} \log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right ) \]

[In]

Int[x^5/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

(4*(2 - 3*x^2)^(1/4))/9 - (2*(2 - 3*x^2)^(5/4))/135 - (4*2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)])/27 + (4*2^(3/4
)*ArcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)])/27 + (2*2^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*
x^2]])/27 - (2*2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/27

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

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 272

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

Rule 454

Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegrand[x^m/((a +
b*x^2)^(3/4)*(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])

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {4 x}{9 \left (2-3 x^2\right )^{3/4}}-\frac {x^3}{3 \left (2-3 x^2\right )^{3/4}}+\frac {16 x}{9 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {x^3}{\left (2-3 x^2\right )^{3/4}} \, dx\right )-\frac {4}{9} \int \frac {x}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac {16}{9} \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx \\ & = \frac {8}{27} \sqrt [4]{2-3 x^2}-\frac {1}{6} \text {Subst}\left (\int \frac {x}{(2-3 x)^{3/4}} \, dx,x,x^2\right )+\frac {8}{9} \text {Subst}\left (\int \frac {1}{(2-3 x)^{3/4} (4-3 x)} \, dx,x,x^2\right ) \\ & = \frac {8}{27} \sqrt [4]{2-3 x^2}-\frac {1}{6} \text {Subst}\left (\int \left (\frac {2}{3 (2-3 x)^{3/4}}-\frac {1}{3} \sqrt [4]{2-3 x}\right ) \, dx,x,x^2\right )-\frac {32}{27} \text {Subst}\left (\int \frac {1}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right ) \\ & = \frac {4}{9} \sqrt [4]{2-3 x^2}-\frac {2}{135} \left (2-3 x^2\right )^{5/4}-\frac {1}{27} \left (8 \sqrt {2}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac {1}{27} \left (8 \sqrt {2}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right ) \\ & = \frac {4}{9} \sqrt [4]{2-3 x^2}-\frac {2}{135} \left (2-3 x^2\right )^{5/4}-\frac {1}{27} \left (4 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac {1}{27} \left (4 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}+2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac {1}{27} \left (2\ 2^{3/4}\right ) \text {Subst}\left (\int \frac {2^{3/4}+2 x}{-\sqrt {2}-2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac {1}{27} \left (2\ 2^{3/4}\right ) \text {Subst}\left (\int \frac {2^{3/4}-2 x}{-\sqrt {2}+2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right ) \\ & = \frac {4}{9} \sqrt [4]{2-3 x^2}-\frac {2}{135} \left (2-3 x^2\right )^{5/4}+\frac {2}{27} 2^{3/4} \log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )-\frac {2}{27} 2^{3/4} \log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )-\frac {1}{27} \left (4\ 2^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{4-6 x^2}\right )+\frac {1}{27} \left (4\ 2^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{4-6 x^2}\right ) \\ & = \frac {4}{9} \sqrt [4]{2-3 x^2}-\frac {2}{135} \left (2-3 x^2\right )^{5/4}-\frac {4}{27} 2^{3/4} \tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )+\frac {4}{27} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )+\frac {2}{27} 2^{3/4} \log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )-\frac {2}{27} 2^{3/4} \log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.61 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {1}{135} \left (2 \sqrt [4]{2-3 x^2} \left (28+3 x^2\right )+20\ 2^{3/4} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-20\ 2^{3/4} \text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )\right ) \]

[In]

Integrate[x^5/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

(2*(2 - 3*x^2)^(1/4)*(28 + 3*x^2) + 20*2^(3/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))]
- 20*2^(3/4)*ArcTanh[(2*(4 - 6*x^2)^(1/4))/(2 + Sqrt[4 - 6*x^2])])/135

Maple [A] (verified)

Time = 4.98 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(\frac {2 x^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{45}+\frac {56 \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{135}-\frac {2 \ln \left (\frac {2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}{-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}\right ) 2^{\frac {3}{4}}}{27}-\frac {4 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right ) 2^{\frac {3}{4}}}{27}-\frac {4 \,2^{\frac {3}{4}} \arctan \left (-1+2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}\right )}{27}\)	\(131\)
trager	\(\left (\frac {2 x^{2}}{45}+\frac {56}{135}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+3 x^{2}}{3 x^{2}-4}\right )}{27}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}-3 x^{2}}{3 x^{2}-4}\right )}{27}\)	\(207\)
risch	\(-\frac {2 \left (3 x^{2}+28\right ) \left (3 x^{2}-2\right )}{135 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}-\frac {\left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {3}{4}}-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{4}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}\, x^{2}-27 x^{6}+24 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}+36 x^{4}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}}-12 x^{2}}{\left (3 x^{2}-4\right ) \left (3 x^{2}-2\right )^{2}}\right )}{27}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {3}{4}}-6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}\, x^{2}+18 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{4}-27 x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}-24 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{2}+36 x^{4}+8 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}}-12 x^{2}}{\left (3 x^{2}-4\right ) \left (3 x^{2}-2\right )^{2}}\right )}{27}\right ) {\left (-\left (3 x^{2}-2\right )^{3}\right )}^{\frac {1}{4}}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}}}\)	\(535\)

[In]

int(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x,method=_RETURNVERBOSE)

[Out]

2/45*x^2*(-3*x^2+2)^(1/4)+56/135*(-3*x^2+2)^(1/4)-2/27*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^(3/4)-4/27*arctan(2^(1/4)*(-3*x^2+2)^(1/4)+1)*2^(3/4)-
4/27*2^(3/4)*arctan(-1+2^(1/4)*(-3*x^2+2)^(1/4))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.58 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {2}{135} \, {\left (3 \, x^{2} + 28\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - \frac {4}{27} \, \left (-2\right )^{\frac {1}{4}} \log \left (\left (-2\right )^{\frac {1}{4}} + {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {4}{27} i \, \left (-2\right )^{\frac {1}{4}} \log \left (i \, \left (-2\right )^{\frac {1}{4}} + {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {4}{27} i \, \left (-2\right )^{\frac {1}{4}} \log \left (-i \, \left (-2\right )^{\frac {1}{4}} + {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {4}{27} \, \left (-2\right )^{\frac {1}{4}} \log \left (-\left (-2\right )^{\frac {1}{4}} + {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) \]

[In]

integrate(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="fricas")

[Out]

2/135*(3*x^2 + 28)*(-3*x^2 + 2)^(1/4) - 4/27*(-2)^(1/4)*log((-2)^(1/4) + (-3*x^2 + 2)^(1/4)) - 4/27*I*(-2)^(1/
4)*log(I*(-2)^(1/4) + (-3*x^2 + 2)^(1/4)) + 4/27*I*(-2)^(1/4)*log(-I*(-2)^(1/4) + (-3*x^2 + 2)^(1/4)) + 4/27*(
-2)^(1/4)*log(-(-2)^(1/4) + (-3*x^2 + 2)^(1/4))

Sympy [F]

\[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{5}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \]

[In]

integrate(x**5/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)

[Out]

-Integral(x**5/(3*x**2*(2 - 3*x**2)**(3/4) - 4*(2 - 3*x**2)**(3/4)), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {4}{27} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {4}{27} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {2}{27} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{27} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {2}{135} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {5}{4}} + \frac {4}{9} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} \]

[In]

integrate(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="maxima")

[Out]

-4/27*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 4/27*2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4)
- 2*(-3*x^2 + 2)^(1/4))) - 2/27*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*2
^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 2/135*(-3*x^2 + 2)^(5/4) + 4/9*(-3*x^2
+ 2)^(1/4)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {4}{27} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {4}{27} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {2}{27} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{27} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {2}{135} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {5}{4}} + \frac {4}{9} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} \]

[In]

integrate(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="giac")

[Out]

-4/27*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 4/27*2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4)
- 2*(-3*x^2 + 2)^(1/4))) - 2/27*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*2
^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 2/135*(-3*x^2 + 2)^(5/4) + 4/9*(-3*x^2
+ 2)^(1/4)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.41 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {4\,{\left (2-3\,x^2\right )}^{1/4}}{9}-\frac {2\,{\left (2-3\,x^2\right )}^{5/4}}{135}+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {4}{27}-\frac {4}{27}{}\mathrm {i}\right )+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {4}{27}+\frac {4}{27}{}\mathrm {i}\right ) \]

[In]

int(-x^5/((2 - 3*x^2)^(3/4)*(3*x^2 - 4)),x)

[Out]

(4*(2 - 3*x^2)^(1/4))/9 - 2^(3/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 + 1i/2))*(4/27 - 4i/27) - 2^(3/4)*atan(2
^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 - 1i/2))*(4/27 + 4i/27) - (2*(2 - 3*x^2)^(5/4))/135