3.1061 \(\int \frac{x^7}{(2-3 x^2)^{3/4} (4-3 x^2)} \, dx\)

Optimal. Leaf size=188 \[ \frac{2}{729} \left (2-3 x^2\right )^{9/4}-\frac{16}{405} \left (2-3 x^2\right )^{5/4}+\frac{56}{81} \sqrt [4]{2-3 x^2}+\frac{8}{81} 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{8}{81} 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{16}{81} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{16}{81} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]

[Out]

(56*(2 - 3*x^2)^(1/4))/81 - (16*(2 - 3*x^2)^(5/4))/405 + (2*(2 - 3*x^2)^(9/4))/729 - (16*2^(3/4)*ArcTan[1 + (4
- 6*x^2)^(1/4)])/81 + (16*2^(3/4)*ArcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)])/81 + (8*2^(3/4)*Log[Sqrt[2] - 2^(3/4
)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/81 - (8*2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x
^2]])/81

________________________________________________________________________________________

Rubi [A]  time = 0.211457, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {443, 261, 266, 43, 444, 63, 211, 1165, 628, 1162, 617, 204} \[ \frac{2}{729} \left (2-3 x^2\right )^{9/4}-\frac{16}{405} \left (2-3 x^2\right )^{5/4}+\frac{56}{81} \sqrt [4]{2-3 x^2}+\frac{8}{81} 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{8}{81} 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{16}{81} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{16}{81} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^7/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

(56*(2 - 3*x^2)^(1/4))/81 - (16*(2 - 3*x^2)^(5/4))/405 + (2*(2 - 3*x^2)^(9/4))/729 - (16*2^(3/4)*ArcTan[1 + (4
- 6*x^2)^(1/4)])/81 + (16*2^(3/4)*ArcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)])/81 + (8*2^(3/4)*Log[Sqrt[2] - 2^(3/4
)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/81 - (8*2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x
^2]])/81

Rule 443

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 261

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 266

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 43

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 444

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 63

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 211

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 1165

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]

Rule 628

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 1162

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 617

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 204

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

Rubi steps

\begin{align*} \int \frac{x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (-\frac{16 x}{27 \left (2-3 x^2\right )^{3/4}}-\frac{4 x^3}{9 \left (2-3 x^2\right )^{3/4}}-\frac{x^5}{3 \left (2-3 x^2\right )^{3/4}}+\frac{64 x}{27 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}\right ) \, dx\\ &=-\left (\frac{1}{3} \int \frac{x^5}{\left (2-3 x^2\right )^{3/4}} \, dx\right )-\frac{4}{9} \int \frac{x^3}{\left (2-3 x^2\right )^{3/4}} \, dx-\frac{16}{27} \int \frac{x}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac{64}{27} \int \frac{x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\\ &=\frac{32}{81} \sqrt [4]{2-3 x^2}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{x^2}{(2-3 x)^{3/4}} \, dx,x,x^2\right )-\frac{2}{9} \operatorname{Subst}\left (\int \frac{x}{(2-3 x)^{3/4}} \, dx,x,x^2\right )+\frac{32}{27} \operatorname{Subst}\left (\int \frac{1}{(2-3 x)^{3/4} (4-3 x)} \, dx,x,x^2\right )\\ &=\frac{32}{81} \sqrt [4]{2-3 x^2}-\frac{1}{6} \operatorname{Subst}\left (\int \left (\frac{4}{9 (2-3 x)^{3/4}}-\frac{4}{9} \sqrt [4]{2-3 x}+\frac{1}{9} (2-3 x)^{5/4}\right ) \, dx,x,x^2\right )-\frac{2}{9} \operatorname{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{128}{81} \operatorname{Subst}\left (\int \frac{1}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac{56}{81} \sqrt [4]{2-3 x^2}-\frac{16}{405} \left (2-3 x^2\right )^{5/4}+\frac{2}{729} \left (2-3 x^2\right )^{9/4}-\frac{1}{81} \left (32 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac{1}{81} \left (32 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac{56}{81} \sqrt [4]{2-3 x^2}-\frac{16}{405} \left (2-3 x^2\right )^{5/4}+\frac{2}{729} \left (2-3 x^2\right )^{9/4}-\frac{1}{81} \left (16 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac{1}{81} \left (16 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac{1}{81} \left (8\ 2^{3/4}\right ) \operatorname{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}{81} \left (8\ 2^{3/4}\right ) \operatorname{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{56}{81} \sqrt [4]{2-3 x^2}-\frac{16}{405} \left (2-3 x^2\right )^{5/4}+\frac{2}{729} \left (2-3 x^2\right )^{9/4}+\frac{8}{81} 2^{3/4} \log \left (\sqrt{2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )-\frac{8}{81} 2^{3/4} \log \left (\sqrt{2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )-\frac{1}{81} \left (16\ 2^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{4-6 x^2}\right )+\frac{1}{81} \left (16\ 2^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{4-6 x^2}\right )\\ &=\frac{56}{81} \sqrt [4]{2-3 x^2}-\frac{16}{405} \left (2-3 x^2\right )^{5/4}+\frac{2}{729} \left (2-3 x^2\right )^{9/4}-\frac{16}{81} 2^{3/4} \tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )+\frac{16}{81} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )+\frac{8}{81} 2^{3/4} \log \left (\sqrt{2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )-\frac{8}{81} 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]  time = 0.0968532, size = 179, normalized size = 0.95 \[ \frac{2 \left (45 \sqrt [4]{2-3 x^2} x^4+156 \sqrt [4]{2-3 x^2} x^2+1136 \sqrt [4]{2-3 x^2}+180\ 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-180\ 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )+360\ 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{4-6 x^2}\right )-360\ 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )\right )}{3645} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

(2*(1136*(2 - 3*x^2)^(1/4) + 156*x^2*(2 - 3*x^2)^(1/4) + 45*x^4*(2 - 3*x^2)^(1/4) + 360*2^(3/4)*ArcTan[1 - (4
- 6*x^2)^(1/4)] - 360*2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)] + 180*2^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1
/4) + Sqrt[2 - 3*x^2]] - 180*2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]]))/3645

________________________________________________________________________________________

Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{7}}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)

[Out]

int(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)

________________________________________________________________________________________

Maxima [A]  time = 1.48939, size = 204, normalized size = 1.09 \begin{align*} \frac{2}{729} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{9}{4}} - \frac{16}{81} \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{16}{81} \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{8}{81} \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{8}{81} \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{16}{405} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{5}{4}} + \frac{56}{81} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="maxima")

[Out]

2/729*(-3*x^2 + 2)^(9/4) - 16/81*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 16/81*2^(3/4)*
arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 8/81*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2)
+ sqrt(-3*x^2 + 2)) + 8/81*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 16/405*(-3*
x^2 + 2)^(5/4) + 56/81*(-3*x^2 + 2)^(1/4)

________________________________________________________________________________________

Fricas [A]  time = 1.70798, size = 622, normalized size = 3.31 \begin{align*} \frac{32}{81} \cdot 2^{\frac{3}{4}} \arctan \left (2^{\frac{1}{4}} \sqrt{2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 1\right ) + \frac{32}{81} \cdot 2^{\frac{3}{4}} \arctan \left (2^{\frac{1}{4}} \sqrt{-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 1\right ) - \frac{8}{81} \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{8}{81} \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}{3645} \,{\left (45 \, x^{4} + 156 \, x^{2} + 1136\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="fricas")

[Out]

32/81*2^(3/4)*arctan(2^(1/4)*sqrt(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 2^(1/4)*(-3*x^2 +
2)^(1/4) - 1) + 32/81*2^(3/4)*arctan(2^(1/4)*sqrt(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) -
2^(1/4)*(-3*x^2 + 2)^(1/4) + 1) - 8/81*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) +
8/81*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/3645*(45*x^4 + 156*x^2 + 1136)
*(-3*x^2 + 2)^(1/4)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{7}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac{3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac{3}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)

[Out]

-Integral(x**7/(3*x**2*(2 - 3*x**2)**(3/4) - 4*(2 - 3*x**2)**(3/4)), x)

________________________________________________________________________________________

Giac [A]  time = 1.20685, size = 216, normalized size = 1.15 \begin{align*} \frac{2}{729} \,{\left (3 \, x^{2} - 2\right )}^{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - \frac{16}{81} \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{16}{81} \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{8}{81} \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{8}{81} \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{16}{405} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{5}{4}} + \frac{56}{81} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="giac")

[Out]

2/729*(3*x^2 - 2)^2*(-3*x^2 + 2)^(1/4) - 16/81*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) -
16/81*2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 8/81*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1
/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 8/81*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2))
- 16/405*(-3*x^2 + 2)^(5/4) + 56/81*(-3*x^2 + 2)^(1/4)