\(\int \frac {x^7}{(-2+3 x^2) \sqrt [4]{-1+3 x^2}} \, dx\) [1042]

Optimal result

Integrand size = 24, antiderivative size = 78 \[ \int \frac {x^7}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {14}{243} \left (-1+3 x^2\right )^{3/4}+\frac {8}{567} \left (-1+3 x^2\right )^{7/4}+\frac {2}{891} \left (-1+3 x^2\right )^{11/4}+\frac {8}{81} \arctan \left (\sqrt [4]{-1+3 x^2}\right )-\frac {8}{81} \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right ) \]

[Out]

14/243*(3*x^2-1)^(3/4)+8/567*(3*x^2-1)^(7/4)+2/891*(3*x^2-1)^(11/4)+8/81*arctan((3*x^2-1)^(1/4))-8/81*arctanh(
(3*x^2-1)^(1/4))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 90, 65, 304, 209, 212} \[ \int \frac {x^7}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {8}{81} \arctan \left (\sqrt [4]{3 x^2-1}\right )-\frac {8}{81} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )+\frac {2}{891} \left (3 x^2-1\right )^{11/4}+\frac {8}{567} \left (3 x^2-1\right )^{7/4}+\frac {14}{243} \left (3 x^2-1\right )^{3/4} \]

[In]

Int[x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]

[Out]

(14*(-1 + 3*x^2)^(3/4))/243 + (8*(-1 + 3*x^2)^(7/4))/567 + (2*(-1 + 3*x^2)^(11/4))/891 + (8*ArcTan[(-1 + 3*x^2
)^(1/4)])/81 - (8*ArcTanh[(-1 + 3*x^2)^(1/4)])/81

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 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 209

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

Rule 212

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

Rule 304

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {7}{27 \sqrt [4]{-1+3 x}}+\frac {8}{27 (-2+3 x) \sqrt [4]{-1+3 x}}+\frac {4}{27} (-1+3 x)^{3/4}+\frac {1}{27} (-1+3 x)^{7/4}\right ) \, dx,x,x^2\right ) \\ & = \frac {14}{243} \left (-1+3 x^2\right )^{3/4}+\frac {8}{567} \left (-1+3 x^2\right )^{7/4}+\frac {2}{891} \left (-1+3 x^2\right )^{11/4}+\frac {4}{27} \text {Subst}\left (\int \frac {1}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right ) \\ & = \frac {14}{243} \left (-1+3 x^2\right )^{3/4}+\frac {8}{567} \left (-1+3 x^2\right )^{7/4}+\frac {2}{891} \left (-1+3 x^2\right )^{11/4}+\frac {16}{81} \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right ) \\ & = \frac {14}{243} \left (-1+3 x^2\right )^{3/4}+\frac {8}{567} \left (-1+3 x^2\right )^{7/4}+\frac {2}{891} \left (-1+3 x^2\right )^{11/4}-\frac {8}{81} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac {8}{81} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right ) \\ & = \frac {14}{243} \left (-1+3 x^2\right )^{3/4}+\frac {8}{567} \left (-1+3 x^2\right )^{7/4}+\frac {2}{891} \left (-1+3 x^2\right )^{11/4}+\frac {8}{81} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {8}{81} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.73 \[ \int \frac {x^7}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2 \left (\left (-1+3 x^2\right )^{3/4} \left (428+270 x^2+189 x^4\right )+924 \arctan \left (\sqrt [4]{-1+3 x^2}\right )-924 \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )\right )}{18711} \]

[In]

Integrate[x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]

[Out]

(2*((-1 + 3*x^2)^(3/4)*(428 + 270*x^2 + 189*x^4) + 924*ArcTan[(-1 + 3*x^2)^(1/4)] - 924*ArcTanh[(-1 + 3*x^2)^(
1/4)]))/18711

Maple [A] (verified)

Time = 4.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {4 \ln \left (-1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{81}-\frac {4 \ln \left (1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{81}+\frac {\left (378 x^{4}+540 x^{2}+856\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}}{18711}+\frac {8 \arctan \left (\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{81}\)	\(65\)
trager	\(\left (\frac {2}{99} x^{4}+\frac {20}{693} x^{2}+\frac {856}{18711}\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}+\frac {4 \ln \left (\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{81}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}+2 \sqrt {3 x^{2}-1}-3 x^{2}}{3 x^{2}-2}\right )}{81}\)	\(146\)
risch	\(\frac {2 \left (189 x^{4}+270 x^{2}+428\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}}{18711}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {3 x^{2}-1}+3 x^{2}}{3 x^{2}-2}\right )}{81}+\frac {4 \ln \left (\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{81}\)	\(148\)

[In]

int(x^7/(3*x^2-2)/(3*x^2-1)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/81*ln(-1+(3*x^2-1)^(1/4))-4/81*ln(1+(3*x^2-1)^(1/4))+1/18711*(378*x^4+540*x^2+856)*(3*x^2-1)^(3/4)+8/81*arct
an((3*x^2-1)^(1/4))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.82 \[ \int \frac {x^7}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2}{18711} \, {\left (189 \, x^{4} + 270 \, x^{2} + 428\right )} {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} + \frac {8}{81} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]

[In]

integrate(x^7/(3*x^2-2)/(3*x^2-1)^(1/4),x, algorithm="fricas")

[Out]

2/18711*(189*x^4 + 270*x^2 + 428)*(3*x^2 - 1)^(3/4) + 8/81*arctan((3*x^2 - 1)^(1/4)) - 4/81*log((3*x^2 - 1)^(1
/4) + 1) + 4/81*log((3*x^2 - 1)^(1/4) - 1)

Sympy [A] (verification not implemented)

Time = 6.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.13 \[ \int \frac {x^7}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2 \left (3 x^{2} - 1\right )^{\frac {11}{4}}}{891} + \frac {8 \left (3 x^{2} - 1\right )^{\frac {7}{4}}}{567} + \frac {14 \left (3 x^{2} - 1\right )^{\frac {3}{4}}}{243} + \frac {4 \log {\left (\sqrt [4]{3 x^{2} - 1} - 1 \right )}}{81} - \frac {4 \log {\left (\sqrt [4]{3 x^{2} - 1} + 1 \right )}}{81} + \frac {8 \operatorname {atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{81} \]

[In]

integrate(x**7/(3*x**2-2)/(3*x**2-1)**(1/4),x)

[Out]

2*(3*x**2 - 1)**(11/4)/891 + 8*(3*x**2 - 1)**(7/4)/567 + 14*(3*x**2 - 1)**(3/4)/243 + 4*log((3*x**2 - 1)**(1/4
) - 1)/81 - 4*log((3*x**2 - 1)**(1/4) + 1)/81 + 8*atan((3*x**2 - 1)**(1/4))/81

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95 \[ \int \frac {x^7}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2}{891} \, {\left (3 \, x^{2} - 1\right )}^{\frac {11}{4}} + \frac {8}{567} \, {\left (3 \, x^{2} - 1\right )}^{\frac {7}{4}} + \frac {14}{243} \, {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} + \frac {8}{81} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]

[In]

integrate(x^7/(3*x^2-2)/(3*x^2-1)^(1/4),x, algorithm="maxima")

[Out]

2/891*(3*x^2 - 1)^(11/4) + 8/567*(3*x^2 - 1)^(7/4) + 14/243*(3*x^2 - 1)^(3/4) + 8/81*arctan((3*x^2 - 1)^(1/4))
- 4/81*log((3*x^2 - 1)^(1/4) + 1) + 4/81*log((3*x^2 - 1)^(1/4) - 1)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96 \[ \int \frac {x^7}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2}{891} \, {\left (3 \, x^{2} - 1\right )}^{\frac {11}{4}} + \frac {8}{567} \, {\left (3 \, x^{2} - 1\right )}^{\frac {7}{4}} + \frac {14}{243} \, {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} + \frac {8}{81} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4}{81} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]

[In]

integrate(x^7/(3*x^2-2)/(3*x^2-1)^(1/4),x, algorithm="giac")

[Out]

2/891*(3*x^2 - 1)^(11/4) + 8/567*(3*x^2 - 1)^(7/4) + 14/243*(3*x^2 - 1)^(3/4) + 8/81*arctan((3*x^2 - 1)^(1/4))
- 4/81*log((3*x^2 - 1)^(1/4) + 1) + 4/81*log(abs((3*x^2 - 1)^(1/4) - 1))

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79 \[ \int \frac {x^7}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {8\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{81}+\frac {14\,{\left (3\,x^2-1\right )}^{3/4}}{243}+\frac {8\,{\left (3\,x^2-1\right )}^{7/4}}{567}+\frac {2\,{\left (3\,x^2-1\right )}^{11/4}}{891}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,8{}\mathrm {i}}{81} \]

[In]

int(x^7/((3*x^2 - 1)^(1/4)*(3*x^2 - 2)),x)

[Out]

(8*atan((3*x^2 - 1)^(1/4)))/81 + (atan((3*x^2 - 1)^(1/4)*1i)*8i)/81 + (14*(3*x^2 - 1)^(3/4))/243 + (8*(3*x^2 -
1)^(7/4))/567 + (2*(3*x^2 - 1)^(11/4))/891