\(\int \frac {1}{x^6 (-2+3 x^2)^{3/4}} \, dx\) [911]

Optimal result

Integrand size = 15, antiderivative size = 140 \[ \int \frac {1}{x^6 \left (-2+3 x^2\right )^{3/4}} \, dx=\frac {\sqrt [4]{-2+3 x^2}}{10 x^5}+\frac {9 \sqrt [4]{-2+3 x^2}}{40 x^3}+\frac {27 \sqrt [4]{-2+3 x^2}}{32 x}+\frac {27 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{64 \sqrt [4]{2} x} \]

[Out]

1/10*(3*x^2-2)^(1/4)/x^5+9/40*(3*x^2-2)^(1/4)/x^3+27/32*(3*x^2-2)^(1/4)/x+27/128*2^(3/4)*(cos(2*arctan(1/2*(3*
x^2-2)^(1/4)*2^(3/4)))^2)^(1/2)/cos(2*arctan(1/2*(3*x^2-2)^(1/4)*2^(3/4)))*EllipticF(sin(2*arctan(1/2*(3*x^2-2
)^(1/4)*2^(3/4))),1/2*2^(1/2))*(2^(1/2)+(3*x^2-2)^(1/2))*(x^2/(2^(1/2)+(3*x^2-2)^(1/2))^2)^(1/2)/x*3^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {331, 240, 226} \[ \int \frac {1}{x^6 \left (-2+3 x^2\right )^{3/4}} \, dx=\frac {27 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{64 \sqrt [4]{2} x}+\frac {27 \sqrt [4]{3 x^2-2}}{32 x}+\frac {\sqrt [4]{3 x^2-2}}{10 x^5}+\frac {9 \sqrt [4]{3 x^2-2}}{40 x^3} \]

[In]

Int[1/(x^6*(-2 + 3*x^2)^(3/4)),x]

[Out]

(-2 + 3*x^2)^(1/4)/(10*x^5) + (9*(-2 + 3*x^2)^(1/4))/(40*x^3) + (27*(-2 + 3*x^2)^(1/4))/(32*x) + (27*Sqrt[3]*S
qrt[x^2/(Sqrt[2] + Sqrt[-2 + 3*x^2])^2]*(Sqrt[2] + Sqrt[-2 + 3*x^2])*EllipticF[2*ArcTan[(-2 + 3*x^2)^(1/4)/2^(
1/4)], 1/2])/(64*2^(1/4)*x)

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 240

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[2*(Sqrt[(-b)*(x^2/a)]/(b*x)), Subst[Int[1/Sqrt[1 - x^4/a],
x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-2+3 x^2}}{10 x^5}+\frac {27}{20} \int \frac {1}{x^4 \left (-2+3 x^2\right )^{3/4}} \, dx \\ & = \frac {\sqrt [4]{-2+3 x^2}}{10 x^5}+\frac {9 \sqrt [4]{-2+3 x^2}}{40 x^3}+\frac {27}{16} \int \frac {1}{x^2 \left (-2+3 x^2\right )^{3/4}} \, dx \\ & = \frac {\sqrt [4]{-2+3 x^2}}{10 x^5}+\frac {9 \sqrt [4]{-2+3 x^2}}{40 x^3}+\frac {27 \sqrt [4]{-2+3 x^2}}{32 x}+\frac {81}{64} \int \frac {1}{\left (-2+3 x^2\right )^{3/4}} \, dx \\ & = \frac {\sqrt [4]{-2+3 x^2}}{10 x^5}+\frac {9 \sqrt [4]{-2+3 x^2}}{40 x^3}+\frac {27 \sqrt [4]{-2+3 x^2}}{32 x}+\frac {\left (27 \sqrt {\frac {3}{2}} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{32 x} \\ & = \frac {\sqrt [4]{-2+3 x^2}}{10 x^5}+\frac {9 \sqrt [4]{-2+3 x^2}}{40 x^3}+\frac {27 \sqrt [4]{-2+3 x^2}}{32 x}+\frac {27 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{64 \sqrt [4]{2} x} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.34 \[ \int \frac {1}{x^6 \left (-2+3 x^2\right )^{3/4}} \, dx=-\frac {\left (1-\frac {3 x^2}{2}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},-\frac {3}{2},\frac {3 x^2}{2}\right )}{5 x^5 \left (-2+3 x^2\right )^{3/4}} \]

[In]

Integrate[1/(x^6*(-2 + 3*x^2)^(3/4)),x]

[Out]

-1/5*((1 - (3*x^2)/2)^(3/4)*Hypergeometric2F1[-5/2, 3/4, -3/2, (3*x^2)/2])/(x^5*(-2 + 3*x^2)^(3/4))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.30

method	result	size
meijerg	\(-\frac {2^{\frac {1}{4}} {\left (-\operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )\right )}^{\frac {3}{4}} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{2},\frac {3}{4};-\frac {3}{2};\frac {3 x^{2}}{2}\right )}{10 \operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )^{\frac {3}{4}} x^{5}}\)	\(42\)
risch	\(\frac {405 x^{6}-162 x^{4}-24 x^{2}-32}{160 x^{5} \left (3 x^{2}-2\right )^{\frac {3}{4}}}+\frac {81 \,2^{\frac {1}{4}} {\left (-\operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )\right )}^{\frac {3}{4}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {3 x^{2}}{2}\right )}{128 \operatorname {signum}\left (-1+\frac {3 x^{2}}{2}\right )^{\frac {3}{4}}}\)	\(72\)

[In]

int(1/x^6/(3*x^2-2)^(3/4),x,method=_RETURNVERBOSE)

[Out]

-1/10*2^(1/4)/signum(-1+3/2*x^2)^(3/4)*(-signum(-1+3/2*x^2))^(3/4)/x^5*hypergeom([-5/2,3/4],[-3/2],3/2*x^2)

Fricas [F]

\[ \int \frac {1}{x^6 \left (-2+3 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}} x^{6}} \,d x } \]

[In]

integrate(1/x^6/(3*x^2-2)^(3/4),x, algorithm="fricas")

[Out]

integral((3*x^2 - 2)^(1/4)/(3*x^8 - 2*x^6), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^6 \left (-2+3 x^2\right )^{3/4}} \, dx=\frac {\sqrt [4]{2} e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{10 x^{5}} \]

[In]

integrate(1/x**6/(3*x**2-2)**(3/4),x)

[Out]

2**(1/4)*exp(I*pi/4)*hyper((-5/2, 3/4), (-3/2,), 3*x**2/2)/(10*x**5)

Maxima [F]

\[ \int \frac {1}{x^6 \left (-2+3 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}} x^{6}} \,d x } \]

[In]

integrate(1/x^6/(3*x^2-2)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((3*x^2 - 2)^(3/4)*x^6), x)

Giac [F]

\[ \int \frac {1}{x^6 \left (-2+3 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}} x^{6}} \,d x } \]

[In]

integrate(1/x^6/(3*x^2-2)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((3*x^2 - 2)^(3/4)*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \left (-2+3 x^2\right )^{3/4}} \, dx=\int \frac {1}{x^6\,{\left (3\,x^2-2\right )}^{3/4}} \,d x \]

[In]

int(1/(x^6*(3*x^2 - 2)^(3/4)),x)

[Out]

int(1/(x^6*(3*x^2 - 2)^(3/4)), x)