\(\int \frac {x}{(a-b x^4)^{3/4}} \, dx\) [1245]

Optimal result

Integrand size = 14, antiderivative size = 59 \[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {\sqrt {a} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{\sqrt {b} \left (a-b x^4\right )^{3/4}} \]

[Out]

(1-b*x^4/a)^(3/4)*(cos(1/2*arcsin(x^2*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arcsin(x^2*b^(1/2)/a^(1/2)))*Elliptic
F(sin(1/2*arcsin(x^2*b^(1/2)/a^(1/2))),2^(1/2))*a^(1/2)/(-b*x^4+a)^(3/4)/b^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {281, 239, 238} \[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {\sqrt {a} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{\sqrt {b} \left (a-b x^4\right )^{3/4}} \]

[In]

Int[x/(a - b*x^4)^(3/4),x]

[Out]

(Sqrt[a]*(1 - (b*x^4)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(Sqrt[b]*(a - b*x^4)^(3/4))

Rule 238

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2]))*EllipticF[(1/2)*ArcSin[Rt[-b/a,
2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rule 239

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right ) \\ & = \frac {\left (1-\frac {b x^4}{a}\right )^{3/4} \text {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{2 \left (a-b x^4\right )^{3/4}} \\ & = \frac {\sqrt {a} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \left (a-b x^4\right )^{3/4}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 7.45 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {x^2 \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {b x^4}{a}\right )}{2 \left (a-b x^4\right )^{3/4}} \]

[In]

Integrate[x/(a - b*x^4)^(3/4),x]

[Out]

(x^2*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^4)/a])/(2*(a - b*x^4)^(3/4))

Maple [F]

\[\int \frac {x}{\left (-b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]

[In]

int(x/(-b*x^4+a)^(3/4),x)

[Out]

int(x/(-b*x^4+a)^(3/4),x)

Fricas [F]

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

[In]

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

[Out]

integral(-(-b*x^4 + a)^(1/4)*x/(b*x^4 - a), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.49 \[ \int \frac {x}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {x^{2} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}}} \]

[In]

integrate(x/(-b*x**4+a)**(3/4),x)

[Out]

x**2*hyper((1/2, 3/4), (3/2,), b*x**4*exp_polar(2*I*pi)/a)/(2*a**(3/4))

Maxima [F]

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

[In]

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

[Out]

integrate(x/(-b*x^4 + a)^(3/4), x)

Giac [F]

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

[In]

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

[Out]

integrate(x/(-b*x^4 + a)^(3/4), x)

Mupad [F(-1)]

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

[In]

int(x/(a - b*x^4)^(3/4),x)

[Out]

int(x/(a - b*x^4)^(3/4), x)