\(\int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx\) [824]

Optimal result

Integrand size = 15, antiderivative size = 110 \[ \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx=-\frac {\sqrt {a+b x^4}}{3 a x^3}-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+b x^4}} \]

[Out]

-1/3*(b*x^4+a)^(1/2)/a/x^3-1/6*b^(3/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/
4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(
1/2))^2)^(1/2)/a^(5/4)/(b*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {331, 226} \[ \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx=-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+b x^4}}-\frac {\sqrt {a+b x^4}}{3 a x^3} \]

[In]

Int[1/(x^4*Sqrt[a + b*x^4]),x]

[Out]

-1/3*Sqrt[a + b*x^4]/(a*x^3) - (b^(3/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*El
lipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(6*a^(5/4)*Sqrt[a + b*x^4])

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 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 {a+b x^4}}{3 a x^3}-\frac {b \int \frac {1}{\sqrt {a+b x^4}} \, dx}{3 a} \\ & = -\frac {\sqrt {a+b x^4}}{3 a x^3}-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+b x^4}} \\ \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 = 51, normalized size of antiderivative = 0.46 \[ \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx=-\frac {\sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {b x^4}{a}\right )}{3 x^3 \sqrt {a+b x^4}} \]

[In]

Integrate[1/(x^4*Sqrt[a + b*x^4]),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {\sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(93\)
risch	\(-\frac {\sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(93\)
elliptic	\(-\frac {\sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(93\)

[In]

int(1/x^4/(b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(b*x^4+a)^(1/2)/a/x^3-1/3*b/a/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1
/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx=\frac {\sqrt {a} x^{3} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - \sqrt {b x^{4} + a}}{3 \, a x^{3}} \]

[In]

integrate(1/x^4/(b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

1/3*(sqrt(a)*x^3*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) - sqrt(b*x^4 + a))/(a*x^3)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.37 \[ \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx=\frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{3} \Gamma \left (\frac {1}{4}\right )} \]

[In]

integrate(1/x**4/(b*x**4+a)**(1/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx=\int { \frac {1}{\sqrt {b x^{4} + a} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^4), x)

Giac [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx=\int { \frac {1}{\sqrt {b x^{4} + a} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx=\int \frac {1}{x^4\,\sqrt {b\,x^4+a}} \,d x \]

[In]

int(1/(x^4*(a + b*x^4)^(1/2)),x)

[Out]

int(1/(x^4*(a + b*x^4)^(1/2)), x)