\(\int \frac {\sqrt {a+c x^4}}{x^4} \, dx\) [778]

Optimal result

Integrand size = 15, antiderivative size = 107 \[ \int \frac {\sqrt {a+c x^4}}{x^4} \, dx=-\frac {\sqrt {a+c x^4}}{3 x^3}+\frac {c^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{a} \sqrt {a+c x^4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 107, 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 = {283, 226} \[ \int \frac {\sqrt {a+c x^4}}{x^4} \, dx=\frac {c^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\sqrt {a+c x^4}}{3 x^3} \]

[In]

Int[Sqrt[a + c*x^4]/x^4,x]

[Out]

-1/3*Sqrt[a + c*x^4]/x^3 + (c^(3/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipt
icF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(3*a^(1/4)*Sqrt[a + c*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 283

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+c x^4}}{3 x^3}+\frac {1}{3} (2 c) \int \frac {1}{\sqrt {a+c x^4}} \, dx \\ & = -\frac {\sqrt {a+c x^4}}{3 x^3}+\frac {c^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{a} \sqrt {a+c 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 = 0.48 \[ \int \frac {\sqrt {a+c x^4}}{x^4} \, dx=-\frac {\sqrt {a+c x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},-\frac {c x^4}{a}\right )}{3 x^3 \sqrt {1+\frac {c x^4}{a}}} \]

[In]

Integrate[Sqrt[a + c*x^4]/x^4,x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {\sqrt {x^{4} c +a}}{3 x^{3}}+\frac {2 c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(87\)
risch	\(-\frac {\sqrt {x^{4} c +a}}{3 x^{3}}+\frac {2 c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(87\)
elliptic	\(-\frac {\sqrt {x^{4} c +a}}{3 x^{3}}+\frac {2 c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(87\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {a+c x^4}}{x^4} \, dx=-\frac {2 \, \sqrt {a} x^{3} \left (-\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + \sqrt {c x^{4} + a}}{3 \, x^{3}} \]

[In]

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

[Out]

-1/3*(2*sqrt(a)*x^3*(-c/a)^(3/4)*elliptic_f(arcsin(x*(-c/a)^(1/4)), -1) + sqrt(c*x^4 + a))/x^3

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {a+c x^4}}{x^4} \, dx=\frac {\sqrt {a} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(c*x^4 + a)/x^4, x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(c*x^4 + a)/x^4, x)

Mupad [F(-1)]

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

[In]

int((a + c*x^4)^(1/2)/x^4,x)

[Out]

int((a + c*x^4)^(1/2)/x^4, x)