\(\int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx\) [166]

Optimal result

Integrand size = 29, antiderivative size = 54 \[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt {-a+c x^4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1214, 1213, 435} \[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt {c x^4-a}} \]

[In]

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

[Out]

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

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 1213

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + e*(x^2/d)]/Sqrt
[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 1214

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4], In
t[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] &&
!GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{\sqrt {-a+c x^4}} \\ & = \frac {\left (\sqrt {a} \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {\sqrt {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}} \, dx}{\sqrt {-a+c x^4}} \\ & = \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \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 = 10.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.59 \[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {\sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+\sqrt {c} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{3 \sqrt {-a+c x^4}} \]

[In]

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

[Out]

(Sqrt[1 - (c*x^4)/a]*(3*Sqrt[a]*x*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + Sqrt[c]*x^3*Hypergeometric2F1[
1/2, 3/4, 7/4, (c*x^4)/a]))/(3*Sqrt[-a + c*x^4])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (42 ) = 84\).

Time = 0.62 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.93

method	result	size
default	\(\frac {\sqrt {a}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}+\frac {\sqrt {a}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}\)	\(158\)
elliptic	\(\frac {\left (\sqrt {a}+x^{2} \sqrt {c}\right ) \sqrt {-\left (-c \,x^{4}+a \right ) c}\, \sqrt {-\left (-c \,x^{4}+a \right ) a}\, \left (\frac {\sqrt {c}\, \sqrt {a}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c^{2} x^{4}-a c}}+\frac {a \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {a c \,x^{4}-a^{2}}}\right )}{\sqrt {c \,x^{4}-a}\, \left (c \,x^{2} \sqrt {-\left (-c \,x^{4}+a \right ) a}+a \sqrt {-\left (-c \,x^{4}+a \right ) c}\right )}\)	\(250\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (41) = 82\).

Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {2 \, a c x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - \sqrt {2} {\left (\sqrt {2} a c x \sqrt {\frac {a}{c}} + \sqrt {2} \sqrt {a} c^{\frac {3}{2}} x \sqrt {\frac {a}{c}}\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 2 \, \sqrt {c x^{4} - a} a \sqrt {c}}{2 \, a c x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=- \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4}}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {i \sqrt {c} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4}}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} \]

[In]

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

[Out]

-I*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4/a)/(4*gamma(5/4)) - I*sqrt(c)*x**3*gamma(3/4)*hyper((1/2, 3/4
), (7/4,), c*x**4/a)/(4*sqrt(a)*gamma(7/4))

Maxima [F]

\[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=\int { \frac {\sqrt {c} x^{2} + \sqrt {a}}{\sqrt {c x^{4} - a}} \,d x } \]

[In]

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

[Out]

integrate((sqrt(c)*x^2 + sqrt(a))/sqrt(c*x^4 - a), x)

Giac [F]

\[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=\int { \frac {\sqrt {c} x^{2} + \sqrt {a}}{\sqrt {c x^{4} - a}} \,d x } \]

[In]

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

[Out]

integrate((sqrt(c)*x^2 + sqrt(a))/sqrt(c*x^4 - a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=\int \frac {\sqrt {a}+\sqrt {c}\,x^2}{\sqrt {c\,x^4-a}} \,d x \]

[In]

int((a^(1/2) + c^(1/2)*x^2)/(c*x^4 - a)^(1/2),x)

[Out]

int((a^(1/2) + c^(1/2)*x^2)/(c*x^4 - a)^(1/2), x)