\(\int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx\) [79]

Optimal result

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

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

Mathematica [C] (verified)

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

Time = 10.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.59 \[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {\sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+\sqrt {c} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{3 \sqrt {-a+c x^4}} \] Input:

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

Output:

(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])
 

Rubi [A] (verified)

Time = 0.36 (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 = {1390, 1389, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1390

\(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c x^4-a}}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {\sqrt {a} \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c x^4-a}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt {c x^4-a}}\)

Input:

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

Output:

(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])
 

Defintions of rubi rules used

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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 1389

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

rule 1390

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

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 = 1.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.93

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

Input:

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

Output:

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. 102 vs. \(2 (41) = 82\).

Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=\frac {a c x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (a c x \sqrt {\frac {a}{c}} + \sqrt {a} c^{\frac {3}{2}} x \sqrt {\frac {a}{c}}\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + \sqrt {c x^{4} - a} a \sqrt {c}}{a c x} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.89 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=- \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4}}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {i \sqrt {c} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4}}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

-I*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4/a)/(4*gamma(5/4)) - I*sqr 
t(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]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a}+\sqrt {c} x^2}{\sqrt {-a+c x^4}} \, dx=-\sqrt {a}\, \left (\int \frac {\sqrt {c \,x^{4}-a}}{-c \,x^{4}+a}d x \right )-\sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}-a}\, x^{2}}{-c \,x^{4}+a}d x \right ) \] Input:

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

Output:

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