\(\int \frac {1+\sqrt {\frac {c}{a}} x^2}{(d+e x^2) \sqrt {a+c x^4}} \, dx\) [82]

Optimal result

Integrand size = 36, antiderivative size = 245 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {\left (\sqrt {\frac {c}{a}} d-e\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \sqrt {e} \sqrt {c d^2+a e^2}}+\frac {\left (\sqrt {\frac {c}{a}} d+e\right ) \left (1+\sqrt {\frac {c}{a}} x^2\right ) \sqrt {\frac {\sqrt {\frac {c}{a}} \left (a+c x^4\right )}{c \left (\frac {1}{\sqrt [4]{\frac {c}{a}}}+\sqrt [4]{\frac {c}{a}} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {\frac {c}{a}} d-e\right )^2}{4 \sqrt {\frac {c}{a}} d e},2 \arctan \left (\sqrt [4]{\frac {c}{a}} x\right ),\frac {1}{2}\right )}{4 \sqrt [4]{\frac {c}{a}} d e \sqrt {a+c x^4}} \] Output:

-1/2*((c/a)^(1/2)*d-e)*arctan((a*e^2+c*d^2)^(1/2)*x/d^(1/2)/e^(1/2)/(c*x^4 
+a)^(1/2))/d^(1/2)/e^(1/2)/(a*e^2+c*d^2)^(1/2)+1/4*((c/a)^(1/2)*d+e)*(1+(c 
/a)^(1/2)*x^2)*((c/a)^(1/2)*(c*x^4+a)/c/(1/(c/a)^(1/4)+(c/a)^(1/4)*x^2)^2) 
^(1/2)*EllipticPi(sin(2*arctan((c/a)^(1/4)*x)),-1/4*((c/a)^(1/2)*d-e)^2/(c 
/a)^(1/2)/d/e,1/2*2^(1/2))/(c/a)^(1/4)/d/e/(c*x^4+a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.39 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.62 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {i \sqrt {1+\frac {c x^4}{a}} \left (\sqrt {\frac {c}{a}} d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\left (-\sqrt {\frac {c}{a}} d+e\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d e \sqrt {a+c x^4}} \] Input:

Integrate[(1 + Sqrt[c/a]*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]),x]
 

Output:

((-I)*Sqrt[1 + (c*x^4)/a]*(Sqrt[c/a]*d*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c] 
)/Sqrt[a]]*x], -1] + (-(Sqrt[c/a]*d) + e)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqr 
t[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/(Sqrt[(I*Sqrt[c])/S 
qrt[a]]*d*e*Sqrt[a + c*x^4])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.93, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2221}

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.

\(\displaystyle \int \frac {x^2 \sqrt {\frac {c}{a}}+1}{\sqrt {a+c x^4} \left (d+e x^2\right )} \, dx\)

\(\Big \downarrow \) 2221

\(\displaystyle \frac {\left (x^2 \sqrt {\frac {c}{a}}+1\right ) \sqrt {\frac {a+c x^4}{a \left (x^2 \sqrt {\frac {c}{a}}+1\right )^2}} \left (d \sqrt {\frac {c}{a}}+e\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {\frac {c}{a}} d-e\right )^2}{4 \sqrt {\frac {c}{a}} d e},2 \arctan \left (\sqrt [4]{\frac {c}{a}} x\right ),\frac {1}{2}\right )}{4 d e \sqrt [4]{\frac {c}{a}} \sqrt {a+c x^4}}-\frac {\left (d \sqrt {\frac {c}{a}}-e\right ) \arctan \left (\frac {x \sqrt {a e^2+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \sqrt {e} \sqrt {a e^2+c d^2}}\)

Input:

Int[(1 + Sqrt[c/a]*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]),x]
 

Output:

-1/2*((Sqrt[c/a]*d - e)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sq 
rt[a + c*x^4])])/(Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 + a*e^2]) + ((Sqrt[c/a]*d + e 
)*(1 + Sqrt[c/a]*x^2)*Sqrt[(a + c*x^4)/(a*(1 + Sqrt[c/a]*x^2)^2)]*Elliptic 
Pi[-1/4*(Sqrt[c/a]*d - e)^2/(Sqrt[c/a]*d*e), 2*ArcTan[(c/a)^(1/4)*x], 1/2] 
)/(4*(c/a)^(1/4)*d*e*Sqrt[a + c*x^4])
 

Defintions of rubi rules used

rule 2221

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e 
) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] 
+ Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* 
d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x 
], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po 
sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.90 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {\sqrt {\frac {c}{a}}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {\left (\sqrt {\frac {c}{a}}\, d -e \right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{e d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\)	\(204\)
elliptic	\(\frac {\sqrt {\frac {\left (c \,x^{4}+a \right ) c}{a}}\, a \left (1+\sqrt {\frac {c}{a}}\, x^{2}\right ) \left (\frac {c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{a e \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {\frac {c^{2} x^{4}}{a}+c}}-\frac {c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{a e \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {\frac {c^{2} x^{4}}{a}+c}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{c \,x^{2} \sqrt {c \,x^{4}+a}+a \sqrt {\frac {\left (c \,x^{4}+a \right ) c}{a}}}\)	\(367\)

Input:

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

Output:

(c/a)^(1/2)/e/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1 
+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/ 
2))^(1/2),I)-((c/a)^(1/2)*d-e)/e/d/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)* 
x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*Ellipti 
cPi(x*(I*c^(1/2)/a^(1/2))^(1/2),I/c^(1/2)*a^(1/2)/d*e,(-I/a^(1/2)*c^(1/2)) 
^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=\text {Timed out} \] Input:

integrate((1+(c/a)^(1/2)*x^2)/(e*x^2+d)/(c*x^4+a)^(1/2),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {x^{2} \sqrt {\frac {c}{a}} + 1}{\sqrt {a + c x^{4}} \left (d + e x^{2}\right )}\, dx \] Input:

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

Output:

Integral((x**2*sqrt(c/a) + 1)/(sqrt(a + c*x**4)*(d + e*x**2)), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {x^{2} \sqrt {\frac {c}{a}} + 1}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:

integrate((1+(c/a)^(1/2)*x^2)/(e*x^2+d)/(c*x^4+a)^(1/2),x, algorithm="maxi 
ma")
 

Output:

integrate((x^2*sqrt(c/a) + 1)/(sqrt(c*x^4 + a)*(e*x^2 + d)), x)
 

Giac [F]

\[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {x^{2} \sqrt {\frac {c}{a}} + 1}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:

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

Output:

integrate((x^2*sqrt(c/a) + 1)/(sqrt(c*x^4 + a)*(e*x^2 + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {x^2\,\sqrt {\frac {c}{a}}+1}{\sqrt {c\,x^4+a}\,\left (e\,x^2+d\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right )+\left (\int \frac {\sqrt {c \,x^{4}+a}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a}{a} \] Input:

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

Output:

(sqrt(c)*sqrt(a)*int((sqrt(a + c*x**4)*x**2)/(a*d + a*e*x**2 + c*d*x**4 + 
c*e*x**6),x) + int(sqrt(a + c*x**4)/(a*d + a*e*x**2 + c*d*x**4 + c*e*x**6) 
,x)*a)/a