\(\int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{(d+e x^2) \sqrt {a+c x^4}} \, dx\) [81]

Optimal result

Integrand size = 37, antiderivative size = 232 \[ \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \sqrt {e} \sqrt {c d^2+a e^2}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )^2}{4 \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{3/4} \sqrt [4]{c} d e \sqrt {a+c x^4}} \] Output:

-1/2*(c^(1/2)*d/a^(1/2)-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^(1/2)*d+a^(1/2)* 
e)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*Ellipti 
cPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*a^(1/2)*(c^(1/2)*d/a^(1/2)-e)^2/ 
c^(1/2)/d/e,1/2*2^(1/2))/a^(3/4)/c^(1/4)/d/e/(c*x^4+a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.54 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.67 \[ \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {\sqrt {c} \sqrt {1+\frac {c x^4}{a}} \left (\sqrt {c} d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\left (-\sqrt {c} d+\sqrt {a} e\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{a \left (\frac {i \sqrt {c}}{\sqrt {a}}\right )^{3/2} d e \sqrt {a+c x^4}} \] Input:

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

Output:

(Sqrt[c]*Sqrt[1 + (c*x^4)/a]*(Sqrt[c]*d*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c 
])/Sqrt[a]]*x], -1] + (-(Sqrt[c]*d) + Sqrt[a]*e)*EllipticPi[((-I)*Sqrt[a]* 
e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/(a*((I*Sqrt[ 
c])/Sqrt[a])^(3/2)*d*e*Sqrt[a + c*x^4])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, 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.

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

\(\Big \downarrow \) 2221

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

Input:

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

Output:

-1/2*(((Sqrt[c]*d)/Sqrt[a] - e)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sq 
rt[e]*Sqrt[a + c*x^4])])/(Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 + a*e^2]) + (((Sqrt[c 
]*d)/Sqrt[a] + e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt 
[c]*x^2)^2]*EllipticPi[-1/4*(Sqrt[a]*((Sqrt[c]*d)/Sqrt[a] - e)^2)/(Sqrt[c] 
*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(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 = 0.34 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\frac {\sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {\left (\sqrt {c}\, d -\sqrt {a}\, e \right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{e d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{\sqrt {a}}\)	\(203\)
elliptic	\(\frac {\sqrt {\left (c \,x^{4}+a \right ) a c}\, \left (\sqrt {a}+\sqrt {c}\, x^{2}\right ) \left (\frac {c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {a \,c^{2} x^{4}+a^{2} c}}-\frac {c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{e \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {a \,c^{2} x^{4}+a^{2} c}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{\left (c \,x^{2} \sqrt {c \,x^{4}+a}+\sqrt {\left (c \,x^{4}+a \right ) a c}\right ) \sqrt {a}}\)	\(359\)

Input:

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

Output:

1/a^(1/2)*(c^(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^(1/2)*d-a^(1/2)*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)*EllipticPi(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(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   catd 
ef: division by zero
 

Sympy [F]

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

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

Output:

(Integral(sqrt(a)/(d*sqrt(a + c*x**4) + e*x**2*sqrt(a + c*x**4)), x) + Int 
egral(sqrt(c)*x**2/(d*sqrt(a + c*x**4) + e*x**2*sqrt(a + c*x**4)), x))/sqr 
t(a)
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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