\(\int \frac {1+\sqrt {\frac {c}{a}} x^2}{(d+e x^2) \sqrt {a-c x^4}} \, dx\) [60]

Optimal result

Integrand size = 37, antiderivative size = 114 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {a \left (\frac {c}{a}\right )^{3/4} \left (1-\sqrt {\frac {c}{a}} x^2\right ) \operatorname {EllipticPi}\left (1-\frac {e}{\sqrt {\frac {c}{a}} d},\arctan \left (\sqrt [4]{\frac {c}{a}} x\right ),2\right )}{c d \sqrt {\frac {1-\sqrt {\frac {c}{a}} x^2}{1+\sqrt {\frac {c}{a}} x^2}} \sqrt {a-c x^4}} \] Output:

a*(c/a)^(3/4)*(1-(c/a)^(1/2)*x^2)*EllipticPi((c/a)^(1/4)*x/(1+(c/a)^(1/2)* 
x^2)^(1/2),1-e/(c/a)^(1/2)/d,2^(1/2))/c/d/((1-(c/a)^(1/2)*x^2)/(1+(c/a)^(1 
/2)*x^2))^(1/2)/(-c*x^4+a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.37 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.28 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {\frac {c}{a}} d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\left (-\sqrt {\frac {c}{a}} d+e\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d e \sqrt {a-c x^4}} \] 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[-(Sqrt[c]/ 
Sqrt[a])]*x], -1] + (-(Sqrt[c/a]*d) + e)*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c] 
*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(Sqrt[-(Sqrt[c]/Sqrt[a] 
)]*d*e*Sqrt[a - c*x^4])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {1787, 414}

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.

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

\(\Big \downarrow \) 1787

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

\(\Big \downarrow \) 414

\(\displaystyle \frac {\left (\frac {c}{a}\right )^{3/4} \left (a-a x^2 \sqrt {\frac {c}{a}}\right ) \operatorname {EllipticPi}\left (1-\frac {e}{\sqrt {\frac {c}{a}} d},\arctan \left (\sqrt [4]{\frac {c}{a}} x\right ),2\right )}{c d \sqrt {\frac {1-x^2 \sqrt {\frac {c}{a}}}{x^2 \sqrt {\frac {c}{a}}+1}} \sqrt {a-c x^4}}\)

Input:

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

Output:

((c/a)^(3/4)*(a - a*Sqrt[c/a]*x^2)*EllipticPi[1 - e/(Sqrt[c/a]*d), ArcTan[ 
(c/a)^(1/4)*x], 2])/(c*d*Sqrt[(1 - Sqrt[c/a]*x^2)/(1 + Sqrt[c/a]*x^2)]*Sqr 
t[a - c*x^4])
 

Defintions of rubi rules used

rule 414

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) 
^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* 
Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ 
Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ 
[d/c]
 

rule 1787

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + ( 
c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + 
e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p])   Int[(d + e*x^n)^(p + q) 
*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p 
, q, r}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.93 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.65

method	result	size
default	\(\frac {\sqrt {\frac {c}{a}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (\sqrt {\frac {c}{a}}\, d -e \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{e d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\)	\(188\)
elliptic	\(\frac {\sqrt {\frac {\left (-c \,x^{4}+a \right ) c}{a}}\, a \left (1+\sqrt {\frac {c}{a}}\, x^{2}\right ) \left (\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{a e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-\frac {c^{2} x^{4}}{a}+c}}-\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{a e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-\frac {c^{2} x^{4}}{a}+c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c \,x^{2} \sqrt {-c \,x^{4}+a}+a \sqrt {\frac {\left (-c \,x^{4}+a \right ) c}{a}}}\)	\(344\)

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

Fricas [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="fri 
cas")
 

Output:

integral(-(sqrt(-c*x^4 + a)*x^2*sqrt(c/a) + sqrt(-c*x^4 + a))/(c*e*x^6 + c 
*d*x^4 - a*e*x^2 - a*d), x)
 

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="max 
ima")
 

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="gia 
c")
 

Output:

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

Mupad [F(-1)]

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