\(\int \frac {1+\sqrt {\frac {c}{a}} x^2}{(d+e x^2) \sqrt {a+b x^2+c x^4}} \, dx\) [181]

Optimal result

Integrand size = 41, antiderivative size = 288 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=-\frac {\left (\sqrt {\frac {c}{a}} d-e\right ) \arctan \left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {d} \sqrt {e} \sqrt {c d^2-b d e+a e^2}}+\frac {\left (\sqrt {\frac {c}{a}} d+e\right ) \left (1+\sqrt {\frac {c}{a}} x^2\right ) \sqrt {\frac {\sqrt {\frac {c}{a}} \left (a+b x^2+c x^4\right )}{c \left (\frac {1}{\sqrt [4]{\frac {c}{a}}}+\sqrt [4]{\frac {c}{a}} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {\frac {c}{a}} d-e\right )^2}{4 \sqrt {\frac {c}{a}} d e},2 \arctan \left (\sqrt [4]{\frac {c}{a}} x\right ),\frac {1}{4} \left (2-\frac {b \sqrt {\frac {c}{a}}}{c}\right )\right )}{4 \sqrt [4]{\frac {c}{a}} d e \sqrt {a+b x^2+c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.57 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.08 \[ \int \frac {1+\sqrt {\frac {c}{a}} x^2}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=-\frac {i \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (\sqrt {\frac {c}{a}} d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+\left (-\sqrt {\frac {c}{a}} d+e\right ) \operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d e \sqrt {a+b x^2+c x^4}} \] Input:

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

Output:

((-I)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt 
[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(Sqrt[c/a]*d*EllipticF[I*ArcSinh[S 
qrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sq 
rt[b^2 - 4*a*c])] + (-(Sqrt[c/a]*d) + e)*EllipticPi[((b + Sqrt[b^2 - 4*a*c 
])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + 
Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^ 
2 - 4*a*c])]*d*e*Sqrt[a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.94, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2220}

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.

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

\(\Big \downarrow \) 2220

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

Input:

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

Output:

-1/2*((Sqrt[c/a]*d - e)*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sq 
rt[e]*Sqrt[a + b*x^2 + c*x^4])])/(Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 - b*d*e + a*e 
^2]) + ((Sqrt[c/a]*d + e)*(1 + Sqrt[c/a]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a* 
(1 + Sqrt[c/a]*x^2)^2)]*EllipticPi[-1/4*(Sqrt[c/a]*d - e)^2/(Sqrt[c/a]*d*e 
), 2*ArcTan[(c/a)^(1/4)*x], (2 - (b*Sqrt[c/a])/c)/4])/(4*(c/a)^(1/4)*d*e*S 
qrt[a + b*x^2 + c*x^4])
 

Defintions of rubi rules used

rule 2220

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

Maple [A] (verified)

Time = 2.01 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.29

method	result	size
default	\(\frac {\sqrt {\frac {c}{a}}\, \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\sqrt {\frac {c}{a}}\, d -e \right ) \sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a e}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{e d \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\)	\(371\)
elliptic	\(\frac {\sqrt {\frac {\left (c \,x^{4}+b \,x^{2}+a \right ) c}{a}}\, a \left (1+\sqrt {\frac {c}{a}}\, x^{2}\right ) \left (\frac {\sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a e}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{d \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {c \sqrt {2}\, \sqrt {4+\frac {2 b \,x^{2}}{a}-\frac {2 x^{2} \sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {4+\frac {2 b \,x^{2}}{a}+\frac {2 x^{2} \sqrt {-4 a c +b^{2}}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 a e \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {\frac {c^{2} x^{4}}{a}+\frac {b c \,x^{2}}{a}+c}}-\frac {c \sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a e}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{a e \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {\frac {c^{2} x^{4}}{a}+\frac {b c \,x^{2}}{a}+c}}\right )}{c \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}+a \sqrt {\frac {\left (c \,x^{4}+b \,x^{2}+a \right ) c}{a}}}\)	\(665\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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