\(\int \frac {A+B x^2}{(d+e x^2) \sqrt {a+c x^4}} \, dx\) [71]

Optimal result

Integrand size = 28, antiderivative size = 372 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {(B d-A e) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \sqrt {e} \sqrt {c d^2+a e^2}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.39 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.37 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {i \sqrt {1+\frac {c x^4}{a}} \left (B d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+(-B d+A e) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d e \sqrt {a+c x^4}} \] Input:

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

Output:

((-I)*Sqrt[1 + (c*x^4)/a]*(B*d*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a 
]]*x], -1] + (-(B*d) + A*e)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*Arc 
Sinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*e*Sq 
rt[a + c*x^4])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2227, 27, 761, 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.

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

\(\Big \downarrow \) 2227

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

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)]
 

rule 2227

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.52

method	result	size
default	\(\frac {B \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\left (A e -B d \right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{e d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\)	\(192\)
elliptic	\(\frac {B \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) A}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) B}{e \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\)	\(290\)

Input:

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

Output:

B/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)+(A*e-B*d)/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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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