\(\int \frac {e+f x^2}{(a+b x^2)^{3/2} \sqrt {c+d x^2}} \, dx\) [30]

Optimal result

Integrand size = 30, antiderivative size = 209 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {(b e-a f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {a} (d e-c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

(-a*f+b*e)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),( 
1-a*d/b/c)^(1/2))/a^(1/2)/b^(1/2)/(-a*d+b*c)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/ 
c/(b*x^2+a))^(1/2)-a^(1/2)*(-c*f+d*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arct 
an(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/(-a*d+b*c)/(b*x^2+a)^(1 
/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.40 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.99 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} (-b e+a f) x \left (c+d x^2\right )+i c (-b e+a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i (-b c+a d) e \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{b (-b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(e + f*x^2)/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
 

Output:

(Sqrt[b/a]*(Sqrt[b/a]*(-(b*e) + a*f)*x*(c + d*x^2) + I*c*(-(b*e) + a*f)*Sq 
rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a 
*d)/(b*c)] - I*(-(b*c) + a*d)*e*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*El 
lipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(b*(-(b*c) + a*d)*Sqrt[a + 
b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {400, 313, 320}

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.

\(\displaystyle \int \frac {e+f x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 400

\(\displaystyle \frac {(b e-a f) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right )^{3/2}}dx}{b c-a d}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\)

Input:

Int[(e + f*x^2)/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
 

Output:

((b*e - a*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a 
*d)/(b*c)])/(Sqrt[a]*Sqrt[b]*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^ 
2))/(c*(a + b*x^2))]) - (Sqrt[c]*(d*e - c*f)*Sqrt[a + b*x^2]*EllipticF[Arc 
Tan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*(b*c - a*d)*Sqrt[(c 
*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 400

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ 
(3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(Sqrt[a + b*x^2]* 
Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[Sqrt[a + b*x^ 
2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & 
& PosQ[d/c]
 

Maple [A] (verified)

Time = 7.77 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.60

method	result	size
default	\(\frac {\left (\sqrt {-\frac {b}{a}}\, a d f \,x^{3}-\sqrt {-\frac {b}{a}}\, b d e \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a d e -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c e -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c f +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c e +\sqrt {-\frac {b}{a}}\, a c f x -\sqrt {-\frac {b}{a}}\, b c e x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{\sqrt {-\frac {b}{a}}\, a \left (a d -b c \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(334\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+b c \right ) x \left (a f -b e \right )}{b a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {f}{b}-\frac {a f -b e}{b a}-\frac {c \left (a f -b e \right )}{a \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {\left (a f -b e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(368\)

Input:

int((f*x^2+e)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((-b/a)^(1/2)*a*d*f*x^3-(-b/a)^(1/2)*b*d*e*x^3+((b*x^2+a)/a)^(1/2)*((d*x^2 
+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*d*e-((b*x^2+a)/a) 
^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c*e 
-((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c 
)^(1/2))*a*c*f+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^ 
(1/2),(a*d/b/c)^(1/2))*b*c*e+(-b/a)^(1/2)*a*c*f*x-(-b/a)^(1/2)*b*c*e*x)*(d 
*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/(-b/a)^(1/2)/a/(a*d-b*c)/(b*d*x^4+a*d*x^2+b* 
c*x^2+a*c)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.21 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {{\left (a b^{2} c e - a^{2} b c f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x - {\left (a b^{2} c e - a^{2} b c f + {\left (b^{3} c e - a b^{2} c f\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (a^{3} + a^{2} b\right )} c f + {\left ({\left (a^{2} b + a b^{2}\right )} c f - {\left (b^{3} c + a^{2} b d\right )} e\right )} x^{2} - {\left (a b^{2} c + a^{3} d\right )} e\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c})}{a^{3} b^{2} c^{2} - a^{4} b c d + {\left (a^{2} b^{3} c^{2} - a^{3} b^{2} c d\right )} x^{2}} \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

((a*b^2*c*e - a^2*b*c*f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x - (a*b^2*c*e - 
a^2*b*c*f + (b^3*c*e - a*b^2*c*f)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arc 
sin(x*sqrt(-b/a)), a*d/(b*c)) - ((a^3 + a^2*b)*c*f + ((a^2*b + a*b^2)*c*f 
- (b^3*c + a^2*b*d)*e)*x^2 - (a*b^2*c + a^3*d)*e)*sqrt(a*c)*sqrt(-b/a)*ell 
iptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)))/(a^3*b^2*c^2 - a^4*b*c*d + (a^2* 
b^3*c^2 - a^3*b^2*c*d)*x^2)
 

Sympy [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {e + f x^{2}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

integrate((f*x**2+e)/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
 

Output:

Integral((e + f*x**2)/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
 

Giac [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {f\,x^2+e}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:

int((e + f*x^2)/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)),x)
 

Output:

int((e + f*x^2)/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) f +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) e \] Input:

int((f*x^2+e)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

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