\(\int \frac {e+f x^2}{\sqrt {a+b x^2} (c+d x^2)^{3/2}} \, dx\) [43]

Optimal result

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.01 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d (d e-c f) x \left (a+b x^2\right )-i b c (-d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c d (-b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (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.

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

\(\Big \downarrow \) 400

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 320

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

Input:

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

Output:

-(((d*e - c*f)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - 
(b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d* 
x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*(b*e - a*f)*Sqrt[a + b*x^2]*EllipticF[ 
ArcTan[(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 = 7.79 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.67

method	result	size
default	\(\frac {\left (-\sqrt {-\frac {b}{a}}\, b c d f \,x^{3}+\sqrt {-\frac {b}{a}}\, b \,d^{2} e \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e -\sqrt {-\frac {b}{a}}\, a c d f x +\sqrt {-\frac {b}{a}}\, a \,d^{2} e x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{\sqrt {-\frac {b}{a}}\, d c \left (a d -b c \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(349\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+a d \right ) x \left (c f -d e \right )}{c d \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {f}{d}-\frac {c f -d e}{d c}+\frac {a \left (c f -d e \right )}{c \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b \left (c f -d e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(369\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.24 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left (a b d^{2} e - a b c d f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x - {\left (b^{2} c d e - b^{2} c^{2} f + {\left (b^{2} d^{2} e - b^{2} c d f\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left ({\left (a b + b^{2}\right )} c d e + {\left ({\left (a b + b^{2}\right )} d^{2} e - {\left (b^{2} c d + a^{2} d^{2}\right )} f\right )} x^{2} - {\left (b^{2} c^{2} + a^{2} c d\right )} f\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c})}{a b^{2} c^{3} d - a^{2} b c^{2} d^{2} + {\left (a b^{2} c^{2} d^{2} - a^{2} b c d^{3}\right )} x^{2}} \] Input:

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

Output:

-((a*b*d^2*e - a*b*c*d*f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x - (b^2*c*d*e - 
 b^2*c^2*f + (b^2*d^2*e - b^2*c*d*f)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e( 
arcsin(x*sqrt(-b/a)), a*d/(b*c)) + ((a*b + b^2)*c*d*e + ((a*b + b^2)*d^2*e 
 - (b^2*c*d + a^2*d^2)*f)*x^2 - (b^2*c^2 + a^2*c*d)*f)*sqrt(a*c)*sqrt(-b/a 
)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)))/(a*b^2*c^3*d - a^2*b*c^2*d^ 
2 + (a*b^2*c^2*d^2 - a^2*b*c*d^3)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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