\(\int \frac {e+f x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx\) [40]

Optimal result

Integrand size = 31, antiderivative size = 190 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\frac {\sqrt {c} f \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\sqrt {c} (b e-a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.92 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.71 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=-\frac {i \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \left (-c f E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+(d e+c f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} d \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {399, 323, 323, 321, 331, 330, 327}

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.

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(Sqrt[c]*f*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x 
)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2 
]) + (Sqrt[c]*(b*e - a*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*Elliptic 
F[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[a + b*x^2] 
*Sqrt[c - d*x^2])
 

Defintions of rubi rules used

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 331

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

Maple [A] (verified)

Time = 5.74 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\left (-\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a f +\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b e +\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a f \right ) \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {b \,x^{2}+a}\, \sqrt {-x^{2} d +c}}{b \sqrt {\frac {d}{c}}\, \left (-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c \right )}\)	\(161\)
elliptic	\(\frac {\sqrt {\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}\, \left (\frac {e \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}-\frac {f a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}\, b}\right )}{\sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}\)	\(258\)

Input:

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

Output:

(-EllipticF(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2))*a*f+EllipticF(x*(1/c*d)^(1/2 
),(-b*c/a/d)^(1/2))*b*e+EllipticE(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2))*a*f)*( 
(b*x^2+a)/a)^(1/2)*((-d*x^2+c)/c)^(1/2)*(b*x^2+a)^(1/2)*(-d*x^2+c)^(1/2)/b 
/(1/c*d)^(1/2)/(-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 = 130, normalized size of antiderivative = 0.68 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=-\frac {\sqrt {-b d} c^{2} f x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) + \sqrt {b x^{2} + a} \sqrt {-d x^{2} + c} c d f - {\left (d^{2} e + c^{2} f\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c})}{b c d^{2} x} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\int \frac {e + f x^{2}}{\sqrt {a + b x^{2}} \sqrt {c - d x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\int { \frac {f x^{2} + e}{\sqrt {b x^{2} + a} \sqrt {-d x^{2} + c}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\int { \frac {f x^{2} + e}{\sqrt {b x^{2} + a} \sqrt {-d x^{2} + c}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\int \frac {f\,x^2+e}{\sqrt {b\,x^2+a}\,\sqrt {c-d\,x^2}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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