\(\int \frac {1}{(a-b x^2)^{3/2} \sqrt {c+d x^2}} \, dx\) [284]

Optimal result

Integrand size = 24, antiderivative size = 224 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {b x \sqrt {c+d x^2}}{a (b c+a d) \sqrt {a-b x^2}}-\frac {\sqrt {b} c \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{a^{3/2} (b c+a d) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}}+\frac {\sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{a^{3/2} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {b x \left (c+d x^2\right )+\frac {a d \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (\arcsin \left (\sqrt {-\frac {d}{c}} x\right )|-\frac {b c}{a d}\right )}{\sqrt {-\frac {d}{c}}}}{a (b c+a d) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[1/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {316, 27, 326, 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.

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

Int[1/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
 

Output:

(b*x*Sqrt[c + d*x^2])/(a*(b*c + a*d)*Sqrt[a - b*x^2]) + (d*(-((Sqrt[a]*Sqr 
t[b]*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt 
[a]], -((a*d)/(b*c))])/(d*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c])) + (Sqrt[a] 
*(b*c + a*d)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqr 
t[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[c + d*x 
^2])))/(a*(b*c + a*d))
 

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 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

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 326

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

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]
 

Maple [A] (verified)

Time = 5.82 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.11

method	result	size
default	\(\frac {\left (\sqrt {\frac {b}{a}}\, b d \,x^{3}+a \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) d +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c +\sqrt {\frac {b}{a}}\, b c x \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{\sqrt {\frac {b}{a}}\, a \left (a d +b c \right ) \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\)	\(248\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) x}{a \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {\left (\frac {1}{a}-\frac {b c}{a \left (a d +b c \right )}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{a \left (a d +b c \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(337\)

Input:

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

Output:

((b/a)^(1/2)*b*d*x^3+a*((-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))*d+((-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-((-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+(b/a)^(1/2)*b* 
c*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 = 0.08 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} a b^{2} c x + {\left (b^{3} c x^{2} - a b^{2} c\right )} \sqrt {a c} \sqrt {\frac {b}{a}} E(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-\frac {a d}{b c}) + {\left (a b^{2} c + a^{3} d - {\left (b^{3} c + a^{2} b d\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {b}{a}} F(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-\frac {a d}{b c})}{a^{3} b^{2} c^{2} + a^{4} b c d - {\left (a^{2} b^{3} c^{2} + a^{3} b^{2} c d\right )} x^{2}} \] Input:

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

Output:

(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*a*b^2*c*x + (b^3*c*x^2 - a*b^2*c)*sqrt(a 
*c)*sqrt(b/a)*elliptic_e(arcsin(x*sqrt(b/a)), -a*d/(b*c)) + (a*b^2*c + a^3 
*d - (b^3*c + a^2*b*d)*x^2)*sqrt(a*c)*sqrt(b/a)*elliptic_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]

\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{\left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

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

Output:

Integral(1/((a - b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:

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

Output:

integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
 

Giac [F]

\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:

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

Output:

integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \] Input:

int(1/(-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))/(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)