\(\int \frac {\sqrt {c+d x^2}}{(a-b x^2)^{3/2}} \, dx\) [157]

Optimal result

Integrand size = 24, antiderivative size = 205 \[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {x \sqrt {c+d x^2}}{a \sqrt {a-b x^2}}-\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {c \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

x*(d*x^2+c)^(1/2)/a/(-b*x^2+a)^(1/2)-(1-b*x^2/a)^(1/2)*(d*x^2+c)^(1/2)*Ell 
ipticE(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/a^(1/2)/b^(1/2)/(-b*x^2+a)^(1/2 
)/(1+d*x^2/c)^(1/2)+c*(1-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^(1/2)/b^(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 = 1.88 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {x \left (c+d x^2\right )+\frac {i c \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{\sqrt {-\frac {b}{a}}}}{a \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {314, 27, 389, 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.

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

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 389

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

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 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, 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 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 389

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

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {\sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}\, \left (\sqrt {\frac {b}{a}}\, d \,x^{3}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) c -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) c +\sqrt {\frac {b}{a}}\, c x \right )}{\left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right ) a \sqrt {\frac {b}{a}}}\)	\(183\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) x}{b a \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {\left (-\frac {d}{b}+\frac {a d +b c}{a b}-\frac {c}{a}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{a \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(328\)

Input:

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

Output:

(d*x^2+c)^(1/2)*(-b*x^2+a)^(1/2)*((b/a)^(1/2)*d*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))*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))*c+ 
(b/a)^(1/2)*c*x)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)/a/(b/a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2}} \, dx=-\frac {\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} a x + {\left (b x^{2} - a\right )} \sqrt {a c} \sqrt {\frac {b}{a}} E(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-\frac {a d}{b c}) - {\left (b x^{2} - a\right )} \sqrt {a c} \sqrt {\frac {b}{a}} F(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-\frac {a d}{b c})}{a^{2} b x^{2} - a^{3}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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