\(\int \sqrt {a c+(b c-a d) x^2-b d x^4} \, dx\) [21]

Optimal result

Integrand size = 27, antiderivative size = 244 \[ \int \sqrt {a c+(b c-a d) x^2-b d x^4} \, dx=\frac {1}{3} x \sqrt {a c+(b c-a d) x^2-b d x^4}+\frac {a \sqrt {c} (b c-a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{3 b \sqrt {d} \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {a \sqrt {c} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{3 b \sqrt {d} \sqrt {a c+(b c-a d) x^2-b d x^4}} \] Output:

1/3*x*(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)+1/3*a*c^(1/2)*(-a*d+b*c)*(1+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)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)+1/3*a*c^(1/2)*(a*d+b*c)*(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)/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.41 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.82 \[ \int \sqrt {a c+(b c-a d) x^2-b d x^4} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c-d x^2\right )-i c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} d \sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \] Input:

Integrate[Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x^4],x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c - d*x^2) - I*c*(-(b*c) + a*d)*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)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[I*Ar 
cSinh[Sqrt[b/a]*x], -((a*d)/(b*c))])/(3*Sqrt[b/a]*d*Sqrt[(a + b*x^2)*(c - 
d*x^2)])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1404, 1514, 399, 321, 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.

\(\displaystyle \int \sqrt {x^2 (b c-a d)+a c-b d x^4} \, dx\)

\(\Big \downarrow \) 1404

\(\displaystyle \frac {1}{3} \int \frac {(b c-a d) x^2+2 a c}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx+\frac {1}{3} x \sqrt {x^2 (b c-a d)+a c-b d x^4}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \int \frac {(b c-a d) x^2+2 a c}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{3 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {1}{3} x \sqrt {x^2 (b c-a d)+a c-b d x^4}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {a (a d+b c) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{b}+\frac {a (b c-a d) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}\right )}{3 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {1}{3} x \sqrt {x^2 (b c-a d)+a c-b d x^4}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {a (b c-a d) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}+\frac {a \sqrt {c} (a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}\right )}{3 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {1}{3} x \sqrt {x^2 (b c-a d)+a c-b d x^4}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {a \sqrt {c} (a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}+\frac {a \sqrt {c} (b c-a d) E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d}}\right )}{3 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {1}{3} x \sqrt {x^2 (b c-a d)+a c-b d x^4}\)

Input:

Int[Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x^4],x]
 

Output:

(x*Sqrt[a*c + (b*c - a*d)*x^2 - b*d*x^4])/3 + (Sqrt[1 + (b*x^2)/a]*Sqrt[1 
- (d*x^2)/c]*((a*Sqrt[c]*(b*c - a*d)*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]] 
, -((b*c)/(a*d))])/(b*Sqrt[d]) + (a*Sqrt[c]*(b*c + a*d)*EllipticF[ArcSin[( 
Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d])))/(3*Sqrt[a*c + (b*c - a 
*d)*x^2 - b*d*x^4])
 

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 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 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])))))
 

rule 1404

Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b 
*x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1))   Int[(2*a + b*x^2)*( 
a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* 
c, 0] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 1514

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt 
[1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4])   Int[(d + e*x^2)/(Sqrt[1 
+ 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, 
c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
 

Maple [A] (verified)

Time = 2.42 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {x \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{3}+\frac {2 a c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{3 \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (-\frac {a d}{3}+\frac {b c}{3}\right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}\)	\(259\)
elliptic	\(\frac {x \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{3}+\frac {2 a c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{3 \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (-\frac {a d}{3}+\frac {b c}{3}\right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}\)	\(259\)
risch	\(\frac {x \left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}{3 \sqrt {-\left (b \,x^{2}+a \right ) \left (d \,x^{2}-c \right )}}+\frac {\left (a d -b c \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{3 \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}+\frac {2 a c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{3 \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}\)	\(267\)

Input:

int((a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.67 \[ \int \sqrt {a c+(b c-a d) x^2-b d x^4} \, dx=-\frac {{\left (b c^{2} - a c d\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) - {\left (b c^{2} - a c d + 2 \, a d^{2}\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) - \sqrt {-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c} {\left (b d^{2} x^{2} - b c d + a d^{2}\right )}}{3 \, b d^{2} x} \] Input:

integrate((a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2),x, algorithm="fricas")
 

Output:

-1/3*((b*c^2 - a*c*d)*sqrt(-b*d)*x*sqrt(c/d)*elliptic_e(arcsin(sqrt(c/d)/x 
), -a*d/(b*c)) - (b*c^2 - a*c*d + 2*a*d^2)*sqrt(-b*d)*x*sqrt(c/d)*elliptic 
_f(arcsin(sqrt(c/d)/x), -a*d/(b*c)) - sqrt(-b*d*x^4 + (b*c - a*d)*x^2 + a* 
c)*(b*d^2*x^2 - b*c*d + a*d^2))/(b*d^2*x)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-b*d*x^4 + (b*c - a*d)*x^2 + a*c), x)
 

Giac [F]

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

integrate((a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-b*d*x^4 + (b*c - a*d)*x^2 + a*c), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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