\(\int \sqrt {e x} \sqrt {1-b x^2} \sqrt {1+b x^2} \, dx\) [1349]

Optimal result

Integrand size = 31, antiderivative size = 60 \[ \int \sqrt {e x} \sqrt {1-b x^2} \sqrt {1+b x^2} \, dx=\frac {2 (e x)^{3/2} \sqrt {1-b^2 x^4}}{7 e}+\frac {8 (e x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},\frac {1}{2},\frac {11}{8},b^2 x^4\right )}{21 e} \] Output:

2/7*(e*x)^(3/2)*(-b^2*x^4+1)^(1/2)/e+8/21*(e*x)^(3/2)*hypergeom([3/8, 1/2] 
,[11/8],b^2*x^4)/e
 

Mathematica [A] (verified)

Time = 10.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.48 \[ \int \sqrt {e x} \sqrt {1-b x^2} \sqrt {1+b x^2} \, dx=\frac {2}{3} x \sqrt {e x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{8},\frac {11}{8},b^2 x^4\right ) \] Input:

Integrate[Sqrt[e*x]*Sqrt[1 - b*x^2]*Sqrt[1 + b*x^2],x]
 

Output:

(2*x*Sqrt[e*x]*Hypergeometric2F1[-1/2, 3/8, 11/8, b^2*x^4])/3
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(457\) vs. \(2(60)=120\).

Time = 0.60 (sec) , antiderivative size = 457, normalized size of antiderivative = 7.62, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {335, 811, 851, 838, 27, 2422}

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.

\(\displaystyle \int \sqrt {1-b x^2} \sqrt {b x^2+1} \sqrt {e x} \, dx\)

\(\Big \downarrow \) 335

\(\displaystyle \int \sqrt {1-b^2 x^4} \sqrt {e x}dx\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {4}{7} \int \frac {\sqrt {e x}}{\sqrt {1-b^2 x^4}}dx+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {8 \int \frac {e x}{\sqrt {1-b^2 x^4}}d\sqrt {e x}}{7 e}+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\)

\(\Big \downarrow \) 838

\(\displaystyle \frac {8 \left (\frac {e \int \frac {\sqrt [4]{-b^2} x e+e}{e \sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 \sqrt [4]{-b^2}}-\frac {e \int \frac {e-\sqrt [4]{-b^2} e x}{e \sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 \sqrt [4]{-b^2}}\right )}{7 e}+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {8 \left (\frac {\int \frac {\sqrt [4]{-b^2} x e+e}{\sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 \sqrt [4]{-b^2}}-\frac {\int \frac {e-\sqrt [4]{-b^2} e x}{\sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 \sqrt [4]{-b^2}}\right )}{7 e}+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\)

\(\Big \downarrow \) 2422

\(\displaystyle \frac {8 \left (-\frac {e (e x)^{3/2} \sqrt {\frac {\left (\sqrt [4]{-b^2} e x+e\right )^2}{\sqrt [4]{-b^2} e^2 x}} \sqrt {-\frac {e^4-b^2 e^4 x^4}{\sqrt {-b^2} e^4 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt {2} \sqrt {-b^2} x^2 e^2-2 \sqrt [4]{-b^2} x e^2+\sqrt {2} e^2}{\sqrt [4]{-b^2} e^2 x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt {1-b^2 x^4} \left (\sqrt [4]{-b^2} e x+e\right )}-\frac {e (e x)^{3/2} \sqrt {-\frac {\left (e-\sqrt [4]{-b^2} e x\right )^2}{\sqrt [4]{-b^2} e^2 x}} \sqrt {-\frac {e^4-b^2 e^4 x^4}{\sqrt {-b^2} e^4 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt {2} \sqrt {-b^2} x^2 e^2+2 \sqrt [4]{-b^2} x e^2+\sqrt {2} e^2}{\sqrt [4]{-b^2} e^2 x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt {1-b^2 x^4} \left (e-\sqrt [4]{-b^2} e x\right )}\right )}{7 e}+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\)

Input:

Int[Sqrt[e*x]*Sqrt[1 - b*x^2]*Sqrt[1 + b*x^2],x]
 

Output:

(2*(e*x)^(3/2)*Sqrt[1 - b^2*x^4])/(7*e) + (8*(-1/2*(e*(e*x)^(3/2)*Sqrt[(e 
+ (-b^2)^(1/4)*e*x)^2/((-b^2)^(1/4)*e^2*x)]*Sqrt[-((e^4 - b^2*e^4*x^4)/(Sq 
rt[-b^2]*e^4*x^2))]*EllipticF[ArcSin[Sqrt[-((Sqrt[2]*e^2 - 2*(-b^2)^(1/4)* 
e^2*x + Sqrt[2]*Sqrt[-b^2]*e^2*x^2)/((-b^2)^(1/4)*e^2*x))]/2], -2*(1 - Sqr 
t[2])])/(Sqrt[2 + Sqrt[2]]*(e + (-b^2)^(1/4)*e*x)*Sqrt[1 - b^2*x^4]) - (e* 
(e*x)^(3/2)*Sqrt[-((e - (-b^2)^(1/4)*e*x)^2/((-b^2)^(1/4)*e^2*x))]*Sqrt[-( 
(e^4 - b^2*e^4*x^4)/(Sqrt[-b^2]*e^4*x^2))]*EllipticF[ArcSin[Sqrt[(Sqrt[2]* 
e^2 + 2*(-b^2)^(1/4)*e^2*x + Sqrt[2]*Sqrt[-b^2]*e^2*x^2)/((-b^2)^(1/4)*e^2 
*x)]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sqrt[2]]*(e - (-b^2)^(1/4)*e*x)*Sq 
rt[1 - b^2*x^4])))/(7*e)
 

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 335

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p 
_.), x_Symbol] :> Int[(e*x)^m*(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] 
))
 

rule 811

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 
))   Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I 
GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m 
, p, x]
 

rule 838

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Simp[1/(2*Rt[b/a, 4]) 
Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] - Simp[1/(2*Rt[b/a, 4]) 
Int[(1 - Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] /; FreeQ[{a, b}, x]
 

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 2422

Int[((c_) + (d_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Simp[(-c) 
*d*x^3*Sqrt[-(c - d*x^2)^2/(c*d*x^2)]*(Sqrt[(-d^2)*((a + b*x^8)/(b*c^2*x^4) 
)]/(Sqrt[2 + Sqrt[2]]*(c - d*x^2)*Sqrt[a + b*x^8]))*EllipticF[ArcSin[(1/2)* 
Sqrt[(Sqrt[2]*c^2 + 2*c*d*x^2 + Sqrt[2]*d^2*x^4)/(c*d*x^2)]], -2*(1 - Sqrt[ 
2])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^4 - a*d^4, 0]
 

Maple [F]

\[\int \sqrt {e x}\, \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}d x\]

Input:

int((e*x)^(1/2)*(-b*x^2+1)^(1/2)*(b*x^2+1)^(1/2),x)
 

Output:

int((e*x)^(1/2)*(-b*x^2+1)^(1/2)*(b*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \sqrt {e x} \sqrt {1-b x^2} \sqrt {1+b x^2} \, dx=\int { \sqrt {b x^{2} + 1} \sqrt {-b x^{2} + 1} \sqrt {e x} \,d x } \] Input:

integrate((e*x)^(1/2)*(-b*x^2+1)^(1/2)*(b*x^2+1)^(1/2),x, algorithm="frica 
s")
 

Output:

integral(sqrt(b*x^2 + 1)*sqrt(-b*x^2 + 1)*sqrt(e*x), x)
 

Sympy [F]

\[ \int \sqrt {e x} \sqrt {1-b x^2} \sqrt {1+b x^2} \, dx=\int \sqrt {e x} \sqrt {- b x^{2} + 1} \sqrt {b x^{2} + 1}\, dx \] Input:

integrate((e*x)**(1/2)*(-b*x**2+1)**(1/2)*(b*x**2+1)**(1/2),x)
 

Output:

Integral(sqrt(e*x)*sqrt(-b*x**2 + 1)*sqrt(b*x**2 + 1), x)
 

Maxima [F]

\[ \int \sqrt {e x} \sqrt {1-b x^2} \sqrt {1+b x^2} \, dx=\int { \sqrt {b x^{2} + 1} \sqrt {-b x^{2} + 1} \sqrt {e x} \,d x } \] Input:

integrate((e*x)^(1/2)*(-b*x^2+1)^(1/2)*(b*x^2+1)^(1/2),x, algorithm="maxim 
a")
 

Output:

integrate(sqrt(b*x^2 + 1)*sqrt(-b*x^2 + 1)*sqrt(e*x), x)
 

Giac [F(-2)]

Exception generated. \[ \int \sqrt {e x} \sqrt {1-b x^2} \sqrt {1+b x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {e x} \sqrt {1-b x^2} \sqrt {1+b x^2} \, dx=\int \sqrt {e\,x}\,\sqrt {1-b\,x^2}\,\sqrt {b\,x^2+1} \,d x \] Input:

int((e*x)^(1/2)*(1 - b*x^2)^(1/2)*(b*x^2 + 1)^(1/2),x)
 

Output:

int((e*x)^(1/2)*(1 - b*x^2)^(1/2)*(b*x^2 + 1)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {e x} \sqrt {1-b x^2} \sqrt {1+b x^2} \, dx=\frac {2 \sqrt {e}\, \left (\sqrt {x}\, \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, x -2 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}}{b^{2} x^{4}-1}d x \right )\right )}{7} \] Input:

int((e*x)^(1/2)*(-b*x^2+1)^(1/2)*(b*x^2+1)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

(2*sqrt(e)*(sqrt(x)*sqrt(b*x**2 + 1)*sqrt( - b*x**2 + 1)*x - 2*int((sqrt(x 
)*sqrt(b*x**2 + 1)*sqrt( - b*x**2 + 1))/(b**2*x**4 - 1),x)))/7