\(\int \sqrt {e x} \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx\) [1359]

Optimal result

Integrand size = 31, antiderivative size = 115 \[ \int \sqrt {e x} \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\frac {2 (e x)^{3/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}{7 e}+\frac {8 a^2 (e x)^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},\frac {1}{2},\frac {11}{8},\frac {b^2 x^4}{a^2}\right )}{21 e \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64 \[ \int \sqrt {e x} \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\frac {2 x \sqrt {e x} \sqrt {a-b x^2} \sqrt {a+b x^2} \, _2F_1\left (-\frac {1}{2},\frac {3}{8};\frac {11}{8};\frac {b^2 x^4}{a^2}\right )}{3 \sqrt {1-\frac {b^2 x^4}{a^2}}} \] Input:

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

Output:

(2*x*Sqrt[e*x]*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*HypergeometricPFQ[{-1/2, 3/ 
8}, {11/8}, (b^2*x^4)/a^2])/(3*Sqrt[1 - (b^2*x^4)/a^2])
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(576\) vs. \(2(115)=230\).

Time = 0.78 (sec) , antiderivative size = 576, normalized size of antiderivative = 5.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {340, 344, 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.

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

\(\Big \downarrow \) 340

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

\(\Big \downarrow \) 344

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

\(\Big \downarrow \) 851

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

\(\Big \downarrow \) 838

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2422

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

Input:

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

Output:

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

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 340

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

rule 344

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(e)*(sqrt(x)*sqrt(a + b*x**2)*sqrt(a - b*x**2)*x + 2*int((sqrt(x)*s 
qrt(a + b*x**2)*sqrt(a - b*x**2))/(a**2 - b**2*x**4),x)*a**2))/7