\(\int \sqrt {e x} (1-b x^2)^{3/2} (1+b x^2)^{3/2} \, dx\) [1354]

Optimal result

Integrand size = 31, antiderivative size = 88 \[ \int \sqrt {e x} \left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2} \, dx=\frac {8 (e x)^{3/2} \sqrt {1-b^2 x^4}}{35 e}+\frac {2 (e x)^{3/2} \left (1-b^2 x^4\right )^{3/2}}{15 e}+\frac {32 (e x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},\frac {1}{2},\frac {11}{8},b^2 x^4\right )}{105 e} \] Output:

8/35*(e*x)^(3/2)*(-b^2*x^4+1)^(1/2)/e+2/15*(e*x)^(3/2)*(-b^2*x^4+1)^(3/2)/ 
e+32/105*(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 = 0.33 \[ \int \sqrt {e x} \left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2} \, dx=\frac {2}{3} x \sqrt {e x} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{8},\frac {11}{8},b^2 x^4\right ) \] Input:

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

Output:

(2*x*Sqrt[e*x]*Hypergeometric2F1[-3/2, 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. \(490\) vs. \(2(88)=176\).

Time = 0.62 (sec) , antiderivative size = 490, normalized size of antiderivative = 5.57, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {335, 811, 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.

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

\(\Big \downarrow \) 335

\(\displaystyle \int \left (1-b^2 x^4\right )^{3/2} \sqrt {e x}dx\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {4}{5} \int \sqrt {e x} \sqrt {1-b^2 x^4}dx+\frac {2 \left (1-b^2 x^4\right )^{3/2} (e x)^{3/2}}{15 e}\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {4}{5} \left (\frac {4}{7} \int \frac {\sqrt {e x}}{\sqrt {1-b^2 x^4}}dx+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} (e x)^{3/2}}{15 e}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {4}{5} \left (\frac {8 \int \frac {e x}{\sqrt {1-b^2 x^4}}d\sqrt {e x}}{7 e}+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} (e x)^{3/2}}{15 e}\)

\(\Big \downarrow \) 838

\(\displaystyle \frac {4}{5} \left (\frac {8 \left (\frac {e \int \frac {\sqrt [4]{-b^2} x e+e}{e \sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 \sqrt [4]{-b^2}}-\frac {e \int \frac {e-\sqrt [4]{-b^2} e x}{e \sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 \sqrt [4]{-b^2}}\right )}{7 e}+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} (e x)^{3/2}}{15 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{5} \left (\frac {8 \left (\frac {\int \frac {\sqrt [4]{-b^2} x e+e}{\sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 \sqrt [4]{-b^2}}-\frac {\int \frac {e-\sqrt [4]{-b^2} e x}{\sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 \sqrt [4]{-b^2}}\right )}{7 e}+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} (e x)^{3/2}}{15 e}\)

\(\Big \downarrow \) 2422

\(\displaystyle \frac {4}{5} \left (\frac {8 \left (-\frac {e (e x)^{3/2} \sqrt {\frac {\left (\sqrt [4]{-b^2} e x+e\right )^2}{\sqrt [4]{-b^2} e^2 x}} \sqrt {-\frac {e^4-b^2 e^4 x^4}{\sqrt {-b^2} e^4 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt {2} \sqrt {-b^2} x^2 e^2-2 \sqrt [4]{-b^2} x e^2+\sqrt {2} e^2}{\sqrt [4]{-b^2} e^2 x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt {1-b^2 x^4} \left (\sqrt [4]{-b^2} e x+e\right )}-\frac {e (e x)^{3/2} \sqrt {-\frac {\left (e-\sqrt [4]{-b^2} e x\right )^2}{\sqrt [4]{-b^2} e^2 x}} \sqrt {-\frac {e^4-b^2 e^4 x^4}{\sqrt {-b^2} e^4 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt {2} \sqrt {-b^2} x^2 e^2+2 \sqrt [4]{-b^2} x e^2+\sqrt {2} e^2}{\sqrt [4]{-b^2} e^2 x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt {1-b^2 x^4} \left (e-\sqrt [4]{-b^2} e x\right )}\right )}{7 e}+\frac {2 \sqrt {1-b^2 x^4} (e x)^{3/2}}{7 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} (e x)^{3/2}}{15 e}\)

Input:

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

Output:

(2*(e*x)^(3/2)*(1 - b^2*x^4)^(3/2))/(15*e) + (4*((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)/(Sqrt[-b^2]*e^4*x^2))]*Ellip 
ticF[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])])/(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)*Sqrt[1 - b^2*x^4])))/(7*e)) 
)/5
 

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]

\[\int \sqrt {e x}\, \left (-b \,x^{2}+1\right )^{\frac {3}{2}} \left (b \,x^{2}+1\right )^{\frac {3}{2}}d x\]

Input:

int((e*x)^(1/2)*(-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2),x)
 

Output:

int((e*x)^(1/2)*(-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2),x)
 

Fricas [F]

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

integrate((e*x)^(1/2)*(-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2),x, algorithm="frica 
s")
 

Output:

integral(-(b^2*x^4 - 1)*sqrt(b*x^2 + 1)*sqrt(-b*x^2 + 1)*sqrt(e*x), x)
 

Sympy [F]

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

integrate((e*x)**(1/2)*(-b*x**2+1)**(3/2)*(b*x**2+1)**(3/2),x)
 

Output:

Integral(sqrt(e*x)*(-b*x**2 + 1)**(3/2)*(b*x**2 + 1)**(3/2), x)
 

Maxima [F]

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

integrate((e*x)^(1/2)*(-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2),x, algorithm="maxim 
a")
 

Output:

integrate((b*x^2 + 1)^(3/2)*(-b*x^2 + 1)^(3/2)*sqrt(e*x), x)
 

Giac [F(-2)]

Exception generated. \[ \int \sqrt {e x} \left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x)^(1/2)*(-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2),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. \[ \int \sqrt {e x} \left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2} \, dx=\int \sqrt {e\,x}\,{\left (1-b\,x^2\right )}^{3/2}\,{\left (b\,x^2+1\right )}^{3/2} \,d x \] Input:

int((e*x)^(1/2)*(1 - b*x^2)^(3/2)*(b*x^2 + 1)^(3/2),x)
 

Output:

int((e*x)^(1/2)*(1 - b*x^2)^(3/2)*(b*x^2 + 1)^(3/2), x)
 

Reduce [F]

\[ \int \sqrt {e x} \left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2} \, dx=\frac {2 \sqrt {e}\, \left (-7 \sqrt {x}\, \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, b^{2} x^{5}+19 \sqrt {x}\, \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, x -24 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}}{b^{2} x^{4}-1}d x \right )\right )}{105} \] Input:

int((e*x)^(1/2)*(-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(2*sqrt(e)*( - 7*sqrt(x)*sqrt(b*x**2 + 1)*sqrt( - b*x**2 + 1)*b**2*x**5 + 
19*sqrt(x)*sqrt(b*x**2 + 1)*sqrt( - b*x**2 + 1)*x - 24*int((sqrt(x)*sqrt(b 
*x**2 + 1)*sqrt( - b*x**2 + 1))/(b**2*x**4 - 1),x)))/105