\(\int \frac {\sqrt {e x}}{(a-b x^2)^{3/2} (a+b x^2)^{3/2}} \, dx\) [1387]

Optimal result

Integrand size = 31, antiderivative size = 118 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {(e x)^{3/2}}{2 a^2 e \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {(e x)^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},\frac {1}{2},\frac {11}{8},\frac {b^2 x^4}{a^2}\right )}{6 a^2 e \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:

1/2*(e*x)^(3/2)/a^2/e/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)+1/6*(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)/a^2/e/(-b*x^2+ 
a)^(1/2)/(b*x^2+a)^(1/2)
 

Mathematica [A] (verified)

Time = 9.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {2 x \sqrt {e x} \sqrt {1-\frac {b^2 x^4}{a^2}} \, _2F_1\left (\frac {3}{8},\frac {3}{2};\frac {11}{8};\frac {b^2 x^4}{a^2}\right )}{3 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Input:

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

Output:

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

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(579\) vs. \(2(118)=236\).

Time = 0.67 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {342, 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.

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

\(\Big \downarrow \) 342

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

\(\Big \downarrow \) 344

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

\(\Big \downarrow \) 851

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

\(\Big \downarrow \) 838

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2422

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

Input:

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

Output:

(e*x)^(3/2)/(2*a^2*e*Sqrt[a - b*x^2]*Sqrt[a + b*x^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))*Sqrt[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))*Sq 
rt[a^2 - b^2*x^4])))/(2*a^2*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 342

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 + 1)*((c + d*x^2)^(p + 
 1)/(4*a*c*e*(p + 1))), x] + Simp[(m + 4*p + 5)/(4*a*c*(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] && LtQ[p, -1] && 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)*sqrt(e*x)/(b^4*x^8 - 2*a^2*b^2*x 
^4 + a^4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 14.45 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=- \frac {i \sqrt {e} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{8}, \frac {11}{8}, 1 & \frac {5}{8}, \frac {13}{8}, \frac {17}{8} \\\frac {7}{8}, \frac {9}{8}, \frac {11}{8}, \frac {13}{8}, \frac {17}{8} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{4}}} \right )}}{4 \pi ^{\frac {3}{2}} a^{\frac {9}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {e} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{8}, \frac {1}{8}, \frac {3}{8}, \frac {5}{8}, \frac {7}{8}, 1 & \\\frac {3}{8}, \frac {7}{8} & - \frac {3}{8}, \frac {1}{8}, \frac {9}{8}, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{4}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi ^{\frac {3}{2}} a^{\frac {9}{4}} b^{\frac {3}{4}}} \] Input:

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

Output:

-I*sqrt(e)*meijerg(((7/8, 11/8, 1), (5/8, 13/8, 17/8)), ((7/8, 9/8, 11/8, 
13/8, 17/8), (0,)), a**2/(b**2*x**4))/(4*pi**(3/2)*a**(9/4)*b**(3/4)) + sq 
rt(e)*meijerg(((-3/8, 1/8, 3/8, 5/8, 7/8, 1), ()), ((3/8, 7/8), (-3/8, 1/8 
, 9/8, 0)), a**2*exp_polar(-2*I*pi)/(b**2*x**4))*exp(I*pi/4)/(4*pi**(3/2)* 
a**(9/4)*b**(3/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{b^{4} x^{8}-2 a^{2} b^{2} x^{4}+a^{4}}d x \right ) \] Input:

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

Output:

sqrt(e)*int((sqrt(x)*sqrt(a + b*x**2)*sqrt(a - b*x**2))/(a**4 - 2*a**2*b** 
2*x**4 + b**4*x**8),x)