\(\int \frac {(e x)^{7/2}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx\) [1379]

Optimal result

Integrand size = 31, antiderivative size = 74 \[ \int \frac {(e x)^{7/2}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\frac {2 (e x)^{9/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{8},\frac {17}{8},\frac {b^2 x^4}{a^2}\right )}{9 e \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:

2/9*(e*x)^(9/2)*(1-b^2*x^4/a^2)^(1/2)*hypergeom([1/2, 9/8],[17/8],b^2*x^4/ 
a^2)/e/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)
 

Mathematica [A] (verified)

Time = 3.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {(e x)^{7/2}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\frac {2 x (e x)^{7/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \, _2F_1\left (\frac {1}{2},\frac {9}{8};\frac {17}{8};\frac {b^2 x^4}{a^2}\right )}{9 \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Input:

Integrate[(e*x)^(7/2)/(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]),x]
 

Output:

(2*x*(e*x)^(7/2)*Sqrt[1 - (b^2*x^4)/a^2]*HypergeometricPFQ[{1/2, 9/8}, {17 
/8}, (b^2*x^4)/a^2])/(9*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. \(602\) vs. \(2(74)=148\).

Time = 0.70 (sec) , antiderivative size = 602, normalized size of antiderivative = 8.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {344, 843, 851, 767, 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.

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

\(\Big \downarrow \) 344

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

\(\Big \downarrow \) 843

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

\(\Big \downarrow \) 851

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

\(\Big \downarrow \) 767

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2422

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

Input:

Int[(e*x)^(7/2)/(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]),x]
 

Output:

(Sqrt[a^2 - b^2*x^4]*((-2*e^3*Sqrt[e*x]*Sqrt[a^2 - b^2*x^4])/(5*b^2) + (2* 
a^2*e^3*((Sqrt[b]*(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]]*(-a^2)^(1/4)*(e + (Sqrt[b]*e*x)/(- 
a^2)^(1/4))*Sqrt[a^2 - b^2*x^4]) - (Sqrt[b]*(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)*(S 
qrt[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]]*(-a^2)^( 
1/4)*(e - (Sqrt[b]*e*x)/(-a^2)^(1/4))*Sqrt[a^2 - b^2*x^4])))/(5*b^2)))/(Sq 
rt[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 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 767

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

rule 843

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

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

Input:

int((e*x)^(7/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x)
 

Output:

int((e*x)^(7/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x)
 

Fricas [F]

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

integrate((e*x)^(7/2)/(-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)*e^3*x^3/(b^2*x^4 - a^ 
2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(7/2)/(-b*x**2+a)**(1/2)/(b*x**2+a)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((e*x)^(7/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="maxim 
a")
 

Output:

integrate((e*x)^(7/2)/(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)), x)
 

Giac [F]

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

integrate((e*x)^(7/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="giac" 
)
 

Output:

integrate((e*x)^(7/2)/(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)), x)
 

Mupad [F(-1)]

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

int((e*x)^(7/2)/((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2)),x)
 

Output:

int((e*x)^(7/2)/((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2)), x)
 

Reduce [F]

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

int((e*x)^(7/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

(sqrt(e)*e**3*( - 2*sqrt(x)*sqrt(a + b*x**2)*sqrt(a - b*x**2) + int((sqrt( 
x)*sqrt(a + b*x**2)*sqrt(a - b*x**2))/(a**2*x - b**2*x**5),x)*a**2))/(5*b* 
*2)