\(\int (e x)^m (a-b x^2)^{3/2} (a+b x^2)^{3/2} \, dx\) [1394]

Optimal result

Integrand size = 29, antiderivative size = 87 \[ \int (e x)^m \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2} \, dx=\frac {a^2 (e x)^{1+m} \sqrt {a-b x^2} \sqrt {a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{4},\frac {5+m}{4},\frac {b^2 x^4}{a^2}\right )}{e (1+m) \sqrt {1-\frac {b^2 x^4}{a^2}}} \] Output:

a^2*(e*x)^(1+m)*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)*hypergeom([-3/2, 1/4+1/4* 
m],[5/4+1/4*m],b^2*x^4/a^2)/e/(1+m)/(1-b^2*x^4/a^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02 \[ \int (e x)^m \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2} \, dx=\frac {a^2 x (e x)^m \sqrt {a-b x^2} \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},\frac {1}{4}+\frac {m}{4};\frac {5}{4}+\frac {m}{4};\frac {b^2 x^4}{a^2}\right )}{(1+m) \sqrt {1-\frac {b^2 x^4}{a^2}}} \] Input:

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

Output:

(a^2*x*(e*x)^m*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*HypergeometricPFQ[{-3/2, 1/ 
4 + m/4}, {5/4 + m/4}, (b^2*x^4)/a^2])/((1 + m)*Sqrt[1 - (b^2*x^4)/a^2])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {344, 889, 888}

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.

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

\(\Big \downarrow \) 344

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {a^2 \sqrt {a-b x^2} \sqrt {a+b x^2} (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {m+1}{4},\frac {m+5}{4},\frac {b^2 x^4}{a^2}\right )}{e (m+1) \sqrt {1-\frac {b^2 x^4}{a^2}}}\)

Input:

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

Output:

(a^2*(e*x)^(1 + m)*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, 
 (1 + m)/4, (5 + m)/4, (b^2*x^4)/a^2])/(e*(1 + m)*Sqrt[1 - (b^2*x^4)/a^2])
 

Defintions of rubi rules used

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 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((e*x)**m*(a - b*x**2)**(3/2)*(a + b*x**2)**(3/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^m \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2} \, dx=\frac {e^{m} \left (x^{m} \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a^{2} m x +9 x^{m} \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a^{2} x -x^{m} \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, b^{2} m \,x^{5}-3 x^{m} \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, b^{2} x^{5}+12 \left (\int \frac {x^{m} \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{-b^{2} m^{2} x^{4}-10 b^{2} m \,x^{4}-21 b^{2} x^{4}+a^{2} m^{2}+10 a^{2} m +21 a^{2}}d x \right ) a^{4} m^{2}+120 \left (\int \frac {x^{m} \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{-b^{2} m^{2} x^{4}-10 b^{2} m \,x^{4}-21 b^{2} x^{4}+a^{2} m^{2}+10 a^{2} m +21 a^{2}}d x \right ) a^{4} m +252 \left (\int \frac {x^{m} \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{-b^{2} m^{2} x^{4}-10 b^{2} m \,x^{4}-21 b^{2} x^{4}+a^{2} m^{2}+10 a^{2} m +21 a^{2}}d x \right ) a^{4}\right )}{m^{2}+10 m +21} \] Input:

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

Output:

(e**m*(x**m*sqrt(a + b*x**2)*sqrt(a - b*x**2)*a**2*m*x + 9*x**m*sqrt(a + b 
*x**2)*sqrt(a - b*x**2)*a**2*x - x**m*sqrt(a + b*x**2)*sqrt(a - b*x**2)*b* 
*2*m*x**5 - 3*x**m*sqrt(a + b*x**2)*sqrt(a - b*x**2)*b**2*x**5 + 12*int((x 
**m*sqrt(a + b*x**2)*sqrt(a - b*x**2))/(a**2*m**2 + 10*a**2*m + 21*a**2 - 
b**2*m**2*x**4 - 10*b**2*m*x**4 - 21*b**2*x**4),x)*a**4*m**2 + 120*int((x* 
*m*sqrt(a + b*x**2)*sqrt(a - b*x**2))/(a**2*m**2 + 10*a**2*m + 21*a**2 - b 
**2*m**2*x**4 - 10*b**2*m*x**4 - 21*b**2*x**4),x)*a**4*m + 252*int((x**m*s 
qrt(a + b*x**2)*sqrt(a - b*x**2))/(a**2*m**2 + 10*a**2*m + 21*a**2 - b**2* 
m**2*x**4 - 10*b**2*m*x**4 - 21*b**2*x**4),x)*a**4))/(m**2 + 10*m + 21)