\(\int (a-b x^n)^{3/2} (a+b x^n)^{3/2} \, dx\) [64]

Optimal result

Integrand size = 24, antiderivative size = 79 \[ \int \left (a-b x^n\right )^{3/2} \left (a+b x^n\right )^{3/2} \, dx=\frac {a^2 x \sqrt {a-b x^n} \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),\frac {b^2 x^{2 n}}{a^2}\right )}{\sqrt {1-\frac {b^2 x^{2 n}}{a^2}}} \] Output:

a^2*x*(a-b*x^n)^(1/2)*(a+b*x^n)^(1/2)*hypergeom([-3/2, 1/2/n],[1+1/2/n],b^ 
2*x^(2*n)/a^2)/(1-b^2*x^(2*n)/a^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \left (a-b x^n\right )^{3/2} \left (a+b x^n\right )^{3/2} \, dx=\frac {a^2 x \sqrt {a-b x^n} \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{2 n},1+\frac {1}{2 n},\frac {b^2 x^{2 n}}{a^2}\right )}{\sqrt {1-\frac {b^2 x^{2 n}}{a^2}}} \] Input:

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

Output:

(a^2*x*Sqrt[a - b*x^n]*Sqrt[a + b*x^n]*Hypergeometric2F1[-3/2, 1/(2*n), 1 
+ 1/(2*n), (b^2*x^(2*n))/a^2])/Sqrt[1 - (b^2*x^(2*n))/a^2]
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {785, 779, 778}

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.

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

\(\Big \downarrow \) 785

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

\(\Big \downarrow \) 779

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

\(\Big \downarrow \) 778

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

Input:

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

Output:

(a^2*x*Sqrt[a - b*x^n]*Sqrt[a + b*x^n]*Hypergeometric2F1[-3/2, 1/(2*n), (2 
 + n^(-1))/2, (b^2*x^(2*n))/a^2])/Sqrt[1 - (b^2*x^(2*n))/a^2]
 

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 785

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Sy 
mbol] :> Simp[(a1 + b1*x^n)^FracPart[p]*((a2 + b2*x^n)^FracPart[p]/(a1*a2 + 
 b1*b2*x^(2*n))^FracPart[p])   Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /; Fre 
eQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a-b x^n\right )^{3/2} \left (a+b x^n\right )^{3/2} \, dx=\frac {-x^{2 n} \sqrt {x^{n} b +a}\, \sqrt {-x^{n} b +a}\, b^{2} n x -x^{2 n} \sqrt {x^{n} b +a}\, \sqrt {-x^{n} b +a}\, b^{2} x +4 \sqrt {x^{n} b +a}\, \sqrt {-x^{n} b +a}\, a^{2} n x +\sqrt {x^{n} b +a}\, \sqrt {-x^{n} b +a}\, a^{2} x -9 \left (\int \frac {\sqrt {x^{n} b +a}\, \sqrt {-x^{n} b +a}}{3 x^{2 n} b^{2} n^{2}+4 x^{2 n} b^{2} n +x^{2 n} b^{2}-3 a^{2} n^{2}-4 a^{2} n -a^{2}}d x \right ) a^{4} n^{4}-12 \left (\int \frac {\sqrt {x^{n} b +a}\, \sqrt {-x^{n} b +a}}{3 x^{2 n} b^{2} n^{2}+4 x^{2 n} b^{2} n +x^{2 n} b^{2}-3 a^{2} n^{2}-4 a^{2} n -a^{2}}d x \right ) a^{4} n^{3}-3 \left (\int \frac {\sqrt {x^{n} b +a}\, \sqrt {-x^{n} b +a}}{3 x^{2 n} b^{2} n^{2}+4 x^{2 n} b^{2} n +x^{2 n} b^{2}-3 a^{2} n^{2}-4 a^{2} n -a^{2}}d x \right ) a^{4} n^{2}}{3 n^{2}+4 n +1} \] Input:

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

Output:

( - x**(2*n)*sqrt(x**n*b + a)*sqrt( - x**n*b + a)*b**2*n*x - x**(2*n)*sqrt 
(x**n*b + a)*sqrt( - x**n*b + a)*b**2*x + 4*sqrt(x**n*b + a)*sqrt( - x**n* 
b + a)*a**2*n*x + sqrt(x**n*b + a)*sqrt( - x**n*b + a)*a**2*x - 9*int((sqr 
t(x**n*b + a)*sqrt( - x**n*b + a))/(3*x**(2*n)*b**2*n**2 + 4*x**(2*n)*b**2 
*n + x**(2*n)*b**2 - 3*a**2*n**2 - 4*a**2*n - a**2),x)*a**4*n**4 - 12*int( 
(sqrt(x**n*b + a)*sqrt( - x**n*b + a))/(3*x**(2*n)*b**2*n**2 + 4*x**(2*n)* 
b**2*n + x**(2*n)*b**2 - 3*a**2*n**2 - 4*a**2*n - a**2),x)*a**4*n**3 - 3*i 
nt((sqrt(x**n*b + a)*sqrt( - x**n*b + a))/(3*x**(2*n)*b**2*n**2 + 4*x**(2* 
n)*b**2*n + x**(2*n)*b**2 - 3*a**2*n**2 - 4*a**2*n - a**2),x)*a**4*n**2)/( 
3*n**2 + 4*n + 1)