\(\int x (-a+b x^n)^p (a+b x^n)^p \, dx\) [577]

Optimal result

Integrand size = 22, antiderivative size = 70 \[ \int x \left (-a+b x^n\right )^p \left (a+b x^n\right )^p \, dx=\frac {1}{2} x^2 \left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},\frac {b^2 x^{2 n}}{a^2}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03 \[ \int x \left (-a+b x^n\right )^p \left (a+b x^n\right )^p \, dx=\frac {1}{2} x^2 \left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};\frac {b^2 x^{2 n}}{a^2}\right ) \] Input:

Integrate[x*(-a + b*x^n)^p*(a + b*x^n)^p,x]
 

Output:

(x^2*(-a + b*x^n)^p*(a + b*x^n)^p*HypergeometricPFQ[{n^(-1), -p}, {1 + n^( 
-1)}, (b^2*x^(2*n))/a^2])/(2*(1 - (b^2*x^(2*n))/a^2)^p)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {890, 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.

\(\displaystyle \int x \left (b x^n-a\right )^p \left (a+b x^n\right )^p \, dx\)

\(\Big \downarrow \) 890

\(\displaystyle \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (b^2 x^{2 n}-a^2\right )^{-p} \int x \left (b^2 x^{2 n}-a^2\right )^pdx\)

\(\Big \downarrow \) 889

\(\displaystyle \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \int x \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^pdx\)

\(\Big \downarrow \) 888

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

Input:

Int[x*(-a + b*x^n)^p*(a + b*x^n)^p,x]
 

Output:

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

Defintions of rubi rules used

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])
 

rule 890

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_) 
^(n_))^(p_), x_Symbol] :> Simp[(a1 + b1*x^n)^FracPart[p]*((a2 + b2*x^n)^Fra 
cPart[p]/(a1*a2 + b1*b2*x^(2*n))^FracPart[p])   Int[(c*x)^m*(a1*a2 + b1*b2* 
x^(2*n))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + 
 a1*b2, 0] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

int(x*(-a+b*x^n)^p*(a+b*x^n)^p,x)
 

Output:

int(x*(-a+b*x^n)^p*(a+b*x^n)^p,x)
 

Fricas [F]

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

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

Output:

integral((b*x^n + a)^p*(b*x^n - a)^p*x, x)
 

Sympy [F]

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

integrate(x*(-a+b*x**n)**p*(a+b*x**n)**p,x)
 

Output:

Integral(x*(-a + b*x**n)**p*(a + b*x**n)**p, x)
 

Maxima [F]

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

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

Output:

integrate((b*x^n + a)^p*(b*x^n - a)^p*x, x)
 

Giac [F]

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

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

Output:

integrate((b*x^n + a)^p*(b*x^n - a)^p*x, x)
 

Mupad [F(-1)]

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

int(x*(a + b*x^n)^p*(b*x^n - a)^p,x)
 

Output:

int(x*(a + b*x^n)^p*(b*x^n - a)^p, x)
 

Reduce [F]

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

int(x*(-a+b*x^n)^p*(a+b*x^n)^p,x)
 

Output:

((x**n*b + a)**p*(x**n*b - a)**p*x**2 - 2*int(((x**n*b + a)**p*(x**n*b - a 
)**p*x)/(x**(2*n)*b**2*n*p + x**(2*n)*b**2 - a**2*n*p - a**2),x)*a**2*n**2 
*p**2 - 2*int(((x**n*b + a)**p*(x**n*b - a)**p*x)/(x**(2*n)*b**2*n*p + x** 
(2*n)*b**2 - a**2*n*p - a**2),x)*a**2*n*p)/(2*(n*p + 1))