\(\int (a-b x^{n/2})^p (a+b x^{n/2})^p (\frac {a^2 d (1+p)}{b^2 (1+\frac {-1-2 n-n p}{n})}+d x^n)^{\frac {-1-2 n-n p}{n}} \, dx\) [6]

Optimal result

Integrand size = 76, antiderivative size = 96 \[ \int \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (\frac {a^2 d (1+p)}{b^2 \left (1+\frac {-1-2 n-n p}{n}\right )}+d x^n\right )^{\frac {-1-2 n-n p}{n}} \, dx=-\frac {b^2 (1+n+n p) x \left (a-b x^{n/2}\right )^{1+p} \left (a+b x^{n/2}\right )^{1+p} \left (-\frac {a^2 d n (1+p)}{b^2 (1+n+n p)}+d x^n\right )^{-\frac {1+n+n p}{n}}}{a^4 d n (1+p)} \] Output:

-b^2*(n*p+n+1)*x*(a-b*x^(1/2*n))^(p+1)*(a+b*x^(1/2*n))^(p+1)/a^4/d/n/(p+1) 
/((-a^2*d*n*(p+1)/b^2/(n*p+n+1)+d*x^n)^((n*p+n+1)/n))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.07 \[ \int \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (\frac {a^2 d (1+p)}{b^2 \left (1+\frac {-1-2 n-n p}{n}\right )}+d x^n\right )^{\frac {-1-2 n-n p}{n}} \, dx=-\frac {b^2 (1+n+n p) x \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (d \left (-\frac {a^2 n (1+p)}{b^2 (1+n+n p)}+x^n\right )\right )^{-\frac {1+n+n p}{n}} \left (a^2-b^2 x^n\right )}{a^4 d n (1+p)} \] Input:

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

Output:

-((b^2*(1 + n + n*p)*x*(a - b*x^(n/2))^p*(a + b*x^(n/2))^p*(a^2 - b^2*x^n) 
)/(a^4*d*n*(1 + p)*(d*(-((a^2*n*(1 + p))/(b^2*(1 + n + n*p))) + x^n))^((1 
+ n + n*p)/n)))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2038, 906}

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.

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

\(\Big \downarrow \) 2038

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

\(\Big \downarrow \) 906

\(\displaystyle -\frac {b^2 x (n p+n+1) \left (a^2-b^2 x^n\right ) \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (d x^n-\frac {a^2 d n (p+1)}{b^2 (n p+n+1)}\right )^{-\frac {1}{n}-p-1}}{a^4 d n (p+1)}\)

Input:

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

Output:

-((b^2*(1 + n + n*p)*x*(a - b*x^(n/2))^p*(a + b*x^(n/2))^p*(a^2 - b^2*x^n) 
*(-((a^2*d*n*(1 + p))/(b^2*(1 + n + n*p))) + d*x^n)^(-1 - n^(-1) - p))/(a^ 
4*d*n*(1 + p)))
 

Defintions of rubi rules used

rule 906

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c)), x] /; FreeQ[{a, 
b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && 
EqQ[a*d*(p + 1) + b*c*(q + 1), 0]
 

rule 2038

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p 
_)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_), x_Symbol] :> Simp[(a1 + b1*x^(n/2)) 
^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPart[p] 
)   Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, 
 b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(Eq 
Q[n, 2] && IGtQ[q, 0])
 

Maple [F]

\[\int \left (a -b \,x^{\frac {n}{2}}\right )^{p} \left (a +b \,x^{\frac {n}{2}}\right )^{p} \left (\frac {a^{2} d \left (p +1\right )}{b^{2} \left (1+\frac {-n p -2 n -1}{n}\right )}+d \,x^{n}\right )^{\frac {-n p -2 n -1}{n}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.88 \[ \int \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (\frac {a^2 d (1+p)}{b^2 \left (1+\frac {-1-2 n-n p}{n}\right )}+d x^n\right )^{\frac {-1-2 n-n p}{n}} \, dx=\frac {{\left ({\left (b^{4} n p + b^{4} n + b^{4}\right )} x x^{2 \, n} - {\left (2 \, a^{2} b^{2} n p + 2 \, a^{2} b^{2} n + a^{2} b^{2}\right )} x x^{n} + {\left (a^{4} n p + a^{4} n\right )} x\right )} {\left (b x^{\frac {1}{2} \, n} + a\right )}^{p} {\left (-b x^{\frac {1}{2} \, n} + a\right )}^{p}}{{\left (a^{4} n p + a^{4} n\right )} \left (-\frac {a^{2} d n p + a^{2} d n - {\left (b^{2} d n p + b^{2} d n + b^{2} d\right )} x^{n}}{b^{2} n p + b^{2} n + b^{2}}\right )^{\frac {n p + 2 \, n + 1}{n}}} \] Input:

integrate((a-b*x^(1/2*n))^p*(a+b*x^(1/2*n))^p*(a^2*d*(p+1)/b^2/(1+(-n*p-2* 
n-1)/n)+d*x^n)^((-n*p-2*n-1)/n),x, algorithm="fricas")
 

Output:

((b^4*n*p + b^4*n + b^4)*x*x^(2*n) - (2*a^2*b^2*n*p + 2*a^2*b^2*n + a^2*b^ 
2)*x*x^n + (a^4*n*p + a^4*n)*x)*(b*x^(1/2*n) + a)^p*(-b*x^(1/2*n) + a)^p/( 
(a^4*n*p + a^4*n)*(-(a^2*d*n*p + a^2*d*n - (b^2*d*n*p + b^2*d*n + b^2*d)*x 
^n)/(b^2*n*p + b^2*n + b^2))^((n*p + 2*n + 1)/n))
 

Sympy [F(-1)]

Timed out. \[ \int \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (\frac {a^2 d (1+p)}{b^2 \left (1+\frac {-1-2 n-n p}{n}\right )}+d x^n\right )^{\frac {-1-2 n-n p}{n}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

integrate((a-b*x^(1/2*n))^p*(a+b*x^(1/2*n))^p*(a^2*d*(p+1)/b^2/(1+(-n*p-2* 
n-1)/n)+d*x^n)^((-n*p-2*n-1)/n),x, algorithm="maxima")
 

Output:

integrate((b*x^(1/2*n) + a)^p*(-b*x^(1/2*n) + a)^p/(d*x^n - a^2*d*(p + 1)/ 
(b^2*((n*p + 2*n + 1)/n - 1)))^((n*p + 2*n + 1)/n), x)
 

Giac [F]

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

integrate((a-b*x^(1/2*n))^p*(a+b*x^(1/2*n))^p*(a^2*d*(p+1)/b^2/(1+(-n*p-2* 
n-1)/n)+d*x^n)^((-n*p-2*n-1)/n),x, algorithm="giac")
 

Output:

integrate((b*x^(1/2*n) + a)^p*(-b*x^(1/2*n) + a)^p/(d*x^n - a^2*d*(p + 1)/ 
(b^2*((n*p + 2*n + 1)/n - 1)))^((n*p + 2*n + 1)/n), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (\frac {a^2 d (1+p)}{b^2 \left (1+\frac {-1-2 n-n p}{n}\right )}+d x^n\right )^{\frac {-1-2 n-n p}{n}} \, dx=\int \frac {{\left (a+b\,x^{n/2}\right )}^p\,{\left (a-b\,x^{n/2}\right )}^p}{{\left (d\,x^n-\frac {a^2\,d\,\left (p+1\right )}{b^2\,\left (\frac {2\,n+n\,p+1}{n}-1\right )}\right )}^{\frac {2\,n+n\,p+1}{n}}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (\frac {a^2 d (1+p)}{b^2 \left (1+\frac {-1-2 n-n p}{n}\right )}+d x^n\right )^{\frac {-1-2 n-n p}{n}} \, dx=\text {too large to display} \] Input:

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

Output:

((b**2*n*p + b**2*n + b**2)**((n*p + 1)/n)*b**4*( - int(((x**(n/2)*b + a)* 
*p*( - x**(n/2)*b + a)**p)/(x**(3*n)*(x**n*b**2*d*n*p + x**n*b**2*d*n + x* 
*n*b**2*d - a**2*d*n*p - a**2*d*n)**((n*p + 1)/n)*b**6*n**2*p**2 + 2*x**(3 
*n)*(x**n*b**2*d*n*p + x**n*b**2*d*n + x**n*b**2*d - a**2*d*n*p - a**2*d*n 
)**((n*p + 1)/n)*b**6*n**2*p + x**(3*n)*(x**n*b**2*d*n*p + x**n*b**2*d*n + 
 x**n*b**2*d - a**2*d*n*p - a**2*d*n)**((n*p + 1)/n)*b**6*n**2 + 2*x**(3*n 
)*(x**n*b**2*d*n*p + x**n*b**2*d*n + x**n*b**2*d - a**2*d*n*p - a**2*d*n)* 
*((n*p + 1)/n)*b**6*n*p + 2*x**(3*n)*(x**n*b**2*d*n*p + x**n*b**2*d*n + x* 
*n*b**2*d - a**2*d*n*p - a**2*d*n)**((n*p + 1)/n)*b**6*n + x**(3*n)*(x**n* 
b**2*d*n*p + x**n*b**2*d*n + x**n*b**2*d - a**2*d*n*p - a**2*d*n)**((n*p + 
 1)/n)*b**6 - 3*x**(2*n)*(x**n*b**2*d*n*p + x**n*b**2*d*n + x**n*b**2*d - 
a**2*d*n*p - a**2*d*n)**((n*p + 1)/n)*a**2*b**4*n**2*p**2 - 6*x**(2*n)*(x* 
*n*b**2*d*n*p + x**n*b**2*d*n + x**n*b**2*d - a**2*d*n*p - a**2*d*n)**((n* 
p + 1)/n)*a**2*b**4*n**2*p - 3*x**(2*n)*(x**n*b**2*d*n*p + x**n*b**2*d*n + 
 x**n*b**2*d - a**2*d*n*p - a**2*d*n)**((n*p + 1)/n)*a**2*b**4*n**2 - 4*x* 
*(2*n)*(x**n*b**2*d*n*p + x**n*b**2*d*n + x**n*b**2*d - a**2*d*n*p - a**2* 
d*n)**((n*p + 1)/n)*a**2*b**4*n*p - 4*x**(2*n)*(x**n*b**2*d*n*p + x**n*b** 
2*d*n + x**n*b**2*d - a**2*d*n*p - a**2*d*n)**((n*p + 1)/n)*a**2*b**4*n - 
x**(2*n)*(x**n*b**2*d*n*p + x**n*b**2*d*n + x**n*b**2*d - a**2*d*n*p - a** 
2*d*n)**((n*p + 1)/n)*a**2*b**4 + 3*x**n*(x**n*b**2*d*n*p + x**n*b**2*d...