\(\int \frac {(a+b x^n)^{5/2} (A+B x^n)}{x^2} \, dx\) [348]

Optimal result

Integrand size = 22, antiderivative size = 105 \[ \int \frac {\left (a+b x^n\right )^{5/2} \left (A+B x^n\right )}{x^2} \, dx=-\frac {2 B \left (a+b x^n\right )^{7/2}}{b (2-7 n) x}-\frac {a^2 \left (A-\frac {2 a B}{2 b-7 b n}\right ) \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{x \sqrt {1+\frac {b x^n}{a}}} \] Output:

-2*B*(a+b*x^n)^(7/2)/b/(2-7*n)/x-a^2*(A-2*a*B/(-7*b*n+2*b))*(a+b*x^n)^(1/2 
)*hypergeom([-5/2, -1/n],[-(1-n)/n],-b*x^n/a)/x/(1+b*x^n/a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^n\right )^{5/2} \left (A+B x^n\right )}{x^2} \, dx=\frac {a^2 \sqrt {a+b x^n} \left ((A-A n) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {1}{n},\frac {-1+n}{n},-\frac {b x^n}{a}\right )+B x^n \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {-1+n}{n},2-\frac {1}{n},-\frac {b x^n}{a}\right )\right )}{(-1+n) x \sqrt {1+\frac {b x^n}{a}}} \] Input:

Integrate[((a + b*x^n)^(5/2)*(A + B*x^n))/x^2,x]
 

Output:

(a^2*Sqrt[a + b*x^n]*((A - A*n)*Hypergeometric2F1[-5/2, -n^(-1), (-1 + n)/ 
n, -((b*x^n)/a)] + B*x^n*Hypergeometric2F1[-5/2, (-1 + n)/n, 2 - n^(-1), - 
((b*x^n)/a)]))/((-1 + n)*x*Sqrt[1 + (b*x^n)/a])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 105, 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 = {959, 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.

\(\displaystyle \int \frac {\left (a+b x^n\right )^{5/2} \left (A+B x^n\right )}{x^2} \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \left (A-\frac {2 a B}{2 b-7 b n}\right ) \int \frac {\left (b x^n+a\right )^{5/2}}{x^2}dx-\frac {2 B \left (a+b x^n\right )^{7/2}}{b (2-7 n) x}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {a^2 \sqrt {a+b x^n} \left (A-\frac {2 a B}{2 b-7 b n}\right ) \int \frac {\left (\frac {b x^n}{a}+1\right )^{5/2}}{x^2}dx}{\sqrt {\frac {b x^n}{a}+1}}-\frac {2 B \left (a+b x^n\right )^{7/2}}{b (2-7 n) x}\)

\(\Big \downarrow \) 888

\(\displaystyle -\frac {a^2 \sqrt {a+b x^n} \left (A-\frac {2 a B}{2 b-7 b n}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{x \sqrt {\frac {b x^n}{a}+1}}-\frac {2 B \left (a+b x^n\right )^{7/2}}{b (2-7 n) x}\)

Input:

Int[((a + b*x^n)^(5/2)*(A + B*x^n))/x^2,x]
 

Output:

(-2*B*(a + b*x^n)^(7/2))/(b*(2 - 7*n)*x) - (a^2*(A - (2*a*B)/(2*b - 7*b*n) 
)*Sqrt[a + b*x^n]*Hypergeometric2F1[-5/2, -n^(-1), -((1 - n)/n), -((b*x^n) 
/a)])/(x*Sqrt[1 + (b*x^n)/a])
 

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 959

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

Maple [F]

\[\int \frac {\left (a +b \,x^{n}\right )^{\frac {5}{2}} \left (A +B \,x^{n}\right )}{x^{2}}d x\]

Input:

int((a+b*x^n)^(5/2)*(A+B*x^n)/x^2,x)
 

Output:

int((a+b*x^n)^(5/2)*(A+B*x^n)/x^2,x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^n\right )^{5/2} \left (A+B x^n\right )}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*x^n)^(5/2)*(A+B*x^n)/x^2,x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.64 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.70 \[ \int \frac {\left (a+b x^n\right )^{5/2} \left (A+B x^n\right )}{x^2} \, dx=\frac {A a^{2} a^{- \frac {1}{n}} a^{\frac {1}{2} + \frac {1}{n}} \Gamma \left (- \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{n} \\ 1 - \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n x \Gamma \left (1 - \frac {1}{n}\right )} + \frac {2 A a a^{- \frac {1}{2} + \frac {1}{n}} a^{1 - \frac {1}{n}} b x^{n - 1} \Gamma \left (1 - \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 - \frac {1}{n} \\ 2 - \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 - \frac {1}{n}\right )} + \frac {A a^{- \frac {3}{2} + \frac {1}{n}} a^{2 - \frac {1}{n}} b^{2} x^{2 n - 1} \Gamma \left (2 - \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 2 - \frac {1}{n} \\ 3 - \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 - \frac {1}{n}\right )} + \frac {B a^{2} a^{- \frac {1}{2} + \frac {1}{n}} a^{1 - \frac {1}{n}} x^{n - 1} \Gamma \left (1 - \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 - \frac {1}{n} \\ 2 - \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 - \frac {1}{n}\right )} + \frac {2 B a a^{- \frac {3}{2} + \frac {1}{n}} a^{2 - \frac {1}{n}} b x^{2 n - 1} \Gamma \left (2 - \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 2 - \frac {1}{n} \\ 3 - \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 - \frac {1}{n}\right )} + \frac {B a^{- \frac {5}{2} + \frac {1}{n}} a^{3 - \frac {1}{n}} b^{2} x^{3 n - 1} \Gamma \left (3 - \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 3 - \frac {1}{n} \\ 4 - \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (4 - \frac {1}{n}\right )} \] Input:

integrate((a+b*x**n)**(5/2)*(A+B*x**n)/x**2,x)
 

Output:

A*a**2*a**(1/2 + 1/n)*gamma(-1/n)*hyper((-1/2, -1/n), (1 - 1/n,), b*x**n*e 
xp_polar(I*pi)/a)/(a**(1/n)*n*x*gamma(1 - 1/n)) + 2*A*a*a**(-1/2 + 1/n)*a* 
*(1 - 1/n)*b*x**(n - 1)*gamma(1 - 1/n)*hyper((-1/2, 1 - 1/n), (2 - 1/n,), 
b*x**n*exp_polar(I*pi)/a)/(n*gamma(2 - 1/n)) + A*a**(-3/2 + 1/n)*a**(2 - 1 
/n)*b**2*x**(2*n - 1)*gamma(2 - 1/n)*hyper((-1/2, 2 - 1/n), (3 - 1/n,), b* 
x**n*exp_polar(I*pi)/a)/(n*gamma(3 - 1/n)) + B*a**2*a**(-1/2 + 1/n)*a**(1 
- 1/n)*x**(n - 1)*gamma(1 - 1/n)*hyper((-1/2, 1 - 1/n), (2 - 1/n,), b*x**n 
*exp_polar(I*pi)/a)/(n*gamma(2 - 1/n)) + 2*B*a*a**(-3/2 + 1/n)*a**(2 - 1/n 
)*b*x**(2*n - 1)*gamma(2 - 1/n)*hyper((-1/2, 2 - 1/n), (3 - 1/n,), b*x**n* 
exp_polar(I*pi)/a)/(n*gamma(3 - 1/n)) + B*a**(-5/2 + 1/n)*a**(3 - 1/n)*b** 
2*x**(3*n - 1)*gamma(3 - 1/n)*hyper((-1/2, 3 - 1/n), (4 - 1/n,), b*x**n*ex 
p_polar(I*pi)/a)/(n*gamma(4 - 1/n))
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\left (a+b x^n\right )^{5/2} \left (A+B x^n\right )}{x^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {5}{2}}}{x^{2}} \,d x } \] Input:

integrate((a+b*x^n)^(5/2)*(A+B*x^n)/x^2,x, algorithm="maxima")
 

Output:

integrate((B*x^n + A)*(b*x^n + a)^(5/2)/x^2, x)
 

Giac [F]

\[ \int \frac {\left (a+b x^n\right )^{5/2} \left (A+B x^n\right )}{x^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {5}{2}}}{x^{2}} \,d x } \] Input:

integrate((a+b*x^n)^(5/2)*(A+B*x^n)/x^2,x, algorithm="giac")
 

Output:

integrate((B*x^n + A)*(b*x^n + a)^(5/2)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^n\right )^{5/2} \left (A+B x^n\right )}{x^2} \, dx=\int \frac {\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^{5/2}}{x^2} \,d x \] Input:

int(((A + B*x^n)*(a + b*x^n)^(5/2))/x^2,x)
 

Output:

int(((A + B*x^n)*(a + b*x^n)^(5/2))/x^2, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^n\right )^{5/2} \left (A+B x^n\right )}{x^2} \, dx =\text {Too large to display} \] Input:

int((a+b*x^n)^(5/2)*(A+B*x^n)/x^2,x)
 

Output:

(30*x**(3*n)*sqrt(x**n*b + a)*b**3*n**3 - 92*x**(3*n)*sqrt(x**n*b + a)*b** 
3*n**2 + 72*x**(3*n)*sqrt(x**n*b + a)*b**3*n - 16*x**(3*n)*sqrt(x**n*b + a 
)*b**3 + 132*x**(2*n)*sqrt(x**n*b + a)*a*b**2*n**3 - 388*x**(2*n)*sqrt(x** 
n*b + a)*a*b**2*n**2 + 272*x**(2*n)*sqrt(x**n*b + a)*a*b**2*n - 48*x**(2*n 
)*sqrt(x**n*b + a)*a*b**2 + 244*x**n*sqrt(x**n*b + a)*a**2*b*n**3 - 640*x* 
*n*sqrt(x**n*b + a)*a**2*b*n**2 + 328*x**n*sqrt(x**n*b + a)*a**2*b*n - 48* 
x**n*sqrt(x**n*b + a)*a**2*b + 352*sqrt(x**n*b + a)*a**3*n**3 - 344*sqrt(x 
**n*b + a)*a**3*n**2 + 128*sqrt(x**n*b + a)*a**3*n - 16*sqrt(x**n*b + a)*a 
**3 + 11025*int(sqrt(x**n*b + a)/(105*x**n*b*n**4*x**2 - 352*x**n*b*n**3*x 
**2 + 344*x**n*b*n**2*x**2 - 128*x**n*b*n*x**2 + 16*x**n*b*x**2 + 105*a*n* 
*4*x**2 - 352*a*n**3*x**2 + 344*a*n**2*x**2 - 128*a*n*x**2 + 16*a*x**2),x) 
*a**4*n**8*x - 36960*int(sqrt(x**n*b + a)/(105*x**n*b*n**4*x**2 - 352*x**n 
*b*n**3*x**2 + 344*x**n*b*n**2*x**2 - 128*x**n*b*n*x**2 + 16*x**n*b*x**2 + 
 105*a*n**4*x**2 - 352*a*n**3*x**2 + 344*a*n**2*x**2 - 128*a*n*x**2 + 16*a 
*x**2),x)*a**4*n**7*x + 36120*int(sqrt(x**n*b + a)/(105*x**n*b*n**4*x**2 - 
 352*x**n*b*n**3*x**2 + 344*x**n*b*n**2*x**2 - 128*x**n*b*n*x**2 + 16*x**n 
*b*x**2 + 105*a*n**4*x**2 - 352*a*n**3*x**2 + 344*a*n**2*x**2 - 128*a*n*x* 
*2 + 16*a*x**2),x)*a**4*n**6*x - 13440*int(sqrt(x**n*b + a)/(105*x**n*b*n* 
*4*x**2 - 352*x**n*b*n**3*x**2 + 344*x**n*b*n**2*x**2 - 128*x**n*b*n*x**2 
+ 16*x**n*b*x**2 + 105*a*n**4*x**2 - 352*a*n**3*x**2 + 344*a*n**2*x**2 ...