\(\int \frac {\sqrt {a+b x^n} (A+B x^n)}{(e x)^{5/2}} \, dx\) [376]

Optimal result

Integrand size = 26, antiderivative size = 128 \[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{(e x)^{5/2}} \, dx=-\frac {2 B \left (a+b x^n\right )^{3/2}}{3 b e (1-n) (e x)^{3/2}}+\frac {2 (a B-A b (1-n)) \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2 n},1-\frac {3}{2 n},-\frac {b x^n}{a}\right )}{3 b e (1-n) (e x)^{3/2} \sqrt {1+\frac {b x^n}{a}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{(e x)^{5/2}} \, dx=\frac {2 x \sqrt {a+b x^n} \left (3 B x^n \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1-\frac {3}{2 n},2-\frac {3}{2 n},-\frac {b x^n}{a}\right )+A (3-2 n) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2 n},1-\frac {3}{2 n},-\frac {b x^n}{a}\right )\right )}{3 (-3+2 n) (e x)^{5/2} \sqrt {1+\frac {b x^n}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, 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.

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

\(\Big \downarrow \) 959

\(\displaystyle -\frac {(a B-A b (1-n)) \int \frac {\sqrt {b x^n+a}}{(e x)^{5/2}}dx}{b (1-n)}-\frac {2 B \left (a+b x^n\right )^{3/2}}{3 b e (1-n) (e x)^{3/2}}\)

\(\Big \downarrow \) 889

\(\displaystyle -\frac {\sqrt {a+b x^n} (a B-A b (1-n)) \int \frac {\sqrt {\frac {b x^n}{a}+1}}{(e x)^{5/2}}dx}{b (1-n) \sqrt {\frac {b x^n}{a}+1}}-\frac {2 B \left (a+b x^n\right )^{3/2}}{3 b e (1-n) (e x)^{3/2}}\)

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

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

Input:

int((a+b*x^n)^(1/2)*(A+B*x^n)/(e*x)^(5/2),x)
 

Output:

int((a+b*x^n)^(1/2)*(A+B*x^n)/(e*x)^(5/2),x)
 

Fricas [F(-2)]

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

integrate((a+b*x^n)^(1/2)*(A+B*x^n)/(e*x)^(5/2),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 = 24.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{(e x)^{5/2}} \, dx=\frac {A a^{- \frac {3}{2 n}} a^{\frac {1}{2} + \frac {3}{2 n}} \Gamma \left (- \frac {3}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {3}{2 n} \\ 1 - \frac {3}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{e^{\frac {5}{2}} n x^{\frac {3}{2}} \Gamma \left (1 - \frac {3}{2 n}\right )} + \frac {B a^{- \frac {1}{2} + \frac {3}{2 n}} a^{1 - \frac {3}{2 n}} x^{n - \frac {3}{2}} \Gamma \left (1 - \frac {3}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 - \frac {3}{2 n} \\ 2 - \frac {3}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{e^{\frac {5}{2}} n \Gamma \left (2 - \frac {3}{2 n}\right )} \] Input:

integrate((a+b*x**n)**(1/2)*(A+B*x**n)/(e*x)**(5/2),x)
 

Output:

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

Maxima [F]

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

integrate((a+b*x^n)^(1/2)*(A+B*x^n)/(e*x)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((a+b*x^n)^(1/2)*(A+B*x^n)/(e*x)^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((A + B*x^n)*(a + b*x^n)^(1/2))/(e*x)^(5/2),x)
 

Output:

int(((A + B*x^n)*(a + b*x^n)^(1/2))/(e*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{(e x)^{5/2}} \, dx=\frac {\sqrt {e}\, \left (2 x^{n} \sqrt {x^{n} b +a}\, b n -6 x^{n} \sqrt {x^{n} b +a}\, b +8 \sqrt {x^{n} b +a}\, a n -6 \sqrt {x^{n} b +a}\, a +3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}}{x^{n} b \,n^{2} x^{3}-4 x^{n} b n \,x^{3}+3 x^{n} b \,x^{3}+a \,n^{2} x^{3}-4 a n \,x^{3}+3 a \,x^{3}}d x \right ) a^{2} n^{4} x -12 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}}{x^{n} b \,n^{2} x^{3}-4 x^{n} b n \,x^{3}+3 x^{n} b \,x^{3}+a \,n^{2} x^{3}-4 a n \,x^{3}+3 a \,x^{3}}d x \right ) a^{2} n^{3} x +9 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}}{x^{n} b \,n^{2} x^{3}-4 x^{n} b n \,x^{3}+3 x^{n} b \,x^{3}+a \,n^{2} x^{3}-4 a n \,x^{3}+3 a \,x^{3}}d x \right ) a^{2} n^{2} x \right )}{3 \sqrt {x}\, e^{3} x \left (n^{2}-4 n +3\right )} \] Input:

int((a+b*x^n)^(1/2)*(A+B*x^n)/(e*x)^(5/2),x)
 

Output:

(sqrt(e)*(2*x**n*sqrt(x**n*b + a)*b*n - 6*x**n*sqrt(x**n*b + a)*b + 8*sqrt 
(x**n*b + a)*a*n - 6*sqrt(x**n*b + a)*a + 3*sqrt(x)*int((sqrt(x)*sqrt(x**n 
*b + a))/(x**n*b*n**2*x**3 - 4*x**n*b*n*x**3 + 3*x**n*b*x**3 + a*n**2*x**3 
 - 4*a*n*x**3 + 3*a*x**3),x)*a**2*n**4*x - 12*sqrt(x)*int((sqrt(x)*sqrt(x* 
*n*b + a))/(x**n*b*n**2*x**3 - 4*x**n*b*n*x**3 + 3*x**n*b*x**3 + a*n**2*x* 
*3 - 4*a*n*x**3 + 3*a*x**3),x)*a**2*n**3*x + 9*sqrt(x)*int((sqrt(x)*sqrt(x 
**n*b + a))/(x**n*b*n**2*x**3 - 4*x**n*b*n*x**3 + 3*x**n*b*x**3 + a*n**2*x 
**3 - 4*a*n*x**3 + 3*a*x**3),x)*a**2*n**2*x))/(3*sqrt(x)*e**3*x*(n**2 - 4* 
n + 3))