\(\int (e x)^{3/2} \sqrt {a+b x^n} (A+B x^n) \, dx\) [372]

Optimal result

Integrand size = 26, antiderivative size = 119 \[ \int (e x)^{3/2} \sqrt {a+b x^n} \left (A+B x^n\right ) \, dx=\frac {2 B (e x)^{5/2} \left (a+b x^n\right )^{3/2}}{b e (5+3 n)}+\frac {2 \left (A-\frac {5 a B}{5 b+3 b n}\right ) (e x)^{5/2} \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {5}{2 n},1+\frac {5}{2 n},-\frac {b x^n}{a}\right )}{5 e \sqrt {1+\frac {b x^n}{a}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99 \[ \int (e x)^{3/2} \sqrt {a+b x^n} \left (A+B x^n\right ) \, dx=\frac {2 x (e x)^{3/2} \sqrt {a+b x^n} \left (A (5+2 n) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {5}{2 n},1+\frac {5}{2 n},-\frac {b x^n}{a}\right )+5 B x^n \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {\frac {5}{2}+n}{n},2+\frac {5}{2 n},-\frac {b x^n}{a}\right )\right )}{5 (5+2 n) \sqrt {1+\frac {b x^n}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 119, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {2 (e x)^{5/2} \sqrt {a+b x^n} \left (A-\frac {5 a B}{3 b n+5 b}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {5}{2 n},1+\frac {5}{2 n},-\frac {b x^n}{a}\right )}{5 e \sqrt {\frac {b x^n}{a}+1}}+\frac {2 B (e x)^{5/2} \left (a+b x^n\right )^{3/2}}{b e (3 n+5)}\)

Input:

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

Output:

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

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

Input:

int((e*x)^(3/2)*(a+b*x^n)^(1/2)*(A+B*x^n),x)
 

Output:

int((e*x)^(3/2)*(a+b*x^n)^(1/2)*(A+B*x^n),x)
 

Fricas [F(-2)]

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

integrate((e*x)^(3/2)*(a+b*x^n)^(1/2)*(A+B*x^n),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 = 33.53 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.26 \[ \int (e x)^{3/2} \sqrt {a+b x^n} \left (A+B x^n\right ) \, dx=\frac {A a^{\frac {5}{2 n}} a^{\frac {1}{2} - \frac {5}{2 n}} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{2 n} \\ 1 + \frac {5}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {5}{2 n}\right )} + \frac {B a^{- \frac {1}{2} - \frac {5}{2 n}} a^{1 + \frac {5}{2 n}} e^{\frac {3}{2}} x^{n + \frac {5}{2}} \Gamma \left (1 + \frac {5}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 + \frac {5}{2 n} \\ 2 + \frac {5}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {5}{2 n}\right )} \] Input:

integrate((e*x)**(3/2)*(a+b*x**n)**(1/2)*(A+B*x**n),x)
 

Output:

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

Maxima [F]

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

integrate((e*x)^(3/2)*(a+b*x^n)^(1/2)*(A+B*x^n),x, algorithm="maxima")
 

Output:

integrate((B*x^n + A)*sqrt(b*x^n + a)*(e*x)^(3/2), x)
 

Giac [F]

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

integrate((e*x)^(3/2)*(a+b*x^n)^(1/2)*(A+B*x^n),x, algorithm="giac")
 

Output:

integrate((B*x^n + A)*sqrt(b*x^n + a)*(e*x)^(3/2), x)
 

Mupad [F(-1)]

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

int((e*x)^(3/2)*(A + B*x^n)*(a + b*x^n)^(1/2),x)
 

Output:

int((e*x)^(3/2)*(A + B*x^n)*(a + b*x^n)^(1/2), x)
 

Reduce [F]

\[ \int (e x)^{3/2} \sqrt {a+b x^n} \left (A+B x^n\right ) \, dx=\frac {\sqrt {e}\, e \left (2 x^{n +\frac {1}{2}} \sqrt {x^{n} b +a}\, b n \,x^{2}+10 x^{n +\frac {1}{2}} \sqrt {x^{n} b +a}\, b \,x^{2}+8 \sqrt {x}\, \sqrt {x^{n} b +a}\, a n \,x^{2}+10 \sqrt {x}\, \sqrt {x^{n} b +a}\, a \,x^{2}+9 \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}\, x}{3 x^{n} b \,n^{2}+20 x^{n} b n +25 x^{n} b +3 a \,n^{2}+20 a n +25 a}d x \right ) a^{2} n^{4}+60 \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}\, x}{3 x^{n} b \,n^{2}+20 x^{n} b n +25 x^{n} b +3 a \,n^{2}+20 a n +25 a}d x \right ) a^{2} n^{3}+75 \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}\, x}{3 x^{n} b \,n^{2}+20 x^{n} b n +25 x^{n} b +3 a \,n^{2}+20 a n +25 a}d x \right ) a^{2} n^{2}\right )}{3 n^{2}+20 n +25} \] Input:

int((e*x)^(3/2)*(a+b*x^n)^(1/2)*(A+B*x^n),x)
 

Output:

(sqrt(e)*e*(2*x**((2*n + 1)/2)*sqrt(x**n*b + a)*b*n*x**2 + 10*x**((2*n + 1 
)/2)*sqrt(x**n*b + a)*b*x**2 + 8*sqrt(x)*sqrt(x**n*b + a)*a*n*x**2 + 10*sq 
rt(x)*sqrt(x**n*b + a)*a*x**2 + 9*int((sqrt(x)*sqrt(x**n*b + a)*x)/(3*x**n 
*b*n**2 + 20*x**n*b*n + 25*x**n*b + 3*a*n**2 + 20*a*n + 25*a),x)*a**2*n**4 
 + 60*int((sqrt(x)*sqrt(x**n*b + a)*x)/(3*x**n*b*n**2 + 20*x**n*b*n + 25*x 
**n*b + 3*a*n**2 + 20*a*n + 25*a),x)*a**2*n**3 + 75*int((sqrt(x)*sqrt(x**n 
*b + a)*x)/(3*x**n*b*n**2 + 20*x**n*b*n + 25*x**n*b + 3*a*n**2 + 20*a*n + 
25*a),x)*a**2*n**2))/(3*n**2 + 20*n + 25)