\(\int \frac {(a+b x^n)^{3/2} (A+B x^n)}{x^4} \, dx\) [343]

Optimal result

Integrand size = 22, antiderivative size = 105 \[ \int \frac {\left (a+b x^n\right )^{3/2} \left (A+B x^n\right )}{x^4} \, dx=-\frac {2 B \left (a+b x^n\right )^{5/2}}{b (6-5 n) x^3}-\frac {a \left (A-\frac {6 a B}{6 b-5 b n}\right ) \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{n},-\frac {3-n}{n},-\frac {b x^n}{a}\right )}{3 x^3 \sqrt {1+\frac {b x^n}{a}}} \] Output:

-2*B*(a+b*x^n)^(5/2)/b/(6-5*n)/x^3-1/3*a*(A-6*a*B/(-5*b*n+6*b))*(a+b*x^n)^ 
(1/2)*hypergeom([-3/2, -3/n],[-(3-n)/n],-b*x^n/a)/x^3/(1+b*x^n/a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^n\right )^{3/2} \left (A+B x^n\right )}{x^4} \, dx=\frac {a \sqrt {a+b x^n} \left (-A (-3+n) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{n},\frac {-3+n}{n},-\frac {b x^n}{a}\right )+3 B x^n \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {-3+n}{n},2-\frac {3}{n},-\frac {b x^n}{a}\right )\right )}{3 (-3+n) x^3 \sqrt {1+\frac {b x^n}{a}}} \] Input:

Integrate[((a + b*x^n)^(3/2)*(A + B*x^n))/x^4,x]
 

Output:

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

\(\displaystyle \int \frac {\left (a+b x^n\right )^{3/2} \left (A+B x^n\right )}{x^4} \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \left (A-\frac {6 a B}{6 b-5 b n}\right ) \int \frac {\left (b x^n+a\right )^{3/2}}{x^4}dx-\frac {2 B \left (a+b x^n\right )^{5/2}}{b (6-5 n) x^3}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {a \sqrt {a+b x^n} \left (A-\frac {6 a B}{6 b-5 b n}\right ) \int \frac {\left (\frac {b x^n}{a}+1\right )^{3/2}}{x^4}dx}{\sqrt {\frac {b x^n}{a}+1}}-\frac {2 B \left (a+b x^n\right )^{5/2}}{b (6-5 n) x^3}\)

\(\Big \downarrow \) 888

\(\displaystyle -\frac {a \sqrt {a+b x^n} \left (A-\frac {6 a B}{6 b-5 b n}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{n},-\frac {3-n}{n},-\frac {b x^n}{a}\right )}{3 x^3 \sqrt {\frac {b x^n}{a}+1}}-\frac {2 B \left (a+b x^n\right )^{5/2}}{b (6-5 n) x^3}\)

Input:

Int[((a + b*x^n)^(3/2)*(A + B*x^n))/x^4,x]
 

Output:

(-2*B*(a + b*x^n)^(5/2))/(b*(6 - 5*n)*x^3) - (a*(A - (6*a*B)/(6*b - 5*b*n) 
)*Sqrt[a + b*x^n]*Hypergeometric2F1[-3/2, -3/n, -((3 - n)/n), -((b*x^n)/a) 
])/(3*x^3*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]

\[\int \frac {\left (a +b \,x^{n}\right )^{\frac {3}{2}} \left (A +B \,x^{n}\right )}{x^{4}}d x\]

Input:

int((a+b*x^n)^(3/2)*(A+B*x^n)/x^4,x)
 

Output:

int((a+b*x^n)^(3/2)*(A+B*x^n)/x^4,x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^n\right )^{3/2} \left (A+B x^n\right )}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*x^n)^(3/2)*(A+B*x^n)/x^4,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 = 6.38 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.36 \[ \int \frac {\left (a+b x^n\right )^{3/2} \left (A+B x^n\right )}{x^4} \, dx=\frac {A a a^{- \frac {3}{n}} a^{\frac {1}{2} + \frac {3}{n}} \Gamma \left (- \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {3}{n} \\ 1 - \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n x^{3} \Gamma \left (1 - \frac {3}{n}\right )} + \frac {A a^{- \frac {1}{2} + \frac {3}{n}} a^{1 - \frac {3}{n}} b x^{n - 3} \Gamma \left (1 - \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 - \frac {3}{n} \\ 2 - \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 - \frac {3}{n}\right )} + \frac {B a a^{- \frac {1}{2} + \frac {3}{n}} a^{1 - \frac {3}{n}} x^{n - 3} \Gamma \left (1 - \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 - \frac {3}{n} \\ 2 - \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 - \frac {3}{n}\right )} + \frac {B a^{- \frac {3}{2} + \frac {3}{n}} a^{2 - \frac {3}{n}} b x^{2 n - 3} \Gamma \left (2 - \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 2 - \frac {3}{n} \\ 3 - \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 - \frac {3}{n}\right )} \] Input:

integrate((a+b*x**n)**(3/2)*(A+B*x**n)/x**4,x)
 

Output:

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

Maxima [F]

\[ \int \frac {\left (a+b x^n\right )^{3/2} \left (A+B x^n\right )}{x^4} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:

integrate((a+b*x^n)^(3/2)*(A+B*x^n)/x^4,x, algorithm="maxima")
 

Output:

integrate((B*x^n + A)*(b*x^n + a)^(3/2)/x^4, x)
 

Giac [F]

\[ \int \frac {\left (a+b x^n\right )^{3/2} \left (A+B x^n\right )}{x^4} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:

integrate((a+b*x^n)^(3/2)*(A+B*x^n)/x^4,x, algorithm="giac")
 

Output:

integrate((B*x^n + A)*(b*x^n + a)^(3/2)/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^n\right )^{3/2} \left (A+B x^n\right )}{x^4} \, dx=\int \frac {\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^{3/2}}{x^4} \,d x \] Input:

int(((A + B*x^n)*(a + b*x^n)^(3/2))/x^4,x)
 

Output:

int(((A + B*x^n)*(a + b*x^n)^(3/2))/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^n\right )^{3/2} \left (A+B x^n\right )}{x^4} \, dx=\frac {6 x^{2 n} \sqrt {x^{n} b +a}\, b^{2} n^{2}-48 x^{2 n} \sqrt {x^{n} b +a}\, b^{2} n +72 x^{2 n} \sqrt {x^{n} b +a}\, b^{2}+22 x^{n} \sqrt {x^{n} b +a}\, a b \,n^{2}-156 x^{n} \sqrt {x^{n} b +a}\, a b n +144 x^{n} \sqrt {x^{n} b +a}\, a b +46 \sqrt {x^{n} b +a}\, a^{2} n^{2}-108 \sqrt {x^{n} b +a}\, a^{2} n +72 \sqrt {x^{n} b +a}\, a^{2}+75 \left (\int \frac {\sqrt {x^{n} b +a}}{5 x^{n} b \,n^{3} x^{4}-46 x^{n} b \,n^{2} x^{4}+108 x^{n} b n \,x^{4}-72 x^{n} b \,x^{4}+5 a \,n^{3} x^{4}-46 a \,n^{2} x^{4}+108 a n \,x^{4}-72 a \,x^{4}}d x \right ) a^{3} n^{6} x^{3}-690 \left (\int \frac {\sqrt {x^{n} b +a}}{5 x^{n} b \,n^{3} x^{4}-46 x^{n} b \,n^{2} x^{4}+108 x^{n} b n \,x^{4}-72 x^{n} b \,x^{4}+5 a \,n^{3} x^{4}-46 a \,n^{2} x^{4}+108 a n \,x^{4}-72 a \,x^{4}}d x \right ) a^{3} n^{5} x^{3}+1620 \left (\int \frac {\sqrt {x^{n} b +a}}{5 x^{n} b \,n^{3} x^{4}-46 x^{n} b \,n^{2} x^{4}+108 x^{n} b n \,x^{4}-72 x^{n} b \,x^{4}+5 a \,n^{3} x^{4}-46 a \,n^{2} x^{4}+108 a n \,x^{4}-72 a \,x^{4}}d x \right ) a^{3} n^{4} x^{3}-1080 \left (\int \frac {\sqrt {x^{n} b +a}}{5 x^{n} b \,n^{3} x^{4}-46 x^{n} b \,n^{2} x^{4}+108 x^{n} b n \,x^{4}-72 x^{n} b \,x^{4}+5 a \,n^{3} x^{4}-46 a \,n^{2} x^{4}+108 a n \,x^{4}-72 a \,x^{4}}d x \right ) a^{3} n^{3} x^{3}}{3 x^{3} \left (5 n^{3}-46 n^{2}+108 n -72\right )} \] Input:

int((a+b*x^n)^(3/2)*(A+B*x^n)/x^4,x)
 

Output:

(6*x**(2*n)*sqrt(x**n*b + a)*b**2*n**2 - 48*x**(2*n)*sqrt(x**n*b + a)*b**2 
*n + 72*x**(2*n)*sqrt(x**n*b + a)*b**2 + 22*x**n*sqrt(x**n*b + a)*a*b*n**2 
 - 156*x**n*sqrt(x**n*b + a)*a*b*n + 144*x**n*sqrt(x**n*b + a)*a*b + 46*sq 
rt(x**n*b + a)*a**2*n**2 - 108*sqrt(x**n*b + a)*a**2*n + 72*sqrt(x**n*b + 
a)*a**2 + 75*int(sqrt(x**n*b + a)/(5*x**n*b*n**3*x**4 - 46*x**n*b*n**2*x** 
4 + 108*x**n*b*n*x**4 - 72*x**n*b*x**4 + 5*a*n**3*x**4 - 46*a*n**2*x**4 + 
108*a*n*x**4 - 72*a*x**4),x)*a**3*n**6*x**3 - 690*int(sqrt(x**n*b + a)/(5* 
x**n*b*n**3*x**4 - 46*x**n*b*n**2*x**4 + 108*x**n*b*n*x**4 - 72*x**n*b*x** 
4 + 5*a*n**3*x**4 - 46*a*n**2*x**4 + 108*a*n*x**4 - 72*a*x**4),x)*a**3*n** 
5*x**3 + 1620*int(sqrt(x**n*b + a)/(5*x**n*b*n**3*x**4 - 46*x**n*b*n**2*x* 
*4 + 108*x**n*b*n*x**4 - 72*x**n*b*x**4 + 5*a*n**3*x**4 - 46*a*n**2*x**4 + 
 108*a*n*x**4 - 72*a*x**4),x)*a**3*n**4*x**3 - 1080*int(sqrt(x**n*b + a)/( 
5*x**n*b*n**3*x**4 - 46*x**n*b*n**2*x**4 + 108*x**n*b*n*x**4 - 72*x**n*b*x 
**4 + 5*a*n**3*x**4 - 46*a*n**2*x**4 + 108*a*n*x**4 - 72*a*x**4),x)*a**3*n 
**3*x**3)/(3*x**3*(5*n**3 - 46*n**2 + 108*n - 72))