\(\int x (a+b x^n)^{5/2} (A+B x^n) \, dx\) [345]

Optimal result

Integrand size = 20, antiderivative size = 104 \[ \int x \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {2 B x^2 \left (a+b x^n\right )^{7/2}}{b (4+7 n)}+\frac {a^2 \left (A-\frac {4 a B}{4 b+7 b n}\right ) x^2 \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2}{n},\frac {2+n}{n},-\frac {b x^n}{a}\right )}{2 \sqrt {1+\frac {b x^n}{a}}} \] Output:

2*B*x^2*(a+b*x^n)^(7/2)/b/(4+7*n)+1/2*a^2*(A-4*a*B/(7*b*n+4*b))*x^2*(a+b*x 
^n)^(1/2)*hypergeom([-5/2, 2/n],[(2+n)/n],-b*x^n/a)/(1+b*x^n/a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int x \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {a^2 x^2 \sqrt {a+b x^n} \left (A (2+n) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2}{n},\frac {2+n}{n},-\frac {b x^n}{a}\right )+2 B x^n \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{n},2 \left (1+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )\right )}{2 (2+n) \sqrt {1+\frac {b x^n}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.

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

\(\Big \downarrow \) 959

\(\displaystyle \left (A-\frac {4 a B}{7 b n+4 b}\right ) \int x \left (b x^n+a\right )^{5/2}dx+\frac {2 B x^2 \left (a+b x^n\right )^{7/2}}{b (7 n+4)}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {a^2 \sqrt {a+b x^n} \left (A-\frac {4 a B}{7 b n+4 b}\right ) \int x \left (\frac {b x^n}{a}+1\right )^{5/2}dx}{\sqrt {\frac {b x^n}{a}+1}}+\frac {2 B x^2 \left (a+b x^n\right )^{7/2}}{b (7 n+4)}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {a^2 x^2 \sqrt {a+b x^n} \left (A-\frac {4 a B}{7 b n+4 b}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2}{n},\frac {n+2}{n},-\frac {b x^n}{a}\right )}{2 \sqrt {\frac {b x^n}{a}+1}}+\frac {2 B x^2 \left (a+b x^n\right )^{7/2}}{b (7 n+4)}\)

Input:

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

Output:

(2*B*x^2*(a + b*x^n)^(7/2))/(b*(4 + 7*n)) + (a^2*(A - (4*a*B)/(4*b + 7*b*n 
))*x^2*Sqrt[a + b*x^n]*Hypergeometric2F1[-5/2, 2/n, (2 + n)/n, -((b*x^n)/a 
)])/(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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate(x*(a+b*x^n)^(5/2)*(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 = 12.03 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.81 \[ \int x \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {A a^{2} a^{\frac {2}{n}} a^{\frac {1}{2} - \frac {2}{n}} x^{2} \Gamma \left (\frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{n} \\ 1 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {2}{n}\right )} + \frac {2 A a a^{- \frac {1}{2} - \frac {2}{n}} a^{1 + \frac {2}{n}} b x^{n + 2} \Gamma \left (1 + \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 + \frac {2}{n} \\ 2 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {2}{n}\right )} + \frac {A a^{- \frac {3}{2} - \frac {2}{n}} a^{2 + \frac {2}{n}} b^{2} x^{2 n + 2} \Gamma \left (2 + \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 2 + \frac {2}{n} \\ 3 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {2}{n}\right )} + \frac {B a^{2} a^{- \frac {1}{2} - \frac {2}{n}} a^{1 + \frac {2}{n}} x^{n + 2} \Gamma \left (1 + \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 + \frac {2}{n} \\ 2 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {2}{n}\right )} + \frac {2 B a a^{- \frac {3}{2} - \frac {2}{n}} a^{2 + \frac {2}{n}} b x^{2 n + 2} \Gamma \left (2 + \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 2 + \frac {2}{n} \\ 3 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {2}{n}\right )} + \frac {B a^{- \frac {5}{2} - \frac {2}{n}} a^{3 + \frac {2}{n}} b^{2} x^{3 n + 2} \Gamma \left (3 + \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 3 + \frac {2}{n} \\ 4 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (4 + \frac {2}{n}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {30 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} n^{3} x^{2}+184 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} n^{2} x^{2}+288 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} n \,x^{2}+128 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} x^{2}+132 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} n^{3} x^{2}+776 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} n^{2} x^{2}+1088 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} n \,x^{2}+384 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} x^{2}+244 x^{n} \sqrt {x^{n} b +a}\, a^{2} b \,n^{3} x^{2}+1280 x^{n} \sqrt {x^{n} b +a}\, a^{2} b \,n^{2} x^{2}+1312 x^{n} \sqrt {x^{n} b +a}\, a^{2} b n \,x^{2}+384 x^{n} \sqrt {x^{n} b +a}\, a^{2} b \,x^{2}+352 \sqrt {x^{n} b +a}\, a^{3} n^{3} x^{2}+688 \sqrt {x^{n} b +a}\, a^{3} n^{2} x^{2}+512 \sqrt {x^{n} b +a}\, a^{3} n \,x^{2}+128 \sqrt {x^{n} b +a}\, a^{3} x^{2}+11025 \left (\int \frac {\sqrt {x^{n} b +a}\, x}{105 x^{n} b \,n^{4}+704 x^{n} b \,n^{3}+1376 x^{n} b \,n^{2}+1024 x^{n} b n +256 x^{n} b +105 a \,n^{4}+704 a \,n^{3}+1376 a \,n^{2}+1024 a n +256 a}d x \right ) a^{4} n^{8}+73920 \left (\int \frac {\sqrt {x^{n} b +a}\, x}{105 x^{n} b \,n^{4}+704 x^{n} b \,n^{3}+1376 x^{n} b \,n^{2}+1024 x^{n} b n +256 x^{n} b +105 a \,n^{4}+704 a \,n^{3}+1376 a \,n^{2}+1024 a n +256 a}d x \right ) a^{4} n^{7}+144480 \left (\int \frac {\sqrt {x^{n} b +a}\, x}{105 x^{n} b \,n^{4}+704 x^{n} b \,n^{3}+1376 x^{n} b \,n^{2}+1024 x^{n} b n +256 x^{n} b +105 a \,n^{4}+704 a \,n^{3}+1376 a \,n^{2}+1024 a n +256 a}d x \right ) a^{4} n^{6}+107520 \left (\int \frac {\sqrt {x^{n} b +a}\, x}{105 x^{n} b \,n^{4}+704 x^{n} b \,n^{3}+1376 x^{n} b \,n^{2}+1024 x^{n} b n +256 x^{n} b +105 a \,n^{4}+704 a \,n^{3}+1376 a \,n^{2}+1024 a n +256 a}d x \right ) a^{4} n^{5}+26880 \left (\int \frac {\sqrt {x^{n} b +a}\, x}{105 x^{n} b \,n^{4}+704 x^{n} b \,n^{3}+1376 x^{n} b \,n^{2}+1024 x^{n} b n +256 x^{n} b +105 a \,n^{4}+704 a \,n^{3}+1376 a \,n^{2}+1024 a n +256 a}d x \right ) a^{4} n^{4}}{105 n^{4}+704 n^{3}+1376 n^{2}+1024 n +256} \] Input:

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

Output:

(30*x**(3*n)*sqrt(x**n*b + a)*b**3*n**3*x**2 + 184*x**(3*n)*sqrt(x**n*b + 
a)*b**3*n**2*x**2 + 288*x**(3*n)*sqrt(x**n*b + a)*b**3*n*x**2 + 128*x**(3* 
n)*sqrt(x**n*b + a)*b**3*x**2 + 132*x**(2*n)*sqrt(x**n*b + a)*a*b**2*n**3* 
x**2 + 776*x**(2*n)*sqrt(x**n*b + a)*a*b**2*n**2*x**2 + 1088*x**(2*n)*sqrt 
(x**n*b + a)*a*b**2*n*x**2 + 384*x**(2*n)*sqrt(x**n*b + a)*a*b**2*x**2 + 2 
44*x**n*sqrt(x**n*b + a)*a**2*b*n**3*x**2 + 1280*x**n*sqrt(x**n*b + a)*a** 
2*b*n**2*x**2 + 1312*x**n*sqrt(x**n*b + a)*a**2*b*n*x**2 + 384*x**n*sqrt(x 
**n*b + a)*a**2*b*x**2 + 352*sqrt(x**n*b + a)*a**3*n**3*x**2 + 688*sqrt(x* 
*n*b + a)*a**3*n**2*x**2 + 512*sqrt(x**n*b + a)*a**3*n*x**2 + 128*sqrt(x** 
n*b + a)*a**3*x**2 + 11025*int((sqrt(x**n*b + a)*x)/(105*x**n*b*n**4 + 704 
*x**n*b*n**3 + 1376*x**n*b*n**2 + 1024*x**n*b*n + 256*x**n*b + 105*a*n**4 
+ 704*a*n**3 + 1376*a*n**2 + 1024*a*n + 256*a),x)*a**4*n**8 + 73920*int((s 
qrt(x**n*b + a)*x)/(105*x**n*b*n**4 + 704*x**n*b*n**3 + 1376*x**n*b*n**2 + 
 1024*x**n*b*n + 256*x**n*b + 105*a*n**4 + 704*a*n**3 + 1376*a*n**2 + 1024 
*a*n + 256*a),x)*a**4*n**7 + 144480*int((sqrt(x**n*b + a)*x)/(105*x**n*b*n 
**4 + 704*x**n*b*n**3 + 1376*x**n*b*n**2 + 1024*x**n*b*n + 256*x**n*b + 10 
5*a*n**4 + 704*a*n**3 + 1376*a*n**2 + 1024*a*n + 256*a),x)*a**4*n**6 + 107 
520*int((sqrt(x**n*b + a)*x)/(105*x**n*b*n**4 + 704*x**n*b*n**3 + 1376*x** 
n*b*n**2 + 1024*x**n*b*n + 256*x**n*b + 105*a*n**4 + 704*a*n**3 + 1376*a*n 
**2 + 1024*a*n + 256*a),x)*a**4*n**5 + 26880*int((sqrt(x**n*b + a)*x)/(...