\(\int \frac {\sqrt {x} (A+B x^n)}{(a+b x^n)^3} \, dx\) [325]

Optimal result

Integrand size = 22, antiderivative size = 95 \[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\frac {(A b-a B) x^{3/2}}{2 a b n \left (a+b x^n\right )^2}+\frac {(3 a B-A b (3-4 n)) x^{3/2} \operatorname {Hypergeometric2F1}\left (2,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {b x^n}{a}\right )}{6 a^3 b n} \] Output:

1/2*(A*b-B*a)*x^(3/2)/a/b/n/(a+b*x^n)^2+1/6*(3*B*a-A*b*(3-4*n))*x^(3/2)*hy 
pergeom([2, 3/2/n],[1+3/2/n],-b*x^n/a)/a^3/b/n
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\frac {x^{3/2} \left (-\frac {6 a^2 (-A b+a B) n}{\left (a+b x^n\right )^2}+\frac {3 a (3 a B+A b (-3+4 n))}{a+b x^n}+(-3+2 n) (3 a B+A b (-3+4 n)) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {b x^n}{a}\right )\right )}{12 a^3 b n^2} \] Input:

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

Output:

(x^(3/2)*((-6*a^2*(-(A*b) + a*B)*n)/(a + b*x^n)^2 + (3*a*(3*a*B + A*b*(-3 
+ 4*n)))/(a + b*x^n) + (-3 + 2*n)*(3*a*B + A*b*(-3 + 4*n))*Hypergeometric2 
F1[1, 3/(2*n), 1 + 3/(2*n), -((b*x^n)/a)]))/(12*a^3*b*n^2)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {957, 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.

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

\(\Big \downarrow \) 957

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {x^{3/2} (3 a B-A b (3-4 n)) \operatorname {Hypergeometric2F1}\left (2,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {b x^n}{a}\right )}{6 a^3 b n}+\frac {x^{3/2} (A b-a B)}{2 a b n \left (a+b x^n\right )^2}\)

Input:

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

Output:

((A*b - a*B)*x^(3/2))/(2*a*b*n*(a + b*x^n)^2) + ((3*a*B - A*b*(3 - 4*n))*x 
^(3/2)*Hypergeometric2F1[2, 3/(2*n), 1 + 3/(2*n), -((b*x^n)/a)])/(6*a^3*b* 
n)
 

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 957

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a 
*b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* 
(p + 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, 
 m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N 
eQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 
, m, (-n)*(p + 1)]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

2*((4*n^2 - 3*n)*A*a*b + 3*B*a^2*n)*integrate(sqrt(x)/((8*n^2 - 10*n + 3)* 
b^5*x^(4*n) + 4*(8*n^2 - 10*n + 3)*a*b^4*x^(3*n) + 6*(8*n^2 - 10*n + 3)*a^ 
2*b^3*x^(2*n) + 4*(8*n^2 - 10*n + 3)*a^3*b^2*x^n + (8*n^2 - 10*n + 3)*a^4* 
b), x) - 2/3*(3*B*b*(2*n - 1)*x*x^n + (A*b*(4*n - 3) + 6*B*a*n)*x)*sqrt(x) 
/((8*n^2 - 10*n + 3)*b^4*x^(3*n) + 3*(8*n^2 - 10*n + 3)*a*b^3*x^(2*n) + 3* 
(8*n^2 - 10*n + 3)*a^2*b^2*x^n + (8*n^2 - 10*n + 3)*a^3*b)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(x)/(x**(2*n)*b**2 + 2*x**n*a*b + a**2),x)