Integrand size = 22, antiderivative size = 88 \[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^2} \, dx=\frac {A b-a B}{a b n \sqrt {x} \left (a+b x^n\right )}+\frac {(a B-A (b+2 b n)) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{a^2 b n \sqrt {x}} \] Output:
(A*b-B*a)/a/b/n/x^(1/2)/(a+b*x^n)+(B*a-A*(2*b*n+b))*hypergeom([1, -1/2/n], [1-1/2/n],-b*x^n/a)/a^2/b/n/x^(1/2)
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^2} \, dx=\frac {a (A b-a B)+(a B-A (b+2 b n)) \left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{a^2 b n \sqrt {x} \left (a+b x^n\right )} \] Input:
Integrate[(A + B*x^n)/(x^(3/2)*(a + b*x^n)^2),x]
Output:
(a*(A*b - a*B) + (a*B - A*(b + 2*b*n))*(a + b*x^n)*Hypergeometric2F1[1, -1 /2*1/n, 1 - 1/(2*n), -((b*x^n)/a)])/(a^2*b*n*Sqrt[x]*(a + b*x^n))
Time = 0.36 (sec) , antiderivative size = 88, 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.
\(\displaystyle \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^2} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {A b-a B}{a b n \sqrt {x} \left (a+b x^n\right )}-\frac {(a B-A (2 b n+b)) \int \frac {1}{x^{3/2} \left (b x^n+a\right )}dx}{2 a b n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {(a B-A (2 b n+b)) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{a^2 b n \sqrt {x}}+\frac {A b-a B}{a b n \sqrt {x} \left (a+b x^n\right )}\) |
Input:
Int[(A + B*x^n)/(x^(3/2)*(a + b*x^n)^2),x]
Output:
(A*b - a*B)/(a*b*n*Sqrt[x]*(a + b*x^n)) + ((a*B - A*(b + 2*b*n))*Hypergeom etric2F1[1, -1/2*1/n, 1 - 1/(2*n), -((b*x^n)/a)])/(a^2*b*n*Sqrt[x])
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])
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)]))
\[\int \frac {A +B \,x^{n}}{x^{\frac {3}{2}} \left (a +b \,x^{n}\right )^{2}}d x\]
Input:
int((A+B*x^n)/x^(3/2)/(a+b*x^n)^2,x)
Output:
int((A+B*x^n)/x^(3/2)/(a+b*x^n)^2,x)
\[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^2} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{2} x^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/x^(3/2)/(a+b*x^n)^2,x, algorithm="fricas")
Output:
integral((B*x^n + A)*sqrt(x)/(b^2*x^2*x^(2*n) + 2*a*b*x^2*x^n + a^2*x^2), x)
Timed out. \[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^2} \, dx=\text {Timed out} \] Input:
integrate((A+B*x**n)/x**(3/2)/(a+b*x**n)**2,x)
Output:
Timed out
\[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^2} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{2} x^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/x^(3/2)/(a+b*x^n)^2,x, algorithm="maxima")
Output:
4*((2*n^2 + n)*A*a*b - B*a^2*n)*integrate(1/(((8*n^2 + 6*n + 1)*b^4*x^(3*n ) + 3*(8*n^2 + 6*n + 1)*a*b^3*x^(2*n) + 3*(8*n^2 + 6*n + 1)*a^2*b^2*x^n + (8*n^2 + 6*n + 1)*a^3*b)*x^(3/2)), x) - 2*(B*b*(4*n + 1)*x*x^n + (A*b*(2*n + 1) + 4*B*a*n)*x)/(((8*n^2 + 6*n + 1)*b^3*x^(2*n) + 2*(8*n^2 + 6*n + 1)* a*b^2*x^n + (8*n^2 + 6*n + 1)*a^2*b)*x^(3/2))
\[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^2} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{2} x^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/x^(3/2)/(a+b*x^n)^2,x, algorithm="giac")
Output:
integrate((B*x^n + A)/((b*x^n + a)^2*x^(3/2)), x)
Timed out. \[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^2} \, dx=\int \frac {A+B\,x^n}{x^{3/2}\,{\left (a+b\,x^n\right )}^2} \,d x \] Input:
int((A + B*x^n)/(x^(3/2)*(a + b*x^n)^2),x)
Output:
int((A + B*x^n)/(x^(3/2)*(a + b*x^n)^2), x)
\[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^2} \, dx=\int \frac {1}{x^{n +\frac {1}{2}} b x +\sqrt {x}\, a x}d x \] Input:
int((A+B*x^n)/x^(3/2)/(a+b*x^n)^2,x)
Output:
int(1/(x**((2*n + 1)/2)*b*x + sqrt(x)*a*x),x)