Integrand size = 22, antiderivative size = 94 \[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^3} \, dx=\frac {A b-a B}{2 a b n \sqrt {x} \left (a+b x^n\right )^2}+\frac {(a B-A (b+4 b n)) \operatorname {Hypergeometric2F1}\left (2,-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{2 a^3 b n \sqrt {x}} \] Output:
1/2*(A*b-B*a)/a/b/n/x^(1/2)/(a+b*x^n)^2+1/2*(B*a-A*(4*b*n+b))*hypergeom([2 , -1/2/n],[1-1/2/n],-b*x^n/a)/a^3/b/n/x^(1/2)
Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^3} \, dx=\frac {\frac {2 a^2 (A b-a B) n}{\left (a+b x^n\right )^2}+\frac {a (-a B+A (b+4 b n))}{a+b x^n}+(1+2 n) (a B-A (b+4 b n)) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{4 a^3 b n^2 \sqrt {x}} \] Input:
Integrate[(A + B*x^n)/(x^(3/2)*(a + b*x^n)^3),x]
Output:
((2*a^2*(A*b - a*B)*n)/(a + b*x^n)^2 + (a*(-(a*B) + A*(b + 4*b*n)))/(a + b *x^n) + (1 + 2*n)*(a*B - A*(b + 4*b*n))*Hypergeometric2F1[1, -1/2*1/n, 1 - 1/(2*n), -((b*x^n)/a)])/(4*a^3*b*n^2*Sqrt[x])
Time = 0.36 (sec) , antiderivative size = 94, 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.
\(\displaystyle \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^3} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {A b-a B}{2 a b n \sqrt {x} \left (a+b x^n\right )^2}-\frac {(a B-A (4 b n+b)) \int \frac {1}{x^{3/2} \left (b x^n+a\right )^2}dx}{4 a b n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {(a B-A (4 b n+b)) \operatorname {Hypergeometric2F1}\left (2,-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{2 a^3 b n \sqrt {x}}+\frac {A b-a B}{2 a b n \sqrt {x} \left (a+b x^n\right )^2}\) |
Input:
Int[(A + B*x^n)/(x^(3/2)*(a + b*x^n)^3),x]
Output:
(A*b - a*B)/(2*a*b*n*Sqrt[x]*(a + b*x^n)^2) + ((a*B - A*(b + 4*b*n))*Hyper geometric2F1[2, -1/2*1/n, 1 - 1/(2*n), -((b*x^n)/a)])/(2*a^3*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)]))
\[\int \frac {A +B \,x^{n}}{x^{\frac {3}{2}} \left (a +b \,x^{n}\right )^{3}}d x\]
Input:
int((A+B*x^n)/x^(3/2)/(a+b*x^n)^3,x)
Output:
int((A+B*x^n)/x^(3/2)/(a+b*x^n)^3,x)
\[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^3} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{3} x^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/x^(3/2)/(a+b*x^n)^3,x, algorithm="fricas")
Output:
integral((B*x^n + A)*sqrt(x)/(b^3*x^2*x^(3*n) + 3*a*b^2*x^2*x^(2*n) + 3*a^ 2*b*x^2*x^n + a^3*x^2), x)
Exception generated. \[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^3} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((A+B*x**n)/x**(3/2)/(a+b*x**n)**3,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^3} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{3} x^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/x^(3/2)/(a+b*x^n)^3,x, algorithm="maxima")
Output:
6*((4*n^2 + n)*A*a*b - B*a^2*n)*integrate(1/(((24*n^2 + 10*n + 1)*b^5*x^(4 *n) + 4*(24*n^2 + 10*n + 1)*a*b^4*x^(3*n) + 6*(24*n^2 + 10*n + 1)*a^2*b^3* x^(2*n) + 4*(24*n^2 + 10*n + 1)*a^3*b^2*x^n + (24*n^2 + 10*n + 1)*a^4*b)*x ^(3/2)), x) - 2*(B*b*(6*n + 1)*x*x^n + (A*b*(4*n + 1) + 6*B*a*n)*x)/(((24* n^2 + 10*n + 1)*b^4*x^(3*n) + 3*(24*n^2 + 10*n + 1)*a*b^3*x^(2*n) + 3*(24* n^2 + 10*n + 1)*a^2*b^2*x^n + (24*n^2 + 10*n + 1)*a^3*b)*x^(3/2))
\[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^3} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{3} x^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/x^(3/2)/(a+b*x^n)^3,x, algorithm="giac")
Output:
integrate((B*x^n + A)/((b*x^n + a)^3*x^(3/2)), x)
Timed out. \[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^3} \, dx=\int \frac {A+B\,x^n}{x^{3/2}\,{\left (a+b\,x^n\right )}^3} \,d x \] Input:
int((A + B*x^n)/(x^(3/2)*(a + b*x^n)^3),x)
Output:
int((A + B*x^n)/(x^(3/2)*(a + b*x^n)^3), x)
\[ \int \frac {A+B x^n}{x^{3/2} \left (a+b x^n\right )^3} \, dx=\int \frac {1}{x^{2 n +\frac {1}{2}} b^{2} x +2 x^{n +\frac {1}{2}} a b x +\sqrt {x}\, a^{2} x}d x \] Input:
int((A+B*x^n)/x^(3/2)/(a+b*x^n)^3,x)
Output:
int(1/(x**((4*n + 1)/2)*b**2*x + 2*x**((2*n + 1)/2)*a*b*x + sqrt(x)*a**2*x ),x)