Integrand size = 22, antiderivative size = 95 \[ \int \frac {x^{3/2} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\frac {(A b-a B) x^{5/2}}{2 a b n \left (a+b x^n\right )^2}+\frac {(5 a B-A b (5-4 n)) x^{5/2} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2 n},1+\frac {5}{2 n},-\frac {b x^n}{a}\right )}{10 a^3 b n} \] Output:
1/2*(A*b-B*a)*x^(5/2)/a/b/n/(a+b*x^n)^2+1/10*(5*B*a-A*b*(5-4*n))*x^(5/2)*h ypergeom([2, 5/2/n],[1+5/2/n],-b*x^n/a)/a^3/b/n
Time = 0.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.19 \[ \int \frac {x^{3/2} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\frac {x^{5/2} \left (-\frac {10 a^2 (-A b+a B) n}{\left (a+b x^n\right )^2}+\frac {5 a (5 a B+A b (-5+4 n))}{a+b x^n}+(-5+2 n) (5 a B+A b (-5+4 n)) \operatorname {Hypergeometric2F1}\left (1,\frac {5}{2 n},1+\frac {5}{2 n},-\frac {b x^n}{a}\right )\right )}{20 a^3 b n^2} \] Input:
Integrate[(x^(3/2)*(A + B*x^n))/(a + b*x^n)^3,x]
Output:
(x^(5/2)*((-10*a^2*(-(A*b) + a*B)*n)/(a + b*x^n)^2 + (5*a*(5*a*B + A*b*(-5 + 4*n)))/(a + b*x^n) + (-5 + 2*n)*(5*a*B + A*b*(-5 + 4*n))*Hypergeometric 2F1[1, 5/(2*n), 1 + 5/(2*n), -((b*x^n)/a)]))/(20*a^3*b*n^2)
Time = 0.37 (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.
\(\displaystyle \int \frac {x^{3/2} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {(5 a B-A b (5-4 n)) \int \frac {x^{3/2}}{\left (b x^n+a\right )^2}dx}{4 a b n}+\frac {x^{5/2} (A b-a B)}{2 a b n \left (a+b x^n\right )^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^{5/2} (5 a B-A b (5-4 n)) \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2 n},1+\frac {5}{2 n},-\frac {b x^n}{a}\right )}{10 a^3 b n}+\frac {x^{5/2} (A b-a B)}{2 a b n \left (a+b x^n\right )^2}\) |
Input:
Int[(x^(3/2)*(A + B*x^n))/(a + b*x^n)^3,x]
Output:
((A*b - a*B)*x^(5/2))/(2*a*b*n*(a + b*x^n)^2) + ((5*a*B - A*b*(5 - 4*n))*x ^(5/2)*Hypergeometric2F1[2, 5/(2*n), 1 + 5/(2*n), -((b*x^n)/a)])/(10*a^3*b *n)
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)]))
\[\int \frac {x^{\frac {3}{2}} \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right )^{3}}d x\]
Input:
int(x^(3/2)*(A+B*x^n)/(a+b*x^n)^3,x)
Output:
int(x^(3/2)*(A+B*x^n)/(a+b*x^n)^3,x)
\[ \int \frac {x^{3/2} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} x^{\frac {3}{2}}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate(x^(3/2)*(A+B*x^n)/(a+b*x^n)^3,x, algorithm="fricas")
Output:
integral((B*x*x^n + A*x)*sqrt(x)/(b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b* x^n + a^3), x)
Exception generated. \[ \int \frac {x^{3/2} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(x**(3/2)*(A+B*x**n)/(a+b*x**n)**3,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {x^{3/2} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} x^{\frac {3}{2}}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate(x^(3/2)*(A+B*x^n)/(a+b*x^n)^3,x, algorithm="maxima")
Output:
6*((4*n^2 - 5*n)*A*a*b + 5*B*a^2*n)*integrate(x^(3/2)/((24*n^2 - 50*n + 25 )*b^5*x^(4*n) + 4*(24*n^2 - 50*n + 25)*a*b^4*x^(3*n) + 6*(24*n^2 - 50*n + 25)*a^2*b^3*x^(2*n) + 4*(24*n^2 - 50*n + 25)*a^3*b^2*x^n + (24*n^2 - 50*n + 25)*a^4*b), x) - 2*(B*b*(6*n - 5)*x*x^n + (A*b*(4*n - 5) + 6*B*a*n)*x)*x ^(3/2)/((24*n^2 - 50*n + 25)*b^4*x^(3*n) + 3*(24*n^2 - 50*n + 25)*a*b^3*x^ (2*n) + 3*(24*n^2 - 50*n + 25)*a^2*b^2*x^n + (24*n^2 - 50*n + 25)*a^3*b)
\[ \int \frac {x^{3/2} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} x^{\frac {3}{2}}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate(x^(3/2)*(A+B*x^n)/(a+b*x^n)^3,x, algorithm="giac")
Output:
integrate((B*x^n + A)*x^(3/2)/(b*x^n + a)^3, x)
Timed out. \[ \int \frac {x^{3/2} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int \frac {x^{3/2}\,\left (A+B\,x^n\right )}{{\left (a+b\,x^n\right )}^3} \,d x \] Input:
int((x^(3/2)*(A + B*x^n))/(a + b*x^n)^3,x)
Output:
int((x^(3/2)*(A + B*x^n))/(a + b*x^n)^3, x)
\[ \int \frac {x^{3/2} \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int \frac {\sqrt {x}\, x}{x^{2 n} b^{2}+2 x^{n} a b +a^{2}}d x \] Input:
int(x^(3/2)*(A+B*x^n)/(a+b*x^n)^3,x)
Output:
int((sqrt(x)*x)/(x**(2*n)*b**2 + 2*x**n*a*b + a**2),x)