Integrand size = 26, antiderivative size = 118 \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{5/2}} \, dx=-\frac {2 B}{b e (1+3 n) \sqrt {e x} \left (a+b x^n\right )^{3/2}}-\frac {2 \left (A-\frac {a B}{b+3 b n}\right ) \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{a^2 e \sqrt {e x} \sqrt {a+b x^n}} \] Output:
-2*B/b/e/(1+3*n)/(e*x)^(1/2)/(a+b*x^n)^(3/2)-2*(A-a*B/(3*b*n+b))*(1+b*x^n/ a)^(1/2)*hypergeom([5/2, -1/2/n],[1-1/2/n],-b*x^n/a)/a^2/e/(e*x)^(1/2)/(a+ b*x^n)^(1/2)
Time = 0.97 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{5/2}} \, dx=\frac {2 x \sqrt {1+\frac {b x^n}{a}} \left (B x^n \operatorname {Hypergeometric2F1}\left (\frac {5}{2},1-\frac {1}{2 n},2-\frac {1}{2 n},-\frac {b x^n}{a}\right )+A (1-2 n) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )\right )}{a^2 (-1+2 n) (e x)^{3/2} \sqrt {a+b x^n}} \] Input:
Integrate[(A + B*x^n)/((e*x)^(3/2)*(a + b*x^n)^(5/2)),x]
Output:
(2*x*Sqrt[1 + (b*x^n)/a]*(B*x^n*Hypergeometric2F1[5/2, 1 - 1/(2*n), 2 - 1/ (2*n), -((b*x^n)/a)] + A*(1 - 2*n)*Hypergeometric2F1[5/2, -1/2*1/n, 1 - 1/ (2*n), -((b*x^n)/a)]))/(a^2*(-1 + 2*n)*(e*x)^(3/2)*Sqrt[a + b*x^n])
Time = 0.42 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, 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.
\(\displaystyle \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \left (A-\frac {a B}{3 b n+b}\right ) \int \frac {1}{(e x)^{3/2} \left (b x^n+a\right )^{5/2}}dx-\frac {2 B}{b e (3 n+1) \sqrt {e x} \left (a+b x^n\right )^{3/2}}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {\sqrt {\frac {b x^n}{a}+1} \left (A-\frac {a B}{3 b n+b}\right ) \int \frac {1}{(e x)^{3/2} \left (\frac {b x^n}{a}+1\right )^{5/2}}dx}{a^2 \sqrt {a+b x^n}}-\frac {2 B}{b e (3 n+1) \sqrt {e x} \left (a+b x^n\right )^{3/2}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -\frac {2 \sqrt {\frac {b x^n}{a}+1} \left (A-\frac {a B}{3 b n+b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{a^2 e \sqrt {e x} \sqrt {a+b x^n}}-\frac {2 B}{b e (3 n+1) \sqrt {e x} \left (a+b x^n\right )^{3/2}}\) |
Input:
Int[(A + B*x^n)/((e*x)^(3/2)*(a + b*x^n)^(5/2)),x]
Output:
(-2*B)/(b*e*(1 + 3*n)*Sqrt[e*x]*(a + b*x^n)^(3/2)) - (2*(A - (a*B)/(b + 3* b*n))*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[5/2, -1/2*1/n, 1 - 1/(2*n), -( (b*x^n)/a)])/(a^2*e*Sqrt[e*x]*Sqrt[a + b*x^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[((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])
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]
\[\int \frac {A +B \,x^{n}}{\left (e x \right )^{\frac {3}{2}} \left (a +b \,x^{n}\right )^{\frac {5}{2}}}d x\]
Input:
int((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(5/2),x)
Output:
int((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(5/2),x)
Exception generated. \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*x**n)/(e*x)**(3/2)/(a+b*x**n)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{5/2}} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x^n + A)/((b*x^n + a)^(5/2)*(e*x)^(3/2)), x)
\[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{5/2}} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(5/2),x, algorithm="giac")
Output:
integrate((B*x^n + A)/((b*x^n + a)^(5/2)*(e*x)^(3/2)), x)
Timed out. \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{5/2}} \, dx=\int \frac {A+B\,x^n}{{\left (e\,x\right )}^{3/2}\,{\left (a+b\,x^n\right )}^{5/2}} \,d x \] Input:
int((A + B*x^n)/((e*x)^(3/2)*(a + b*x^n)^(5/2)),x)
Output:
int((A + B*x^n)/((e*x)^(3/2)*(a + b*x^n)^(5/2)), x)
\[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{5/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {x^{n} b +a}}{x^{2 n +\frac {1}{2}} b^{2} x +2 x^{n +\frac {1}{2}} a b x +\sqrt {x}\, a^{2} x}d x \right )}{e^{2}} \] Input:
int((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(5/2),x)
Output:
(sqrt(e)*int(sqrt(x**n*b + a)/(x**((4*n + 1)/2)*b**2*x + 2*x**((2*n + 1)/2 )*a*b*x + sqrt(x)*a**2*x),x))/e**2