Integrand size = 26, antiderivative size = 115 \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{3/2}} \, dx=-\frac {2 B}{b e (1+n) \sqrt {e x} \sqrt {a+b x^n}}-\frac {2 \left (\frac {A}{a}-\frac {B}{b+b n}\right ) \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{e \sqrt {e x} \sqrt {a+b x^n}} \] Output:
-2*B/b/e/(1+n)/(e*x)^(1/2)/(a+b*x^n)^(1/2)-2*(A/a-B/(b*n+b))*(1+b*x^n/a)^( 1/2)*hypergeom([3/2, -1/2/n],[1-1/2/n],-b*x^n/a)/e/(e*x)^(1/2)/(a+b*x^n)^( 1/2)
Time = 0.76 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{3/2}} \, dx=\frac {2 x \sqrt {1+\frac {b x^n}{a}} \left (B x^n \operatorname {Hypergeometric2F1}\left (\frac {3}{2},1-\frac {1}{2 n},2-\frac {1}{2 n},-\frac {b x^n}{a}\right )+A (1-2 n) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )\right )}{a (-1+2 n) (e x)^{3/2} \sqrt {a+b x^n}} \] Input:
Integrate[(A + B*x^n)/((e*x)^(3/2)*(a + b*x^n)^(3/2)),x]
Output:
(2*x*Sqrt[1 + (b*x^n)/a]*(B*x^n*Hypergeometric2F1[3/2, 1 - 1/(2*n), 2 - 1/ (2*n), -((b*x^n)/a)] + A*(1 - 2*n)*Hypergeometric2F1[3/2, -1/2*1/n, 1 - 1/ (2*n), -((b*x^n)/a)]))/(a*(-1 + 2*n)*(e*x)^(3/2)*Sqrt[a + b*x^n])
Time = 0.41 (sec) , antiderivative size = 115, 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.
\(\displaystyle \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \left (A-\frac {a B}{b n+b}\right ) \int \frac {1}{(e x)^{3/2} \left (b x^n+a\right )^{3/2}}dx-\frac {2 B}{b e (n+1) \sqrt {e x} \sqrt {a+b x^n}}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {\sqrt {\frac {b x^n}{a}+1} \left (A-\frac {a B}{b n+b}\right ) \int \frac {1}{(e x)^{3/2} \left (\frac {b x^n}{a}+1\right )^{3/2}}dx}{a \sqrt {a+b x^n}}-\frac {2 B}{b e (n+1) \sqrt {e x} \sqrt {a+b x^n}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -\frac {2 \sqrt {\frac {b x^n}{a}+1} \left (A-\frac {a B}{b n+b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2 n},1-\frac {1}{2 n},-\frac {b x^n}{a}\right )}{a e \sqrt {e x} \sqrt {a+b x^n}}-\frac {2 B}{b e (n+1) \sqrt {e x} \sqrt {a+b x^n}}\) |
Input:
Int[(A + B*x^n)/((e*x)^(3/2)*(a + b*x^n)^(3/2)),x]
Output:
(-2*B)/(b*e*(1 + n)*Sqrt[e*x]*Sqrt[a + b*x^n]) - (2*(A - (a*B)/(b + b*n))* Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[3/2, -1/2*1/n, 1 - 1/(2*n), -((b*x^n )/a)])/(a*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]
\[\int \frac {A +B \,x^{n}}{\left (e x \right )^{\frac {3}{2}} \left (a +b \,x^{n}\right )^{\frac {3}{2}}}d x\]
Input:
int((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(3/2),x)
Output:
int((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(3/2),x)
Exception generated. \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Result contains complex when optimal does not.
Time = 37.89 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{3/2}} \, dx=\frac {A a^{- \frac {1}{2 n}} a^{- \frac {3}{2} + \frac {1}{2 n}} \Gamma \left (- \frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - \frac {1}{2 n} \\ 1 - \frac {1}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{e^{\frac {3}{2}} n \sqrt {x} \Gamma \left (1 - \frac {1}{2 n}\right )} + \frac {B a^{- \frac {5}{2} + \frac {1}{2 n}} a^{1 - \frac {1}{2 n}} x^{n - \frac {1}{2}} \Gamma \left (1 - \frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, 1 - \frac {1}{2 n} \\ 2 - \frac {1}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{e^{\frac {3}{2}} n \Gamma \left (2 - \frac {1}{2 n}\right )} \] Input:
integrate((A+B*x**n)/(e*x)**(3/2)/(a+b*x**n)**(3/2),x)
Output:
A*a**(-3/2 + 1/(2*n))*gamma(-1/(2*n))*hyper((3/2, -1/(2*n)), (1 - 1/(2*n), ), b*x**n*exp_polar(I*pi)/a)/(a**(1/(2*n))*e**(3/2)*n*sqrt(x)*gamma(1 - 1/ (2*n))) + B*a**(-5/2 + 1/(2*n))*a**(1 - 1/(2*n))*x**(n - 1/2)*gamma(1 - 1/ (2*n))*hyper((3/2, 1 - 1/(2*n)), (2 - 1/(2*n),), b*x**n*exp_polar(I*pi)/a) /(e**(3/2)*n*gamma(2 - 1/(2*n)))
\[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{3/2}} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^n + A)/((b*x^n + a)^(3/2)*(e*x)^(3/2)), x)
\[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{3/2}} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^n + A)/((b*x^n + a)^(3/2)*(e*x)^(3/2)), x)
Timed out. \[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{3/2}} \, dx=\int \frac {A+B\,x^n}{{\left (e\,x\right )}^{3/2}\,{\left (a+b\,x^n\right )}^{3/2}} \,d x \] Input:
int((A + B*x^n)/((e*x)^(3/2)*(a + b*x^n)^(3/2)),x)
Output:
int((A + B*x^n)/((e*x)^(3/2)*(a + b*x^n)^(3/2)), x)
\[ \int \frac {A+B x^n}{(e x)^{3/2} \left (a+b x^n\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}}{x^{n} b \,x^{2}+a \,x^{2}}d x \right )}{e^{2}} \] Input:
int((A+B*x^n)/(e*x)^(3/2)/(a+b*x^n)^(3/2),x)
Output:
(sqrt(e)*int((sqrt(x)*sqrt(x**n*b + a))/(x**n*b*x**2 + a*x**2),x))/e**2