Integrand size = 26, antiderivative size = 112 \[ \int \frac {A+B x^n}{\sqrt {e x} \sqrt {a+b x^n}} \, dx=\frac {2 B \sqrt {e x} \sqrt {a+b x^n}}{b e (1+n)}+\frac {2 \left (A-\frac {a B}{b+b n}\right ) \sqrt {e x} \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )}{e \sqrt {a+b x^n}} \] Output:
2*B*(e*x)^(1/2)*(a+b*x^n)^(1/2)/b/e/(1+n)+2*(A-a*B/(b*n+b))*(e*x)^(1/2)*(1 +b*x^n/a)^(1/2)*hypergeom([1/2, 1/2/n],[1+1/2/n],-b*x^n/a)/e/(a+b*x^n)^(1/ 2)
Time = 0.40 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^n}{\sqrt {e x} \sqrt {a+b x^n}} \, dx=\frac {2 x \sqrt {1+\frac {b x^n}{a}} \left ((A+2 A n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )+B x^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {\frac {1}{2}+n}{n},\frac {1}{2} \left (4+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )\right )}{(1+2 n) \sqrt {e x} \sqrt {a+b x^n}} \] Input:
Integrate[(A + B*x^n)/(Sqrt[e*x]*Sqrt[a + b*x^n]),x]
Output:
(2*x*Sqrt[1 + (b*x^n)/a]*((A + 2*A*n)*Hypergeometric2F1[1/2, 1/(2*n), (2 + n^(-1))/2, -((b*x^n)/a)] + B*x^n*Hypergeometric2F1[1/2, (1/2 + n)/n, (4 + n^(-1))/2, -((b*x^n)/a)]))/((1 + 2*n)*Sqrt[e*x]*Sqrt[a + b*x^n])
Time = 0.41 (sec) , antiderivative size = 112, 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.
\(\displaystyle \int \frac {A+B x^n}{\sqrt {e x} \sqrt {a+b x^n}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \left (A-\frac {a B}{b n+b}\right ) \int \frac {1}{\sqrt {e x} \sqrt {b x^n+a}}dx+\frac {2 B \sqrt {e x} \sqrt {a+b x^n}}{b e (n+1)}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {\sqrt {\frac {b x^n}{a}+1} \left (A-\frac {a B}{b n+b}\right ) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x^n}{a}+1}}dx}{\sqrt {a+b x^n}}+\frac {2 B \sqrt {e x} \sqrt {a+b x^n}}{b e (n+1)}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 \sqrt {e x} \sqrt {\frac {b x^n}{a}+1} \left (A-\frac {a B}{b n+b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )}{e \sqrt {a+b x^n}}+\frac {2 B \sqrt {e x} \sqrt {a+b x^n}}{b e (n+1)}\) |
Input:
Int[(A + B*x^n)/(Sqrt[e*x]*Sqrt[a + b*x^n]),x]
Output:
(2*B*Sqrt[e*x]*Sqrt[a + b*x^n])/(b*e*(1 + n)) + (2*(A - (a*B)/(b + b*n))*S qrt[e*x]*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, 1/(2*n), (2 + n^(-1))/ 2, -((b*x^n)/a)])/(e*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]
\[\int \frac {A +B \,x^{n}}{\sqrt {e x}\, \sqrt {a +b \,x^{n}}}d x\]
Input:
int((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(1/2),x)
Output:
int((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(1/2),x)
Exception generated. \[ \int \frac {A+B x^n}{\sqrt {e x} \sqrt {a+b x^n}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Result contains complex when optimal does not.
Time = 2.67 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x^n}{\sqrt {e x} \sqrt {a+b x^n}} \, dx=\frac {A a^{\frac {1}{2 n}} a^{- \frac {1}{2} - \frac {1}{2 n}} \sqrt {x} \Gamma \left (\frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2 n} \\ 1 + \frac {1}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {e} n \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {B a^{- \frac {3}{2} - \frac {1}{2 n}} a^{1 + \frac {1}{2 n}} x^{n + \frac {1}{2}} \Gamma \left (1 + \frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, 1 + \frac {1}{2 n} \\ 2 + \frac {1}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {e} n \Gamma \left (2 + \frac {1}{2 n}\right )} \] Input:
integrate((A+B*x**n)/(e*x)**(1/2)/(a+b*x**n)**(1/2),x)
Output:
A*a**(1/(2*n))*a**(-1/2 - 1/(2*n))*sqrt(x)*gamma(1/(2*n))*hyper((1/2, 1/(2 *n)), (1 + 1/(2*n),), b*x**n*exp_polar(I*pi)/a)/(sqrt(e)*n*gamma(1 + 1/(2* n))) + B*a**(-3/2 - 1/(2*n))*a**(1 + 1/(2*n))*x**(n + 1/2)*gamma(1 + 1/(2* n))*hyper((1/2, 1 + 1/(2*n)), (2 + 1/(2*n),), b*x**n*exp_polar(I*pi)/a)/(s qrt(e)*n*gamma(2 + 1/(2*n)))
\[ \int \frac {A+B x^n}{\sqrt {e x} \sqrt {a+b x^n}} \, dx=\int { \frac {B x^{n} + A}{\sqrt {b x^{n} + a} \sqrt {e x}} \,d x } \] Input:
integrate((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(1/2),x, algorithm="maxima")
Output:
integrate((B*x^n + A)/(sqrt(b*x^n + a)*sqrt(e*x)), x)
\[ \int \frac {A+B x^n}{\sqrt {e x} \sqrt {a+b x^n}} \, dx=\int { \frac {B x^{n} + A}{\sqrt {b x^{n} + a} \sqrt {e x}} \,d x } \] Input:
integrate((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(1/2),x, algorithm="giac")
Output:
integrate((B*x^n + A)/(sqrt(b*x^n + a)*sqrt(e*x)), x)
Timed out. \[ \int \frac {A+B x^n}{\sqrt {e x} \sqrt {a+b x^n}} \, dx=\int \frac {A+B\,x^n}{\sqrt {e\,x}\,\sqrt {a+b\,x^n}} \,d x \] Input:
int((A + B*x^n)/((e*x)^(1/2)*(a + b*x^n)^(1/2)),x)
Output:
int((A + B*x^n)/((e*x)^(1/2)*(a + b*x^n)^(1/2)), x)
\[ \int \frac {A+B x^n}{\sqrt {e x} \sqrt {a+b x^n}} \, dx=\frac {\sqrt {e}\, \left (2 \sqrt {x}\, \sqrt {x^{n} b +a}+\left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}}{x^{n} b n x +x^{n} b x +a n x +a x}d x \right ) a \,n^{2}+\left (\int \frac {\sqrt {x}\, \sqrt {x^{n} b +a}}{x^{n} b n x +x^{n} b x +a n x +a x}d x \right ) a n \right )}{e \left (n +1\right )} \] Input:
int((A+B*x^n)/(e*x)^(1/2)/(a+b*x^n)^(1/2),x)
Output:
(sqrt(e)*(2*sqrt(x)*sqrt(x**n*b + a) + int((sqrt(x)*sqrt(x**n*b + a))/(x** n*b*n*x + x**n*b*x + a*n*x + a*x),x)*a*n**2 + int((sqrt(x)*sqrt(x**n*b + a ))/(x**n*b*n*x + x**n*b*x + a*n*x + a*x),x)*a*n))/(e*(n + 1))