Integrand size = 22, antiderivative size = 104 \[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{x^3} \, dx=-\frac {2 B \left (a+b x^n\right )^{3/2}}{b (4-3 n) x^2}-\frac {\left (A-\frac {4 a B}{4 b-3 b n}\right ) \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {1+\frac {b x^n}{a}}} \] Output:
-2*B*(a+b*x^n)^(3/2)/b/(4-3*n)/x^2-1/2*(A-4*a*B/(-3*b*n+4*b))*(a+b*x^n)^(1 /2)*hypergeom([-1/2, -2/n],[-(2-n)/n],-b*x^n/a)/x^2/(1+b*x^n/a)^(1/2)
Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{x^3} \, dx=\frac {\sqrt {a+b x^n} \left (-A (-2+n) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{n},\frac {-2+n}{n},-\frac {b x^n}{a}\right )+2 B x^n \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {-2+n}{n},2-\frac {2}{n},-\frac {b x^n}{a}\right )\right )}{2 (-2+n) x^2 \sqrt {1+\frac {b x^n}{a}}} \] Input:
Integrate[(Sqrt[a + b*x^n]*(A + B*x^n))/x^3,x]
Output:
(Sqrt[a + b*x^n]*(-(A*(-2 + n)*Hypergeometric2F1[-1/2, -2/n, (-2 + n)/n, - ((b*x^n)/a)]) + 2*B*x^n*Hypergeometric2F1[-1/2, (-2 + n)/n, 2 - 2/n, -((b* x^n)/a)]))/(2*(-2 + n)*x^2*Sqrt[1 + (b*x^n)/a])
Time = 0.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.
\(\displaystyle \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \left (A-\frac {4 a B}{4 b-3 b n}\right ) \int \frac {\sqrt {b x^n+a}}{x^3}dx-\frac {2 B \left (a+b x^n\right )^{3/2}}{b (4-3 n) x^2}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {\sqrt {a+b x^n} \left (A-\frac {4 a B}{4 b-3 b n}\right ) \int \frac {\sqrt {\frac {b x^n}{a}+1}}{x^3}dx}{\sqrt {\frac {b x^n}{a}+1}}-\frac {2 B \left (a+b x^n\right )^{3/2}}{b (4-3 n) x^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -\frac {\sqrt {a+b x^n} \left (A-\frac {4 a B}{4 b-3 b n}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {\frac {b x^n}{a}+1}}-\frac {2 B \left (a+b x^n\right )^{3/2}}{b (4-3 n) x^2}\) |
Input:
Int[(Sqrt[a + b*x^n]*(A + B*x^n))/x^3,x]
Output:
(-2*B*(a + b*x^n)^(3/2))/(b*(4 - 3*n)*x^2) - ((A - (4*a*B)/(4*b - 3*b*n))* Sqrt[a + b*x^n]*Hypergeometric2F1[-1/2, -2/n, -((2 - n)/n), -((b*x^n)/a)]) /(2*x^2*Sqrt[1 + (b*x^n)/a])
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]
\[\int \frac {\sqrt {a +b \,x^{n}}\, \left (A +B \,x^{n}\right )}{x^{3}}d x\]
Input:
int((a+b*x^n)^(1/2)*(A+B*x^n)/x^3,x)
Output:
int((a+b*x^n)^(1/2)*(A+B*x^n)/x^3,x)
Exception generated. \[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*x^n)^(1/2)*(A+B*x^n)/x^3,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 = 2.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{x^3} \, dx=\frac {A a^{- \frac {2}{n}} a^{\frac {1}{2} + \frac {2}{n}} \Gamma \left (- \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {2}{n} \\ 1 - \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n x^{2} \Gamma \left (1 - \frac {2}{n}\right )} + \frac {B a^{- \frac {1}{2} + \frac {2}{n}} a^{1 - \frac {2}{n}} x^{n - 2} \Gamma \left (1 - \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 - \frac {2}{n} \\ 2 - \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 - \frac {2}{n}\right )} \] Input:
integrate((a+b*x**n)**(1/2)*(A+B*x**n)/x**3,x)
Output:
A*a**(1/2 + 2/n)*gamma(-2/n)*hyper((-1/2, -2/n), (1 - 2/n,), b*x**n*exp_po lar(I*pi)/a)/(a**(2/n)*n*x**2*gamma(1 - 2/n)) + B*a**(-1/2 + 2/n)*a**(1 - 2/n)*x**(n - 2)*gamma(1 - 2/n)*hyper((-1/2, 1 - 2/n), (2 - 2/n,), b*x**n*e xp_polar(I*pi)/a)/(n*gamma(2 - 2/n))
\[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{x^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \sqrt {b x^{n} + a}}{x^{3}} \,d x } \] Input:
integrate((a+b*x^n)^(1/2)*(A+B*x^n)/x^3,x, algorithm="maxima")
Output:
integrate((B*x^n + A)*sqrt(b*x^n + a)/x^3, x)
\[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{x^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \sqrt {b x^{n} + a}}{x^{3}} \,d x } \] Input:
integrate((a+b*x^n)^(1/2)*(A+B*x^n)/x^3,x, algorithm="giac")
Output:
integrate((B*x^n + A)*sqrt(b*x^n + a)/x^3, x)
Timed out. \[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{x^3} \, dx=\int \frac {\left (A+B\,x^n\right )\,\sqrt {a+b\,x^n}}{x^3} \,d x \] Input:
int(((A + B*x^n)*(a + b*x^n)^(1/2))/x^3,x)
Output:
int(((A + B*x^n)*(a + b*x^n)^(1/2))/x^3, x)
\[ \int \frac {\sqrt {a+b x^n} \left (A+B x^n\right )}{x^3} \, dx=\frac {2 x^{n} \sqrt {x^{n} b +a}\, b n -8 x^{n} \sqrt {x^{n} b +a}\, b +8 \sqrt {x^{n} b +a}\, a n -8 \sqrt {x^{n} b +a}\, a +9 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2} x^{3}-16 x^{n} b n \,x^{3}+16 x^{n} b \,x^{3}+3 a \,n^{2} x^{3}-16 a n \,x^{3}+16 a \,x^{3}}d x \right ) a^{2} n^{4} x^{2}-48 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2} x^{3}-16 x^{n} b n \,x^{3}+16 x^{n} b \,x^{3}+3 a \,n^{2} x^{3}-16 a n \,x^{3}+16 a \,x^{3}}d x \right ) a^{2} n^{3} x^{2}+48 \left (\int \frac {\sqrt {x^{n} b +a}}{3 x^{n} b \,n^{2} x^{3}-16 x^{n} b n \,x^{3}+16 x^{n} b \,x^{3}+3 a \,n^{2} x^{3}-16 a n \,x^{3}+16 a \,x^{3}}d x \right ) a^{2} n^{2} x^{2}}{x^{2} \left (3 n^{2}-16 n +16\right )} \] Input:
int((a+b*x^n)^(1/2)*(A+B*x^n)/x^3,x)
Output:
(2*x**n*sqrt(x**n*b + a)*b*n - 8*x**n*sqrt(x**n*b + a)*b + 8*sqrt(x**n*b + a)*a*n - 8*sqrt(x**n*b + a)*a + 9*int(sqrt(x**n*b + a)/(3*x**n*b*n**2*x** 3 - 16*x**n*b*n*x**3 + 16*x**n*b*x**3 + 3*a*n**2*x**3 - 16*a*n*x**3 + 16*a *x**3),x)*a**2*n**4*x**2 - 48*int(sqrt(x**n*b + a)/(3*x**n*b*n**2*x**3 - 1 6*x**n*b*n*x**3 + 16*x**n*b*x**3 + 3*a*n**2*x**3 - 16*a*n*x**3 + 16*a*x**3 ),x)*a**2*n**3*x**2 + 48*int(sqrt(x**n*b + a)/(3*x**n*b*n**2*x**3 - 16*x** n*b*n*x**3 + 16*x**n*b*x**3 + 3*a*n**2*x**3 - 16*a*n*x**3 + 16*a*x**3),x)* a**2*n**2*x**2)/(x**2*(3*n**2 - 16*n + 16))