Integrand size = 22, antiderivative size = 88 \[ \int \frac {A+B x^n}{\sqrt {x} \left (a+b x^n\right )^2} \, dx=\frac {(A b-a B) \sqrt {x}}{a b n \left (a+b x^n\right )}+\frac {(a B-A b (1-2 n)) \sqrt {x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )}{a^2 b n} \] Output:
(A*b-B*a)*x^(1/2)/a/b/n/(a+b*x^n)+(B*a-A*b*(1-2*n))*x^(1/2)*hypergeom([1, 1/2/n],[1+1/2/n],-b*x^n/a)/a^2/b/n
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^n}{\sqrt {x} \left (a+b x^n\right )^2} \, dx=\frac {\sqrt {x} \left (a (A b-a B)+(a B+A b (-1+2 n)) \left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},1+\frac {1}{2 n},-\frac {b x^n}{a}\right )\right )}{a^2 b n \left (a+b x^n\right )} \] Input:
Integrate[(A + B*x^n)/(Sqrt[x]*(a + b*x^n)^2),x]
Output:
(Sqrt[x]*(a*(A*b - a*B) + (a*B + A*b*(-1 + 2*n))*(a + b*x^n)*Hypergeometri c2F1[1, 1/(2*n), 1 + 1/(2*n), -((b*x^n)/a)]))/(a^2*b*n*(a + b*x^n))
Time = 0.36 (sec) , antiderivative size = 88, 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.
\(\displaystyle \int \frac {A+B x^n}{\sqrt {x} \left (a+b x^n\right )^2} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {(a B-A b (1-2 n)) \int \frac {1}{\sqrt {x} \left (b x^n+a\right )}dx}{2 a b n}+\frac {\sqrt {x} (A b-a B)}{a b n \left (a+b x^n\right )}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {\sqrt {x} (a B-A b (1-2 n)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )}{a^2 b n}+\frac {\sqrt {x} (A b-a B)}{a b n \left (a+b x^n\right )}\) |
Input:
Int[(A + B*x^n)/(Sqrt[x]*(a + b*x^n)^2),x]
Output:
((A*b - a*B)*Sqrt[x])/(a*b*n*(a + b*x^n)) + ((a*B - A*b*(1 - 2*n))*Sqrt[x] *Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((b*x^n)/a)])/(a^2*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)]))
\[\int \frac {A +B \,x^{n}}{\sqrt {x}\, \left (a +b \,x^{n}\right )^{2}}d x\]
Input:
int((A+B*x^n)/x^(1/2)/(a+b*x^n)^2,x)
Output:
int((A+B*x^n)/x^(1/2)/(a+b*x^n)^2,x)
\[ \int \frac {A+B x^n}{\sqrt {x} \left (a+b x^n\right )^2} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{2} \sqrt {x}} \,d x } \] Input:
integrate((A+B*x^n)/x^(1/2)/(a+b*x^n)^2,x, algorithm="fricas")
Output:
integral((B*x^n + A)*sqrt(x)/(b^2*x*x^(2*n) + 2*a*b*x*x^n + a^2*x), x)
Result contains complex when optimal does not.
Time = 52.37 (sec) , antiderivative size = 925, normalized size of antiderivative = 10.51 \[ \int \frac {A+B x^n}{\sqrt {x} \left (a+b x^n\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((A+B*x**n)/x**(1/2)/(a+b*x**n)**2,x)
Output:
A*(2*a*a**(1/(2*n))*a**(-2 - 1/(2*n))*n*sqrt(x)*lerchphi(b*x**n*exp_polar( I*pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(4*a*n**3*gamma(1 + 1/(2*n)) + 4*b*n** 3*x**n*gamma(1 + 1/(2*n))) + 2*a*a**(1/(2*n))*a**(-2 - 1/(2*n))*n*sqrt(x)* gamma(1/(2*n))/(4*a*n**3*gamma(1 + 1/(2*n)) + 4*b*n**3*x**n*gamma(1 + 1/(2 *n))) - a*a**(1/(2*n))*a**(-2 - 1/(2*n))*sqrt(x)*lerchphi(b*x**n*exp_polar (I*pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(4*a*n**3*gamma(1 + 1/(2*n)) + 4*b*n* *3*x**n*gamma(1 + 1/(2*n))) + 2*a**(1/(2*n))*a**(-2 - 1/(2*n))*b*n*sqrt(x) *x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(4*a*n **3*gamma(1 + 1/(2*n)) + 4*b*n**3*x**n*gamma(1 + 1/(2*n))) - a**(1/(2*n))* a**(-2 - 1/(2*n))*b*sqrt(x)*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/( 2*n))*gamma(1/(2*n))/(4*a*n**3*gamma(1 + 1/(2*n)) + 4*b*n**3*x**n*gamma(1 + 1/(2*n)))) + B*(4*a*a**(-3 - 1/(2*n))*a**(1 + 1/(2*n))*n**2*x**(n + 1/2) *gamma(1 + 1/(2*n))/(4*a*n**3*gamma(2 + 1/(2*n)) + 4*b*n**3*x**n*gamma(2 + 1/(2*n))) - 2*a*a**(-3 - 1/(2*n))*a**(1 + 1/(2*n))*n*x**(n + 1/2)*lerchph i(b*x**n*exp_polar(I*pi)/a, 1, 1 + 1/(2*n))*gamma(1 + 1/(2*n))/(4*a*n**3*g amma(2 + 1/(2*n)) + 4*b*n**3*x**n*gamma(2 + 1/(2*n))) + 2*a*a**(-3 - 1/(2* n))*a**(1 + 1/(2*n))*n*x**(n + 1/2)*gamma(1 + 1/(2*n))/(4*a*n**3*gamma(2 + 1/(2*n)) + 4*b*n**3*x**n*gamma(2 + 1/(2*n))) - a*a**(-3 - 1/(2*n))*a**(1 + 1/(2*n))*x**(n + 1/2)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1 + 1/(2*n)) *gamma(1 + 1/(2*n))/(4*a*n**3*gamma(2 + 1/(2*n)) + 4*b*n**3*x**n*gamma(...
\[ \int \frac {A+B x^n}{\sqrt {x} \left (a+b x^n\right )^2} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{2} \sqrt {x}} \,d x } \] Input:
integrate((A+B*x^n)/x^(1/2)/(a+b*x^n)^2,x, algorithm="maxima")
Output:
4*((2*n^2 - n)*A*a*b + B*a^2*n)*integrate(1/(((8*n^2 - 6*n + 1)*b^4*x^(3*n ) + 3*(8*n^2 - 6*n + 1)*a*b^3*x^(2*n) + 3*(8*n^2 - 6*n + 1)*a^2*b^2*x^n + (8*n^2 - 6*n + 1)*a^3*b)*sqrt(x)), x) - 2*(B*b*(4*n - 1)*x*x^n + (A*b*(2*n - 1) + 4*B*a*n)*x)/(((8*n^2 - 6*n + 1)*b^3*x^(2*n) + 2*(8*n^2 - 6*n + 1)* a*b^2*x^n + (8*n^2 - 6*n + 1)*a^2*b)*sqrt(x))
\[ \int \frac {A+B x^n}{\sqrt {x} \left (a+b x^n\right )^2} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{2} \sqrt {x}} \,d x } \] Input:
integrate((A+B*x^n)/x^(1/2)/(a+b*x^n)^2,x, algorithm="giac")
Output:
integrate((B*x^n + A)/((b*x^n + a)^2*sqrt(x)), x)
Timed out. \[ \int \frac {A+B x^n}{\sqrt {x} \left (a+b x^n\right )^2} \, dx=\int \frac {A+B\,x^n}{\sqrt {x}\,{\left (a+b\,x^n\right )}^2} \,d x \] Input:
int((A + B*x^n)/(x^(1/2)*(a + b*x^n)^2),x)
Output:
int((A + B*x^n)/(x^(1/2)*(a + b*x^n)^2), x)
\[ \int \frac {A+B x^n}{\sqrt {x} \left (a+b x^n\right )^2} \, dx=\int \frac {1}{x^{n +\frac {1}{2}} b +\sqrt {x}\, a}d x \] Input:
int((A+B*x^n)/x^(1/2)/(a+b*x^n)^2,x)
Output:
int(1/(x**((2*n + 1)/2)*b + sqrt(x)*a),x)