Integrand size = 17, antiderivative size = 61 \[ \int \frac {A+B x^n}{\left (a+b x^n\right )^3} \, dx=\frac {B x}{b (1-2 n) \left (a+b x^n\right )^2}+\frac {\left (A-\frac {a B}{b-2 b n}\right ) x \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^3} \] Output:
B*x/b/(1-2*n)/(a+b*x^n)^2+(A-a*B/(-2*b*n+b))*x*hypergeom([3, 1/n],[1+1/n], -b*x^n/a)/a^3
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x^n}{\left (a+b x^n\right )^3} \, dx=\frac {x \left (\frac {B}{\left (a+b x^n\right )^2}-\frac {(a B+A b (-1+2 n)) \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^3}\right )}{b-2 b n} \] Input:
Integrate[(A + B*x^n)/(a + b*x^n)^3,x]
Output:
(x*(B/(a + b*x^n)^2 - ((a*B + A*b*(-1 + 2*n))*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^3))/(b - 2*b*n)
Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.28, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {910, 778}
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.
\(\displaystyle \int \frac {A+B x^n}{\left (a+b x^n\right )^3} \, dx\) |
\(\Big \downarrow \) 910 |
\(\displaystyle \frac {(a B-A b (1-2 n)) \int \frac {1}{\left (b x^n+a\right )^2}dx}{2 a b n}+\frac {x (A b-a B)}{2 a b n \left (a+b x^n\right )^2}\) |
\(\Big \downarrow \) 778 |
\(\displaystyle \frac {x (a B-A b (1-2 n)) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{2 a^3 b n}+\frac {x (A b-a B)}{2 a b n \left (a+b x^n\right )^2}\) |
Input:
Int[(A + B*x^n)/(a + b*x^n)^3,x]
Output:
((A*b - a*B)*x)/(2*a*b*n*(a + b*x^n)^2) + ((a*B - A*b*(1 - 2*n))*x*Hyperge ometric2F1[2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(2*a^3*b*n)
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
\[\int \frac {A +B \,x^{n}}{\left (a +b \,x^{n}\right )^{3}}d x\]
Input:
int((A+B*x^n)/(a+b*x^n)^3,x)
Output:
int((A+B*x^n)/(a+b*x^n)^3,x)
\[ \int \frac {A+B x^n}{\left (a+b x^n\right )^3} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate((A+B*x^n)/(a+b*x^n)^3,x, algorithm="fricas")
Output:
integral((B*x^n + A)/(b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n + a^3), x)
Result contains complex when optimal does not.
Time = 49.66 (sec) , antiderivative size = 2319, normalized size of antiderivative = 38.02 \[ \int \frac {A+B x^n}{\left (a+b x^n\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((A+B*x**n)/(a+b*x**n)**3,x)
Output:
A*(2*a**2*a**(1/n)*a**(-3 - 1/n)*n**2*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(2*a**2*n**4*gamma(1 + 1/n) + 4*a*b*n**4*x**n*gamma(1 + 1/n) + 2*b**2*n**4*x**(2*n)*gamma(1 + 1/n)) + 3*a**2*a**(1/n)*a**(-3 - 1 /n)*n**2*x*gamma(1/n)/(2*a**2*n**4*gamma(1 + 1/n) + 4*a*b*n**4*x**n*gamma( 1 + 1/n) + 2*b**2*n**4*x**(2*n)*gamma(1 + 1/n)) - 3*a**2*a**(1/n)*a**(-3 - 1/n)*n*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(2*a**2*n* *4*gamma(1 + 1/n) + 4*a*b*n**4*x**n*gamma(1 + 1/n) + 2*b**2*n**4*x**(2*n)* gamma(1 + 1/n)) - a**2*a**(1/n)*a**(-3 - 1/n)*n*x*gamma(1/n)/(2*a**2*n**4* gamma(1 + 1/n) + 4*a*b*n**4*x**n*gamma(1 + 1/n) + 2*b**2*n**4*x**(2*n)*gam ma(1 + 1/n)) + a**2*a**(1/n)*a**(-3 - 1/n)*x*lerchphi(b*x**n*exp_polar(I*p i)/a, 1, 1/n)*gamma(1/n)/(2*a**2*n**4*gamma(1 + 1/n) + 4*a*b*n**4*x**n*gam ma(1 + 1/n) + 2*b**2*n**4*x**(2*n)*gamma(1 + 1/n)) + 4*a*a**(1/n)*a**(-3 - 1/n)*b*n**2*x*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/ (2*a**2*n**4*gamma(1 + 1/n) + 4*a*b*n**4*x**n*gamma(1 + 1/n) + 2*b**2*n**4 *x**(2*n)*gamma(1 + 1/n)) + 2*a*a**(1/n)*a**(-3 - 1/n)*b*n**2*x*x**n*gamma (1/n)/(2*a**2*n**4*gamma(1 + 1/n) + 4*a*b*n**4*x**n*gamma(1 + 1/n) + 2*b** 2*n**4*x**(2*n)*gamma(1 + 1/n)) - 6*a*a**(1/n)*a**(-3 - 1/n)*b*n*x*x**n*le rchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(2*a**2*n**4*gamma(1 + 1/n) + 4*a*b*n**4*x**n*gamma(1 + 1/n) + 2*b**2*n**4*x**(2*n)*gamma(1 + 1/ n)) - a*a**(1/n)*a**(-3 - 1/n)*b*n*x*x**n*gamma(1/n)/(2*a**2*n**4*gamma...
\[ \int \frac {A+B x^n}{\left (a+b x^n\right )^3} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate((A+B*x^n)/(a+b*x^n)^3,x, algorithm="maxima")
Output:
((2*n^2 - 3*n + 1)*A*b + B*a*(n - 1))*integrate(1/2/(a^2*b^2*n^2*x^n + a^3 *b*n^2), x) + 1/2*((A*b^2*(2*n - 1) + B*a*b)*x*x^n + (A*a*b*(3*n - 1) - B* a^2*(n - 1))*x)/(a^2*b^3*n^2*x^(2*n) + 2*a^3*b^2*n^2*x^n + a^4*b*n^2)
\[ \int \frac {A+B x^n}{\left (a+b x^n\right )^3} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate((A+B*x^n)/(a+b*x^n)^3,x, algorithm="giac")
Output:
integrate((B*x^n + A)/(b*x^n + a)^3, x)
Timed out. \[ \int \frac {A+B x^n}{\left (a+b x^n\right )^3} \, dx=\int \frac {A+B\,x^n}{{\left (a+b\,x^n\right )}^3} \,d x \] Input:
int((A + B*x^n)/(a + b*x^n)^3,x)
Output:
int((A + B*x^n)/(a + b*x^n)^3, x)
\[ \int \frac {A+B x^n}{\left (a+b x^n\right )^3} \, dx=\int \frac {1}{x^{2 n} b^{2}+2 x^{n} a b +a^{2}}d x \] Input:
int((A+B*x^n)/(a+b*x^n)^3,x)
Output:
int(1/(x**(2*n)*b**2 + 2*x**n*a*b + a**2),x)