Integrand size = 20, antiderivative size = 85 \[ \int \frac {A+B x^n}{x^3 \left (a+b x^n\right )^2} \, dx=\frac {A b-a B}{a b n x^2 \left (a+b x^n\right )}+\frac {(2 a B-A b (2+n)) \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 a^2 b n x^2} \] Output:
(A*b-B*a)/a/b/n/x^2/(a+b*x^n)+1/2*(2*B*a-A*b*(2+n))*hypergeom([1, -2/n],[- (2-n)/n],-b*x^n/a)/a^2/b/n/x^2
Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x^n}{x^3 \left (a+b x^n\right )^2} \, dx=\frac {-A (-2+n) \operatorname {Hypergeometric2F1}\left (2,-\frac {2}{n},\frac {-2+n}{n},-\frac {b x^n}{a}\right )+2 B x^n \operatorname {Hypergeometric2F1}\left (2,\frac {-2+n}{n},2-\frac {2}{n},-\frac {b x^n}{a}\right )}{2 a^2 (-2+n) x^2} \] Input:
Integrate[(A + B*x^n)/(x^3*(a + b*x^n)^2),x]
Output:
(-(A*(-2 + n)*Hypergeometric2F1[2, -2/n, (-2 + n)/n, -((b*x^n)/a)]) + 2*B* x^n*Hypergeometric2F1[2, (-2 + n)/n, 2 - 2/n, -((b*x^n)/a)])/(2*a^2*(-2 + n)*x^2)
Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
\(\displaystyle \int \frac {A+B x^n}{x^3 \left (a+b x^n\right )^2} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {A b-a B}{a b n x^2 \left (a+b x^n\right )}-\frac {(2 a B-A b (n+2)) \int \frac {1}{x^3 \left (b x^n+a\right )}dx}{a b n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {(2 a B-A b (n+2)) \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 a^2 b n x^2}+\frac {A b-a B}{a b n x^2 \left (a+b x^n\right )}\) |
Input:
Int[(A + B*x^n)/(x^3*(a + b*x^n)^2),x]
Output:
(A*b - a*B)/(a*b*n*x^2*(a + b*x^n)) + ((2*a*B - A*b*(2 + n))*Hypergeometri c2F1[1, -2/n, -((2 - n)/n), -((b*x^n)/a)])/(2*a^2*b*n*x^2)
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)]))
\[\int \frac {A +B \,x^{n}}{x^{3} \left (a +b \,x^{n}\right )^{2}}d x\]
Input:
int((A+B*x^n)/x^3/(a+b*x^n)^2,x)
Output:
int((A+B*x^n)/x^3/(a+b*x^n)^2,x)
\[ \int \frac {A+B x^n}{x^3 \left (a+b x^n\right )^2} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{2} x^{3}} \,d x } \] Input:
integrate((A+B*x^n)/x^3/(a+b*x^n)^2,x, algorithm="fricas")
Output:
integral((B*x^n + A)/(b^2*x^3*x^(2*n) + 2*a*b*x^3*x^n + a^2*x^3), x)
Result contains complex when optimal does not.
Time = 27.41 (sec) , antiderivative size = 821, normalized size of antiderivative = 9.66 \[ \int \frac {A+B x^n}{x^3 \left (a+b x^n\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((A+B*x**n)/x**3/(a+b*x**n)**2,x)
Output:
A*(-2*a*a**(-2 + 2/n)*n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar( I*pi)/n)*gamma(-2/n)/(a*a**(2/n)*n**3*x**2*gamma(1 - 2/n) + a**(2/n)*b*n** 3*x**2*x**n*gamma(1 - 2/n)) - 2*a*a**(-2 + 2/n)*n*gamma(-2/n)/(a*a**(2/n)* n**3*x**2*gamma(1 - 2/n) + a**(2/n)*b*n**3*x**2*x**n*gamma(1 - 2/n)) - 4*a *a**(-2 + 2/n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)* gamma(-2/n)/(a*a**(2/n)*n**3*x**2*gamma(1 - 2/n) + a**(2/n)*b*n**3*x**2*x* *n*gamma(1 - 2/n)) - 2*a**(-2 + 2/n)*b*n*x**n*lerchphi(b*x**n*exp_polar(I* pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a*a**(2/n)*n**3*x**2*gamma(1 - 2/n) + a**(2/n)*b*n**3*x**2*x**n*gamma(1 - 2/n)) - 4*a**(-2 + 2/n)*b*x**n *lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a *a**(2/n)*n**3*x**2*gamma(1 - 2/n) + a**(2/n)*b*n**3*x**2*x**n*gamma(1 - 2 /n))) + B*(a*a**(-3 + 2/n)*a**(1 - 2/n)*n**2*x**(n - 2)*gamma(1 - 2/n)/(a* n**3*gamma(2 - 2/n) + b*n**3*x**n*gamma(2 - 2/n)) + 2*a*a**(-3 + 2/n)*a**( 1 - 2/n)*n*x**(n - 2)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1 - 2/n)*gamma (1 - 2/n)/(a*n**3*gamma(2 - 2/n) + b*n**3*x**n*gamma(2 - 2/n)) - 2*a*a**(- 3 + 2/n)*a**(1 - 2/n)*n*x**(n - 2)*gamma(1 - 2/n)/(a*n**3*gamma(2 - 2/n) + b*n**3*x**n*gamma(2 - 2/n)) - 4*a*a**(-3 + 2/n)*a**(1 - 2/n)*x**(n - 2)*l erchphi(b*x**n*exp_polar(I*pi)/a, 1, 1 - 2/n)*gamma(1 - 2/n)/(a*n**3*gamma (2 - 2/n) + b*n**3*x**n*gamma(2 - 2/n)) + 2*a**(-3 + 2/n)*a**(1 - 2/n)*b*n *x**n*x**(n - 2)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1 - 2/n)*gamma(1...
\[ \int \frac {A+B x^n}{x^3 \left (a+b x^n\right )^2} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{2} x^{3}} \,d x } \] Input:
integrate((A+B*x^n)/x^3/(a+b*x^n)^2,x, algorithm="maxima")
Output:
(A*b*(n + 2) - 2*B*a)*integrate(1/(a*b^2*n*x^3*x^n + a^2*b*n*x^3), x) - (B *a - A*b)/(a*b^2*n*x^2*x^n + a^2*b*n*x^2)
\[ \int \frac {A+B x^n}{x^3 \left (a+b x^n\right )^2} \, dx=\int { \frac {B x^{n} + A}{{\left (b x^{n} + a\right )}^{2} x^{3}} \,d x } \] Input:
integrate((A+B*x^n)/x^3/(a+b*x^n)^2,x, algorithm="giac")
Output:
integrate((B*x^n + A)/((b*x^n + a)^2*x^3), x)
Timed out. \[ \int \frac {A+B x^n}{x^3 \left (a+b x^n\right )^2} \, dx=\int \frac {A+B\,x^n}{x^3\,{\left (a+b\,x^n\right )}^2} \,d x \] Input:
int((A + B*x^n)/(x^3*(a + b*x^n)^2),x)
Output:
int((A + B*x^n)/(x^3*(a + b*x^n)^2), x)
\[ \int \frac {A+B x^n}{x^3 \left (a+b x^n\right )^2} \, dx=\int \frac {1}{x^{n} b \,x^{3}+a \,x^{3}}d x \] Input:
int((A+B*x^n)/x^3/(a+b*x^n)^2,x)
Output:
int(1/(x**n*b*x**3 + a*x**3),x)