Integrand size = 18, antiderivative size = 86 \[ \int \frac {x \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\frac {(A b-a B) x^2}{2 a b n \left (a+b x^n\right )^2}+\frac {(a B-A b (1-n)) x^2 \operatorname {Hypergeometric2F1}\left (2,\frac {2}{n},\frac {2+n}{n},-\frac {b x^n}{a}\right )}{2 a^3 b n} \] Output:
1/2*(A*b-B*a)*x^2/a/b/n/(a+b*x^n)^2+1/2*(B*a-A*b*(1-n))*x^2*hypergeom([2, 2/n],[(2+n)/n],-b*x^n/a)/a^3/b/n
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {x \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\frac {A x^2 \operatorname {Hypergeometric2F1}\left (3,\frac {2}{n},\frac {2+n}{n},-\frac {b x^n}{a}\right )}{2 a^3}+\frac {B x^{2+n} \operatorname {Hypergeometric2F1}\left (3,\frac {2+n}{n},2 \left (1+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )}{a^3 (2+n)} \] Input:
Integrate[(x*(A + B*x^n))/(a + b*x^n)^3,x]
Output:
(A*x^2*Hypergeometric2F1[3, 2/n, (2 + n)/n, -((b*x^n)/a)])/(2*a^3) + (B*x^ (2 + n)*Hypergeometric2F1[3, (2 + n)/n, 2*(1 + n^(-1)), -((b*x^n)/a)])/(a^ 3*(2 + n))
Time = 0.35 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
\(\displaystyle \int \frac {x \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {(a B-A b (1-n)) \int \frac {x}{\left (b x^n+a\right )^2}dx}{a b n}+\frac {x^2 (A b-a B)}{2 a b n \left (a+b x^n\right )^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^2 (a B-A b (1-n)) \operatorname {Hypergeometric2F1}\left (2,\frac {2}{n},\frac {n+2}{n},-\frac {b x^n}{a}\right )}{2 a^3 b n}+\frac {x^2 (A b-a B)}{2 a b n \left (a+b x^n\right )^2}\) |
Input:
Int[(x*(A + B*x^n))/(a + b*x^n)^3,x]
Output:
((A*b - a*B)*x^2)/(2*a*b*n*(a + b*x^n)^2) + ((a*B - A*b*(1 - n))*x^2*Hyper geometric2F1[2, 2/n, (2 + n)/n, -((b*x^n)/a)])/(2*a^3*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)]))
\[\int \frac {x \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right )^{3}}d x\]
Input:
int(x*(A+B*x^n)/(a+b*x^n)^3,x)
Output:
int(x*(A+B*x^n)/(a+b*x^n)^3,x)
\[ \int \frac {x \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} x}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate(x*(A+B*x^n)/(a+b*x^n)^3,x, algorithm="fricas")
Output:
integral((B*x*x^n + A*x)/(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 = 49.54 (sec) , antiderivative size = 2315, normalized size of antiderivative = 26.92 \[ \int \frac {x \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x*(A+B*x**n)/(a+b*x**n)**3,x)
Output:
A*(2*a**2*a**(2/n)*a**(-3 - 2/n)*n**2*x**2*lerchphi(b*x**n*exp_polar(I*pi) /a, 1, 2/n)*gamma(2/n)/(a**2*n**4*gamma(1 + 2/n) + 2*a*b*n**4*x**n*gamma(1 + 2/n) + b**2*n**4*x**(2*n)*gamma(1 + 2/n)) + 3*a**2*a**(2/n)*a**(-3 - 2/ n)*n**2*x**2*gamma(2/n)/(a**2*n**4*gamma(1 + 2/n) + 2*a*b*n**4*x**n*gamma( 1 + 2/n) + b**2*n**4*x**(2*n)*gamma(1 + 2/n)) - 6*a**2*a**(2/n)*a**(-3 - 2 /n)*n*x**2*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**2*n** 4*gamma(1 + 2/n) + 2*a*b*n**4*x**n*gamma(1 + 2/n) + b**2*n**4*x**(2*n)*gam ma(1 + 2/n)) - 2*a**2*a**(2/n)*a**(-3 - 2/n)*n*x**2*gamma(2/n)/(a**2*n**4* gamma(1 + 2/n) + 2*a*b*n**4*x**n*gamma(1 + 2/n) + b**2*n**4*x**(2*n)*gamma (1 + 2/n)) + 4*a**2*a**(2/n)*a**(-3 - 2/n)*x**2*lerchphi(b*x**n*exp_polar( I*pi)/a, 1, 2/n)*gamma(2/n)/(a**2*n**4*gamma(1 + 2/n) + 2*a*b*n**4*x**n*ga mma(1 + 2/n) + b**2*n**4*x**(2*n)*gamma(1 + 2/n)) + 4*a*a**(2/n)*a**(-3 - 2/n)*b*n**2*x**2*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n )/(a**2*n**4*gamma(1 + 2/n) + 2*a*b*n**4*x**n*gamma(1 + 2/n) + b**2*n**4*x **(2*n)*gamma(1 + 2/n)) + 2*a*a**(2/n)*a**(-3 - 2/n)*b*n**2*x**2*x**n*gamm a(2/n)/(a**2*n**4*gamma(1 + 2/n) + 2*a*b*n**4*x**n*gamma(1 + 2/n) + b**2*n **4*x**(2*n)*gamma(1 + 2/n)) - 12*a*a**(2/n)*a**(-3 - 2/n)*b*n*x**2*x**n*l erchphi(b*x**n*exp_polar(I*pi)/a, 1, 2/n)*gamma(2/n)/(a**2*n**4*gamma(1 + 2/n) + 2*a*b*n**4*x**n*gamma(1 + 2/n) + b**2*n**4*x**(2*n)*gamma(1 + 2/n)) - 2*a*a**(2/n)*a**(-3 - 2/n)*b*n*x**2*x**n*gamma(2/n)/(a**2*n**4*gamma...
\[ \int \frac {x \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} x}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate(x*(A+B*x^n)/(a+b*x^n)^3,x, algorithm="maxima")
Output:
((n^2 - 3*n + 2)*A*b + B*a*(n - 2))*integrate(x/(a^2*b^2*n^2*x^n + a^3*b*n ^2), x) + 1/2*(2*(A*b^2*(n - 1) + B*a*b)*x^2*x^n + (A*a*b*(3*n - 2) - B*a^ 2*(n - 2))*x^2)/(a^2*b^3*n^2*x^(2*n) + 2*a^3*b^2*n^2*x^n + a^4*b*n^2)
\[ \int \frac {x \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} x}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate(x*(A+B*x^n)/(a+b*x^n)^3,x, algorithm="giac")
Output:
integrate((B*x^n + A)*x/(b*x^n + a)^3, x)
Timed out. \[ \int \frac {x \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int \frac {x\,\left (A+B\,x^n\right )}{{\left (a+b\,x^n\right )}^3} \,d x \] Input:
int((x*(A + B*x^n))/(a + b*x^n)^3,x)
Output:
int((x*(A + B*x^n))/(a + b*x^n)^3, x)
\[ \int \frac {x \left (A+B x^n\right )}{\left (a+b x^n\right )^3} \, dx=\int \frac {x}{x^{2 n} b^{2}+2 x^{n} a b +a^{2}}d x \] Input:
int(x*(A+B*x^n)/(a+b*x^n)^3,x)
Output:
int(x/(x**(2*n)*b**2 + 2*x**n*a*b + a**2),x)