Integrand size = 18, antiderivative size = 136 \[ \int \frac {x}{a+b x^n+c x^{2 n}} \, dx=-\frac {c x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {2+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {2+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}} \] Output:
-c*x^2*hypergeom([1, 2/n],[(2+n)/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))/(b^2- 4*a*c-b*(-4*a*c+b^2)^(1/2))-c*x^2*hypergeom([1, 2/n],[(2+n)/n],-2*c*x^n/(b +(-4*a*c+b^2)^(1/2)))/(b*(-4*a*c+b^2)^(1/2)-4*a*c+b^2)
Time = 0.81 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.93 \[ \int \frac {x}{a+b x^n+c x^{2 n}} \, dx=-c x^2 \left (\frac {1-\left (\frac {x^n}{-\frac {-b+\sqrt {b^2-4 a c}}{2 c}+x^n}\right )^{-2/n} \operatorname {Hypergeometric2F1}\left (-\frac {2}{n},-\frac {2}{n},\frac {-2+n}{n},\frac {b-\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}+2 c x^n}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {1-4^{-1/n} \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-2/n} \operatorname {Hypergeometric2F1}\left (-\frac {2}{n},-\frac {2}{n},\frac {-2+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )}\right ) \] Input:
Integrate[x/(a + b*x^n + c*x^(2*n)),x]
Output:
-(c*x^2*((1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b - Sqrt[b^2 - 4* a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c ])/c + x^n))^(2/n))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hypergeomet ric2F1[-2/n, -2/n, (-2 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a *c] + 2*c*x^n)]/(4^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(2/n )))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c]))))
Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1719, 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.
\(\displaystyle \int \frac {x}{a+b x^n+c x^{2 n}} \, dx\) |
\(\Big \downarrow \) 1719 |
\(\displaystyle \frac {2 c \int \frac {x}{2 c x^n+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {x}{2 c x^n+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {c x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {n+2}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {c x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {n+2}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}\) |
Input:
Int[x/(a + b*x^n + c*x^(2*n)),x]
Output:
(c*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a *c])])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])) - (c*x^2*Hypergeometric 2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4* a*c]*(b + Sqrt[b^2 - 4*a*c]))
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[((d_.)*(x_))^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symb ol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int[(d*x)^m/(b - q + 2 *c*x^n), x], x] - Simp[2*(c/q) Int[(d*x)^m/(b + q + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
\[\int \frac {x}{a +b \,x^{n}+c \,x^{2 n}}d x\]
Input:
int(x/(a+b*x^n+c*x^(2*n)),x)
Output:
int(x/(a+b*x^n+c*x^(2*n)),x)
\[ \int \frac {x}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x}{c x^{2 \, n} + b x^{n} + a} \,d x } \] Input:
integrate(x/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")
Output:
integral(x/(c*x^(2*n) + b*x^n + a), x)
\[ \int \frac {x}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x}{a + b x^{n} + c x^{2 n}}\, dx \] Input:
integrate(x/(a+b*x**n+c*x**(2*n)),x)
Output:
Integral(x/(a + b*x**n + c*x**(2*n)), x)
\[ \int \frac {x}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x}{c x^{2 \, n} + b x^{n} + a} \,d x } \] Input:
integrate(x/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")
Output:
integrate(x/(c*x^(2*n) + b*x^n + a), x)
\[ \int \frac {x}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x}{c x^{2 \, n} + b x^{n} + a} \,d x } \] Input:
integrate(x/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")
Output:
integrate(x/(c*x^(2*n) + b*x^n + a), x)
Timed out. \[ \int \frac {x}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x}{a+b\,x^n+c\,x^{2\,n}} \,d x \] Input:
int(x/(a + b*x^n + c*x^(2*n)),x)
Output:
int(x/(a + b*x^n + c*x^(2*n)), x)
\[ \int \frac {x}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x}{x^{2 n} c +x^{n} b +a}d x \] Input:
int(x/(a+b*x^n+c*x^(2*n)),x)
Output:
int(x/(x**(2*n)*c + x**n*b + a),x)