Integrand size = 22, antiderivative size = 89 \[ \int \frac {x^2 \left (d+e x^n\right )}{a+c x^{2 n}} \, dx=\frac {d x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {c x^{2 n}}{a}\right )}{3 a}+\frac {e x^{3+n} \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{2 n},\frac {3 (1+n)}{2 n},-\frac {c x^{2 n}}{a}\right )}{a (3+n)} \] Output:
1/3*d*x^3*hypergeom([1, 3/2/n],[1+3/2/n],-c*x^(2*n)/a)/a+e*x^(3+n)*hyperge om([1, 1/2*(3+n)/n],[3/2*(1+n)/n],-c*x^(2*n)/a)/a/(3+n)
Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (d+e x^n\right )}{a+c x^{2 n}} \, dx=\frac {d x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {c x^{2 n}}{a}\right )}{3 a}+\frac {e x^{3+n} \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{2 n},\frac {3 (1+n)}{2 n},-\frac {c x^{2 n}}{a}\right )}{a (3+n)} \] Input:
Integrate[(x^2*(d + e*x^n))/(a + c*x^(2*n)),x]
Output:
(d*x^3*Hypergeometric2F1[1, 3/(2*n), 1 + 3/(2*n), -((c*x^(2*n))/a)])/(3*a) + (e*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/(2*n), (3*(1 + n))/(2*n), -(( c*x^(2*n))/a)])/(a*(3 + n))
Time = 0.24 (sec) , antiderivative size = 89, 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 = {1885, 2009}
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.
\(\displaystyle \int \frac {x^2 \left (d+e x^n\right )}{a+c x^{2 n}} \, dx\) |
\(\Big \downarrow \) 1885 |
\(\displaystyle \int \left (\frac {d x^2}{a+c x^{2 n}}+\frac {e x^{n+2}}{a+c x^{2 n}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {c x^{2 n}}{a}\right )}{3 a}+\frac {e x^{n+3} \operatorname {Hypergeometric2F1}\left (1,\frac {n+3}{2 n},\frac {3 (n+1)}{2 n},-\frac {c x^{2 n}}{a}\right )}{a (n+3)}\) |
Input:
Int[(x^2*(d + e*x^n))/(a + c*x^(2*n)),x]
Output:
(d*x^3*Hypergeometric2F1[1, 3/(2*n), 1 + 3/(2*n), -((c*x^(2*n))/a)])/(3*a) + (e*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/(2*n), (3*(1 + n))/(2*n), -(( c*x^(2*n))/a)])/(a*(3 + n))
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^ (n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + c* x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && !RationalQ[n] && (IGtQ[p, 0] || IGtQ[q, 0])
\[\int \frac {x^{2} \left (d +e \,x^{n}\right )}{a +c \,x^{2 n}}d x\]
Input:
int(x^2*(d+e*x^n)/(a+c*x^(2*n)),x)
Output:
int(x^2*(d+e*x^n)/(a+c*x^(2*n)),x)
\[ \int \frac {x^2 \left (d+e x^n\right )}{a+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )} x^{2}}{c x^{2 \, n} + a} \,d x } \] Input:
integrate(x^2*(d+e*x^n)/(a+c*x^(2*n)),x, algorithm="fricas")
Output:
integral((e*x^2*x^n + d*x^2)/(c*x^(2*n) + a), x)
Result contains complex when optimal does not.
Time = 3.91 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.40 \[ \int \frac {x^2 \left (d+e x^n\right )}{a+c x^{2 n}} \, dx=\frac {3 a^{\frac {3}{2 n}} a^{-1 - \frac {3}{2 n}} d x^{3} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {3}{2 n}\right ) \Gamma \left (\frac {3}{2 n}\right )}{4 n^{2} \Gamma \left (1 + \frac {3}{2 n}\right )} + \frac {a^{- \frac {3}{2} - \frac {3}{2 n}} a^{\frac {1}{2} + \frac {3}{2 n}} e x^{n + 3} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {3}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {3}{2 n}\right )}{4 n \Gamma \left (\frac {3}{2} + \frac {3}{2 n}\right )} + \frac {3 a^{- \frac {3}{2} - \frac {3}{2 n}} a^{\frac {1}{2} + \frac {3}{2 n}} e x^{n + 3} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {3}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {3}{2 n}\right )}{4 n^{2} \Gamma \left (\frac {3}{2} + \frac {3}{2 n}\right )} \] Input:
integrate(x**2*(d+e*x**n)/(a+c*x**(2*n)),x)
Output:
3*a**(3/(2*n))*a**(-1 - 3/(2*n))*d*x**3*lerchphi(c*x**(2*n)*exp_polar(I*pi )/a, 1, 3/(2*n))*gamma(3/(2*n))/(4*n**2*gamma(1 + 3/(2*n))) + a**(-3/2 - 3 /(2*n))*a**(1/2 + 3/(2*n))*e*x**(n + 3)*lerchphi(c*x**(2*n)*exp_polar(I*pi )/a, 1, 1/2 + 3/(2*n))*gamma(1/2 + 3/(2*n))/(4*n*gamma(3/2 + 3/(2*n))) + 3 *a**(-3/2 - 3/(2*n))*a**(1/2 + 3/(2*n))*e*x**(n + 3)*lerchphi(c*x**(2*n)*e xp_polar(I*pi)/a, 1, 1/2 + 3/(2*n))*gamma(1/2 + 3/(2*n))/(4*n**2*gamma(3/2 + 3/(2*n)))
\[ \int \frac {x^2 \left (d+e x^n\right )}{a+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )} x^{2}}{c x^{2 \, n} + a} \,d x } \] Input:
integrate(x^2*(d+e*x^n)/(a+c*x^(2*n)),x, algorithm="maxima")
Output:
integrate((e*x^n + d)*x^2/(c*x^(2*n) + a), x)
\[ \int \frac {x^2 \left (d+e x^n\right )}{a+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )} x^{2}}{c x^{2 \, n} + a} \,d x } \] Input:
integrate(x^2*(d+e*x^n)/(a+c*x^(2*n)),x, algorithm="giac")
Output:
integrate((e*x^n + d)*x^2/(c*x^(2*n) + a), x)
Timed out. \[ \int \frac {x^2 \left (d+e x^n\right )}{a+c x^{2 n}} \, dx=\int \frac {x^2\,\left (d+e\,x^n\right )}{a+c\,x^{2\,n}} \,d x \] Input:
int((x^2*(d + e*x^n))/(a + c*x^(2*n)),x)
Output:
int((x^2*(d + e*x^n))/(a + c*x^(2*n)), x)
\[ \int \frac {x^2 \left (d+e x^n\right )}{a+c x^{2 n}} \, dx=\left (\int \frac {x^{2}}{x^{2 n} c +a}d x \right ) d +\left (\int \frac {x^{n} x^{2}}{x^{2 n} c +a}d x \right ) e \] Input:
int(x^2*(d+e*x^n)/(a+c*x^(2*n)),x)
Output:
int(x**2/(x**(2*n)*c + a),x)*d + int((x**n*x**2)/(x**(2*n)*c + a),x)*e