\(\int \frac {x^2}{(d+e x^n) (a+c x^{2 n})} \, dx\) [12]

Optimal result

Integrand size = 24, antiderivative size = 167 \[ \int \frac {x^2}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\frac {c d x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {c x^{2 n}}{a}\right )}{3 a \left (c d^2+a e^2\right )}+\frac {e^2 x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {e x^n}{d}\right )}{3 d \left (c d^2+a e^2\right )}-\frac {c e x^{3+n} \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{2 n},\frac {3 (1+n)}{2 n},-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right ) (3+n)} \] Output:

1/3*c*d*x^3*hypergeom([1, 3/2/n],[1+3/2/n],-c*x^(2*n)/a)/a/(a*e^2+c*d^2)+1 
/3*e^2*x^3*hypergeom([1, 3/n],[(3+n)/n],-e*x^n/d)/d/(a*e^2+c*d^2)-c*e*x^(3 
+n)*hypergeom([1, 1/2*(3+n)/n],[3/2*(1+n)/n],-c*x^(2*n)/a)/a/(a*e^2+c*d^2) 
/(3+n)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\frac {x^3 \left (c d^2 (3+n) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {c x^{2 n}}{a}\right )+e \left (a e (3+n) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {e x^n}{d}\right )-3 c d x^n \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{2 n},\frac {3 (1+n)}{2 n},-\frac {c x^{2 n}}{a}\right )\right )\right )}{3 a d \left (c d^2+a e^2\right ) (3+n)} \] Input:

Integrate[x^2/((d + e*x^n)*(a + c*x^(2*n))),x]
 

Output:

(x^3*(c*d^2*(3 + n)*Hypergeometric2F1[1, 3/(2*n), 1 + 3/(2*n), -((c*x^(2*n 
))/a)] + e*(a*e*(3 + n)*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((e*x^n)/d)] 
 - 3*c*d*x^n*Hypergeometric2F1[1, (3 + n)/(2*n), (3*(1 + n))/(2*n), -((c*x 
^(2*n))/a)])))/(3*a*d*(c*d^2 + a*e^2)*(3 + n))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.68, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1881, 1010, 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.

\(\displaystyle \int \frac {x^2}{\left (a+c x^{2 n}\right ) \left (d+e x^n\right )} \, dx\)

\(\Big \downarrow \) 1881

\(\displaystyle -\frac {\sqrt {c} \int \frac {x^2}{\left (\sqrt {-a} \sqrt {c}-c x^n\right ) \left (e x^n+d\right )}dx}{2 \sqrt {-a}}-\frac {\sqrt {c} \int \frac {x^2}{\left (c x^n+\sqrt {-a} \sqrt {c}\right ) \left (e x^n+d\right )}dx}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 1010

\(\displaystyle -\frac {\sqrt {c} \left (\frac {\sqrt {c} \int \frac {x^2}{c x^n+\sqrt {-a} \sqrt {c}}dx}{\sqrt {c} d-\sqrt {-a} e}-\frac {e \int \frac {x^2}{e x^n+d}dx}{\sqrt {c} \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )}{2 \sqrt {-a}}-\frac {\sqrt {c} \left (\frac {\sqrt {c} \int \frac {x^2}{\sqrt {-a} \sqrt {c}-c x^n}dx}{\sqrt {-a} e+\sqrt {c} d}+\frac {e \int \frac {x^2}{e x^n+d}dx}{\sqrt {c} \left (\sqrt {-a} e+\sqrt {c} d\right )}\right )}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 888

\(\displaystyle -\frac {\sqrt {c} \left (\frac {x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {\sqrt {c} x^n}{\sqrt {-a}}\right )}{3 \sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {e x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {e x^n}{d}\right )}{3 \sqrt {c} d \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )}{2 \sqrt {-a}}-\frac {\sqrt {c} \left (\frac {x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},\frac {\sqrt {c} x^n}{\sqrt {-a}}\right )}{3 \sqrt {-a} \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {e x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {e x^n}{d}\right )}{3 \sqrt {c} d \left (\sqrt {-a} e+\sqrt {c} d\right )}\right )}{2 \sqrt {-a}}\)

Input:

Int[x^2/((d + e*x^n)*(a + c*x^(2*n))),x]
 

Output:

-1/2*(Sqrt[c]*((x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((Sqrt[c]*x^n)/S 
qrt[-a])])/(3*Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)) - (e*x^3*Hypergeometric2F 
1[1, 3/n, (3 + n)/n, -((e*x^n)/d)])/(3*Sqrt[c]*d*(Sqrt[c]*d - Sqrt[-a]*e)) 
))/Sqrt[-a] - (Sqrt[c]*((x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (Sqrt[c] 
*x^n)/Sqrt[-a]])/(3*Sqrt[-a]*(Sqrt[c]*d + Sqrt[-a]*e)) + (e*x^3*Hypergeome 
tric2F1[1, 3/n, (3 + n)/n, -((e*x^n)/d)])/(3*Sqrt[c]*d*(Sqrt[c]*d + Sqrt[- 
a]*e))))/(2*Sqrt[-a])
 

Defintions of rubi rules used

rule 888

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])
 

rule 1010

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), 
 x_Symbol] :> Simp[b/(b*c - a*d)   Int[(e*x)^m/(a + b*x^n), x], x] - Simp[d 
/(b*c - a*d)   Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, 
m}, x] && NeQ[b*c - a*d, 0]
 

rule 1881

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^( 
n2_.)), x_Symbol] :> With[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(f*x)^m* 
((d + e*x^n)^q/(r - c*x^n)), x], x] - Simp[c/(2*r)   Int[(f*x)^m*((d + e*x^ 
n)^q/(r + c*x^n)), x], x]] /; FreeQ[{a, c, d, e, f, m, n, q}, x] && EqQ[n2, 
 2*n] &&  !RationalQ[n]
 

Maple [F]

\[\int \frac {x^{2}}{\left (d +e \,x^{n}\right ) \left (a +c \,x^{2 n}\right )}d x\]

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)
 

Fricas [F]

\[ \int \frac {x^2}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\int { \frac {x^{2}}{{\left (c x^{2 \, n} + a\right )} {\left (e x^{n} + d\right )}} \,d x } \] Input:

integrate(x^2/(d+e*x^n)/(a+c*x^(2*n)),x, algorithm="fricas")
 

Output:

integral(x^2/(a*e*x^n + a*d + (c*e*x^n + c*d)*x^(2*n)), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {x^2}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**2/(d+e*x**n)/(a+c*x**(2*n)),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {x^2}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\int { \frac {x^{2}}{{\left (c x^{2 \, n} + a\right )} {\left (e x^{n} + d\right )}} \,d x } \] Input:

integrate(x^2/(d+e*x^n)/(a+c*x^(2*n)),x, algorithm="maxima")
 

Output:

integrate(x^2/((c*x^(2*n) + a)*(e*x^n + d)), x)
 

Giac [F]

\[ \int \frac {x^2}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\int { \frac {x^{2}}{{\left (c x^{2 \, n} + a\right )} {\left (e x^{n} + d\right )}} \,d x } \] Input:

integrate(x^2/(d+e*x^n)/(a+c*x^(2*n)),x, algorithm="giac")
 

Output:

integrate(x^2/((c*x^(2*n) + a)*(e*x^n + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\int \frac {x^2}{\left (a+c\,x^{2\,n}\right )\,\left (d+e\,x^n\right )} \,d x \] Input:

int(x^2/((a + c*x^(2*n))*(d + e*x^n)),x)
 

Output:

int(x^2/((a + c*x^(2*n))*(d + e*x^n)), x)
 

Reduce [F]

\[ \int \frac {x^2}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )} \, dx=\int \frac {x^{2}}{x^{3 n} c e +x^{2 n} c d +x^{n} a e +a d}d x \] Input:

int(x^2/(d+e*x^n)/(a+c*x^(2*n)),x)
 

Output:

int(x**2/(x**(3*n)*c*e + x**(2*n)*c*d + x**n*a*e + a*d),x)