\(\int \frac {1}{(d+e x^n) (a+b x^n+c x^{2 n})} \, dx\) [66]

Optimal result

Integrand size = 26, antiderivative size = 243 \[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx=-\frac {c \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )}-\frac {c \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )}+\frac {e^2 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )} \] Output:

-c*(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)*x*hypergeom([1, 1/n],[1+1/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))/(a*e^2-b*d*e+c*d^ 
2)-c*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*x*hypergeom([1, 1/n],[1+1/n],-2*c 
*x^n/(b+(-4*a*c+b^2)^(1/2)))/(b+(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)+e^ 
2*x*hypergeom([1, 1/n],[1+1/n],-e*x^n/d)/d/(a*e^2-b*d*e+c*d^2)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx=\frac {x \left (-\frac {c \left (e+\frac {-2 c d+b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{b-\sqrt {b^2-4 a c}}-\frac {c \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b+\sqrt {b^2-4 a c}}+\frac {e^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d}\right )}{c d^2+e (-b d+a e)} \] Input:

Integrate[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))),x]
 

Output:

(x*(-((c*(e + (-2*c*d + b*e)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, n^(-1 
), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(b - Sqrt[b^2 - 4*a*c] 
)) - (c*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, n^(-1), 
 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b + Sqrt[b^2 - 4*a*c]) 
+ (e^2*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d))/(c*d^2 
+ e*(-(b*d) + a*e))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1754, 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.

\(\displaystyle \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx\)

\(\Big \downarrow \) 1754

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c x \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {c x \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (\sqrt {b^2-4 a c}+b\right ) \left (a e^2-b d e+c d^2\right )}+\frac {e^2 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))),x]
 

Output:

-((c*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x*Hypergeometric2F1[1, n^(-1), 1 
+ n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 
- 4*a*c])*(c*d^2 - b*d*e + a*e^2))) - (c*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a 
*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 
 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)) + (e^2*x*Hype 
rgeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a* 
e^2))
 

Defintions of rubi rules used

rule 1754

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
)), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), 
 x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 
 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {1}{\left (d +e \,x^{n}\right ) \left (a +b \,x^{n}+c \,x^{2 n}\right )}d x\]

Input:

int(1/(d+e*x^n)/(a+b*x^n+c*x^(2*n)),x)
 

Output:

int(1/(d+e*x^n)/(a+b*x^n+c*x^(2*n)),x)
 

Fricas [F]

\[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (e x^{n} + d\right )}} \,d x } \] Input:

integrate(1/(d+e*x^n)/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")
 

Output:

integral(1/(b*e*x^(2*n) + a*d + (c*e*x^n + c*d)*x^(2*n) + (b*d + a*e)*x^n) 
, x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(1/(d+e*x**n)/(a+b*x**n+c*x**(2*n)),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (e x^{n} + d\right )}} \,d x } \] Input:

integrate(1/(d+e*x^n)/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^(2*n) + b*x^n + a)*(e*x^n + d)), x)
 

Giac [F]

\[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (e x^{n} + d\right )}} \,d x } \] Input:

integrate(1/(d+e*x^n)/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")
 

Output:

integrate(1/((c*x^(2*n) + b*x^n + a)*(e*x^n + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx=\int \frac {1}{\left (d+e\,x^n\right )\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \] Input:

int(1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))),x)
 

Output:

int(1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))), x)
 

Reduce [F]

\[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx=\int \frac {1}{x^{3 n} c e +x^{2 n} b e +x^{2 n} c d +x^{n} a e +x^{n} b d +a d}d x \] Input:

int(1/(d+e*x^n)/(a+b*x^n+c*x^(2*n)),x)
 

Output:

int(1/(x**(3*n)*c*e + x**(2*n)*b*e + x**(2*n)*c*d + x**n*a*e + x**n*b*d + 
a*d),x)