\(\int \frac {(d+e x^n)^q}{x (a+c x^{2 n})} \, dx\) [55]

Optimal result

Integrand size = 24, antiderivative size = 228 \[ \int \frac {\left (d+e x^n\right )^q}{x \left (a+c x^{2 n}\right )} \, dx=\frac {\sqrt {c} \left (d+e x^n\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,1+q,2+q,\frac {\sqrt {c} \left (d+e x^n\right )}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 a \left (\sqrt {c} d-\sqrt {-a} e\right ) n (1+q)}+\frac {\sqrt {c} \left (d+e x^n\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,1+q,2+q,\frac {\sqrt {c} \left (d+e x^n\right )}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 a \left (\sqrt {c} d+\sqrt {-a} e\right ) n (1+q)}-\frac {\left (d+e x^n\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,1+q,2+q,1+\frac {e x^n}{d}\right )}{a d n (1+q)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1803, 615, 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.

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

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x 
)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && 
 EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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