\(\int \frac {(d+e x^n)^q}{a+c x^{2 n}} \, dx\) [54]

Optimal result

Integrand size = 21, antiderivative size = 140 \[ \int \frac {\left (d+e x^n\right )^q}{a+c x^{2 n}} \, dx=\frac {x \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},-q,1,1+\frac {1}{n},-\frac {e x^n}{d},-\frac {\sqrt {c} x^n}{\sqrt {-a}}\right )}{2 a}+\frac {x \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},-q,1,1+\frac {1}{n},-\frac {e x^n}{d},\frac {\sqrt {c} x^n}{\sqrt {-a}}\right )}{2 a} \] Output:

1/2*x*(d+e*x^n)^q*AppellF1(1/n,1,-q,1+1/n,-c^(1/2)*x^n/(-a)^(1/2),-e*x^n/d 
)/a/((1+e*x^n/d)^q)+1/2*x*(d+e*x^n)^q*AppellF1(1/n,1,-q,1+1/n,c^(1/2)*x^n/ 
(-a)^(1/2),-e*x^n/d)/a/((1+e*x^n/d)^q)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1759, 937, 936}

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.

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

\(\Big \downarrow \) 1759

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

\(\displaystyle \frac {x \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},1,-q,1+\frac {1}{n},-\frac {\sqrt {c} x^n}{\sqrt {-a}},-\frac {e x^n}{d}\right )}{2 a}+\frac {x \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},1,-q,1+\frac {1}{n},\frac {\sqrt {c} x^n}{\sqrt {-a}},-\frac {e x^n}{d}\right )}{2 a}\)

Input:

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

Output:

(x*(d + e*x^n)^q*AppellF1[n^(-1), 1, -q, 1 + n^(-1), -((Sqrt[c]*x^n)/Sqrt[ 
-a]), -((e*x^n)/d)])/(2*a*(1 + (e*x^n)/d)^q) + (x*(d + e*x^n)^q*AppellF1[n 
^(-1), 1, -q, 1 + n^(-1), (Sqrt[c]*x^n)/Sqrt[-a], -((e*x^n)/d)])/(2*a*(1 + 
 (e*x^n)/d)^q)
 

Defintions of rubi rules used

rule 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 937

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) 
  Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 1759

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> W 
ith[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(d + e*x^n)^q/(r - c*x^n), x], 
 x] - Simp[c/(2*r)   Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a, c, 
 d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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