\(\int \frac {1}{(d+e x^n) \sqrt {a+c x^{2 n}}} \, dx\) [46]

Optimal result

Integrand size = 23, antiderivative size = 171 \[ \int \frac {1}{\left (d+e x^n\right ) \sqrt {a+c x^{2 n}}} \, dx=\frac {x \sqrt {1+\frac {c x^{2 n}}{a}} \operatorname {AppellF1}\left (\frac {1}{2 n},\frac {1}{2},1,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d \sqrt {a+c x^{2 n}}}-\frac {e x^{1+n} \sqrt {1+\frac {c x^{2 n}}{a}} \operatorname {AppellF1}\left (\frac {1+n}{2 n},\frac {1}{2},1,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+n) \sqrt {a+c x^{2 n}}} \] Output:

x*(1+c*x^(2*n)/a)^(1/2)*AppellF1(1/2/n,1,1/2,1+1/2/n,e^2*x^(2*n)/d^2,-c*x^ 
(2*n)/a)/d/(a+c*x^(2*n))^(1/2)-e*x^(1+n)*(1+c*x^(2*n)/a)^(1/2)*AppellF1(1/ 
2*(1+n)/n,1,1/2,3/2+1/2/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^2/(1+n)/(a+c*x^( 
2*n))^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1768, 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.

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

\(\Big \downarrow \) 1768

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x \sqrt {\frac {c x^{2 n}}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2 n},\frac {1}{2},1,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d \sqrt {a+c x^{2 n}}}-\frac {e x^{n+1} \sqrt {\frac {c x^{2 n}}{a}+1} \operatorname {AppellF1}\left (\frac {n+1}{2 n},\frac {1}{2},1,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1) \sqrt {a+c x^{2 n}}}\)

Input:

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

Output:

(x*Sqrt[1 + (c*x^(2*n))/a]*AppellF1[1/(2*n), 1/2, 1, (2 + n^(-1))/2, -((c* 
x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d*Sqrt[a + c*x^(2*n)]) - (e*x^(1 + n)*Sq 
rt[1 + (c*x^(2*n))/a]*AppellF1[(1 + n)/(2*n), 1/2, 1, (3 + n^(-1))/2, -((c 
*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + n)*Sqrt[a + c*x^(2*n)])
 

Defintions of rubi rules used

rule 1768

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

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(1/(sqrt(a + c*x**(2*n))*(d + e*x**n)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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