\(\int \frac {1}{\sqrt {d+e x^n} (a+c x^{2 n})} \, dx\) [44]

Optimal result

Integrand size = 23, antiderivative size = 144 \[ \int \frac {1}{\sqrt {d+e x^n} \left (a+c x^{2 n}\right )} \, dx=\frac {x \sqrt {1+\frac {e x^n}{d}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {e x^n}{d},-\frac {\sqrt {c} x^n}{\sqrt {-a}}\right )}{2 a \sqrt {d+e x^n}}+\frac {x \sqrt {1+\frac {e x^n}{d}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {e x^n}{d},\frac {\sqrt {c} x^n}{\sqrt {-a}}\right )}{2 a \sqrt {d+e x^n}} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.

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

\(\Big \downarrow \) 1759

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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