\(\int \frac {1}{\sqrt {d+e x^n} (a+b x^n+c x^{2 n})} \, dx\) [81]

Optimal result

Integrand size = 28, antiderivative size = 198 \[ \int \frac {1}{\sqrt {d+e x^n} \left (a+b x^n+c x^{2 n}\right )} \, dx=-\frac {2 c x \sqrt {1+\frac {e x^n}{d}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {e x^n}{d},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) \sqrt {d+e x^n}}-\frac {2 c x \sqrt {1+\frac {e x^n}{d}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {e x^n}{d},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) \sqrt {d+e x^n}} \] Output:

-2*c*x*(1+e*x^n/d)^(1/2)*AppellF1(1/n,1,1/2,1+1/n,-2*c*x^n/(b-(-4*a*c+b^2) 
^(1/2)),-e*x^n/d)/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))/(d+e*x^n)^(1/2)-2*c*x*( 
1+e*x^n/d)^(1/2)*AppellF1(1/n,1,1/2,1+1/n,-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)), 
-e*x^n/d)/(b*(-4*a*c+b^2)^(1/2)-4*a*c+b^2)/(d+e*x^n)^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1758, 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.

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

\(\Big \downarrow \) 1758

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

\(\displaystyle \frac {2 c x \sqrt {\frac {e x^n}{d}+1} \operatorname {AppellF1}\left (\frac {1}{n},1,\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) \sqrt {d+e x^n}}-\frac {2 c x \sqrt {\frac {e x^n}{d}+1} \operatorname {AppellF1}\left (\frac {1}{n},1,\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right ) \sqrt {d+e x^n}}\)

Input:

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

Output:

(2*c*x*Sqrt[1 + (e*x^n)/d]*AppellF1[n^(-1), 1, 1/2, 1 + n^(-1), (-2*c*x^n) 
/(b - Sqrt[b^2 - 4*a*c]), -((e*x^n)/d)])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 
- 4*a*c])*Sqrt[d + e*x^n]) - (2*c*x*Sqrt[1 + (e*x^n)/d]*AppellF1[n^(-1), 1 
, 1/2, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^n)/d)])/(Sqr 
t[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*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 1758

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(e*x^n + d)/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(x**n*e + d)/(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)