\(\int \frac {\sqrt {d+e x^n}}{a+b x^n+c x^{2 n}} \, dx\) [80]

Optimal result

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

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

Mathematica [F]

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

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

Output:

Integrate[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.

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

\(\Big \downarrow \) 1758

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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]

\[\int \frac {\sqrt {d +e \,x^{n}}}{a +b \,x^{n}+c \,x^{2 n}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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