\(\int \frac {1}{\sqrt {a+b x^n} (c+d x^n)^2} \, dx\) [102]

Optimal result

Integrand size = 21, antiderivative size = 61 \[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )^2} \, dx=\frac {x \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},2,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2 \sqrt {a+b x^n}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(391\) vs. \(2(61)=122\).

Time = 0.84 (sec) , antiderivative size = 391, normalized size of antiderivative = 6.41 \[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )^2} \, dx=\frac {x \left (\frac {b d (-2+n) x^n \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{(-b c+a d) (1+n)}-\frac {2 c \left (2 a d^2 n x^n \left (a+b x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b c d n x^n \left (a+b x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {3}{2},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-2 a c (1+n) \left (-b c n+a d n+b d x^n\right ) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{(b c-a d) \left (c+d x^n\right ) \left (2 a d n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b c n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {3}{2},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-2 a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}\right )}{2 c^2 n \sqrt {a+b x^n}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(b*x^n + a)/(b*d^2*x^(3*n) + a*c^2 + (2*b*c*d + a*d^2)*x^(2*n 
) + (b*c^2 + 2*a*c*d)*x^n), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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