Integrand size = 21, antiderivative size = 61 \[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\frac {x \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c \sqrt {a+b x^n}} \] Output:
x*(1+b*x^n/a)^(1/2)*AppellF1(1/n,1/2,1,1+1/n,-b*x^n/a,-d*x^n/c)/c/(a+b*x^n )^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(181\) vs. \(2(61)=122\).
Time = 0.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.97 \[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=-\frac {2 a c (1+n) x \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (2 a d n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b c n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {3}{2},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-2 a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )} \] Input:
Integrate[1/(Sqrt[a + b*x^n]*(c + d*x^n)),x]
Output:
(-2*a*c*(1 + n)*x*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -((d* x^n)/c)])/(Sqrt[a + b*x^n]*(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)*Appe llF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]))
Time = 0.33 (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.
\(\displaystyle \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt {\frac {b x^n}{a}+1} \int \frac {1}{\sqrt {\frac {b x^n}{a}+1} \left (d x^n+c\right )}dx}{\sqrt {a+b x^n}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {x \sqrt {\frac {b x^n}{a}+1} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c \sqrt {a+b x^n}}\) |
Input:
Int[1/(Sqrt[a + b*x^n]*(c + d*x^n)),x]
Output:
(x*Sqrt[1 + (b*x^n)/a]*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/(c*Sqrt[a + b*x^n])
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])
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])
\[\int \frac {1}{\sqrt {a +b \,x^{n}}\, \left (c +d \,x^{n}\right )}d x\]
Input:
int(1/(a+b*x^n)^(1/2)/(c+d*x^n),x)
Output:
int(1/(a+b*x^n)^(1/2)/(c+d*x^n),x)
\[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int { \frac {1}{\sqrt {b x^{n} + a} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate(1/(a+b*x^n)^(1/2)/(c+d*x^n),x, algorithm="fricas")
Output:
integral(sqrt(b*x^n + a)/(b*d*x^(2*n) + a*c + (b*c + a*d)*x^n), x)
\[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int \frac {1}{\sqrt {a + b x^{n}} \left (c + d x^{n}\right )}\, dx \] Input:
integrate(1/(a+b*x**n)**(1/2)/(c+d*x**n),x)
Output:
Integral(1/(sqrt(a + b*x**n)*(c + d*x**n)), x)
\[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int { \frac {1}{\sqrt {b x^{n} + a} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate(1/(a+b*x^n)^(1/2)/(c+d*x^n),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x^n + a)*(d*x^n + c)), x)
\[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int { \frac {1}{\sqrt {b x^{n} + a} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate(1/(a+b*x^n)^(1/2)/(c+d*x^n),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x^n + a)*(d*x^n + c)), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int \frac {1}{\sqrt {a+b\,x^n}\,\left (c+d\,x^n\right )} \,d x \] Input:
int(1/((a + b*x^n)^(1/2)*(c + d*x^n)),x)
Output:
int(1/((a + b*x^n)^(1/2)*(c + d*x^n)), x)
\[ \int \frac {1}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int \frac {\sqrt {x^{n} b +a}}{x^{2 n} b d +x^{n} a d +x^{n} b c +a c}d x \] Input:
int(1/(a+b*x^n)^(1/2)/(c+d*x^n),x)
Output:
int(sqrt(x**n*b + a)/(x**(2*n)*b*d + x**n*a*d + x**n*b*c + a*c),x)