Integrand size = 24, antiderivative size = 77 \[ \int \frac {x^m}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\frac {x^{1+m} \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1+m}{n},\frac {1}{2},1,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c (1+m) \sqrt {a+b x^n}} \] Output:
x^(1+m)*(1+b*x^n/a)^(1/2)*AppellF1((1+m)/n,1/2,1,(1+m+n)/n,-b*x^n/a,-d*x^n /c)/c/(1+m)/(a+b*x^n)^(1/2)
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {x^m}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\frac {x^{1+m} \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1+m}{n},\frac {1}{2},1,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{(c+c m) \sqrt {a+b x^n}} \] Input:
Integrate[x^m/(Sqrt[a + b*x^n]*(c + d*x^n)),x]
Output:
(x^(1 + m)*Sqrt[1 + (b*x^n)/a]*AppellF1[(1 + m)/n, 1/2, 1, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)])/((c + c*m)*Sqrt[a + b*x^n])
Time = 0.39 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1013, 1012}
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.
\(\displaystyle \int \frac {x^m}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt {\frac {b x^n}{a}+1} \int \frac {x^m}{\sqrt {\frac {b x^n}{a}+1} \left (d x^n+c\right )}dx}{\sqrt {a+b x^n}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {x^{m+1} \sqrt {\frac {b x^n}{a}+1} \operatorname {AppellF1}\left (\frac {m+1}{n},\frac {1}{2},1,\frac {m+n+1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c (m+1) \sqrt {a+b x^n}}\) |
Input:
Int[x^m/(Sqrt[a + b*x^n]*(c + d*x^n)),x]
Output:
(x^(1 + m)*Sqrt[1 + (b*x^n)/a]*AppellF1[(1 + m)/n, 1/2, 1, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(c*(1 + m)*Sqrt[a + b*x^n])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
\[\int \frac {x^{m}}{\sqrt {a +b \,x^{n}}\, \left (c +d \,x^{n}\right )}d x\]
Input:
int(x^m/(a+b*x^n)^(1/2)/(c+d*x^n),x)
Output:
int(x^m/(a+b*x^n)^(1/2)/(c+d*x^n),x)
\[ \int \frac {x^m}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int { \frac {x^{m}}{\sqrt {b x^{n} + a} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate(x^m/(a+b*x^n)^(1/2)/(c+d*x^n),x, algorithm="fricas")
Output:
integral(sqrt(b*x^n + a)*x^m/(b*d*x^(2*n) + a*c + (b*c + a*d)*x^n), x)
\[ \int \frac {x^m}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int \frac {x^{m}}{\sqrt {a + b x^{n}} \left (c + d x^{n}\right )}\, dx \] Input:
integrate(x**m/(a+b*x**n)**(1/2)/(c+d*x**n),x)
Output:
Integral(x**m/(sqrt(a + b*x**n)*(c + d*x**n)), x)
\[ \int \frac {x^m}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int { \frac {x^{m}}{\sqrt {b x^{n} + a} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate(x^m/(a+b*x^n)^(1/2)/(c+d*x^n),x, algorithm="maxima")
Output:
integrate(x^m/(sqrt(b*x^n + a)*(d*x^n + c)), x)
\[ \int \frac {x^m}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int { \frac {x^{m}}{\sqrt {b x^{n} + a} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate(x^m/(a+b*x^n)^(1/2)/(c+d*x^n),x, algorithm="giac")
Output:
integrate(x^m/(sqrt(b*x^n + a)*(d*x^n + c)), x)
Timed out. \[ \int \frac {x^m}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int \frac {x^m}{\sqrt {a+b\,x^n}\,\left (c+d\,x^n\right )} \,d x \] Input:
int(x^m/((a + b*x^n)^(1/2)*(c + d*x^n)),x)
Output:
int(x^m/((a + b*x^n)^(1/2)*(c + d*x^n)), x)
\[ \int \frac {x^m}{\sqrt {a+b x^n} \left (c+d x^n\right )} \, dx=\int \frac {x^{m} \sqrt {x^{n} b +a}}{x^{2 n} b d +x^{n} a d +x^{n} b c +a c}d x \] Input:
int(x^m/(a+b*x^n)^(1/2)/(c+d*x^n),x)
Output:
int((x**m*sqrt(x**n*b + a))/(x**(2*n)*b*d + x**n*a*d + x**n*b*c + a*c),x)