Integrand size = 19, antiderivative size = 99 \[ \int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx=\frac {x^2 \sqrt {1+\frac {a d x^n}{b+a c}} \operatorname {AppellF1}\left (\frac {2}{n},-\frac {1}{2},\frac {1}{2},\frac {2+n}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )}{2 \sqrt {1+\frac {d x^n}{c}} \sqrt {a+\frac {b}{c+d x^n}}} \] Output:
1/2*x^2*(1+a*d*x^n/(a*c+b))^(1/2)*AppellF1(2/n,-1/2,1/2,(2+n)/n,-d*x^n/c,- a*d*x^n/(a*c+b))/(1+d*x^n/c)^(1/2)/(a+b/(c+d*x^n))^(1/2)
\[ \int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx=\int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx \] Input:
Integrate[x/Sqrt[a + b/(c + d*x^n)],x]
Output:
Integrate[x/Sqrt[a + b/(c + d*x^n)], x]
Time = 0.61 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2057, 2058, 1013, 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.
\(\displaystyle \int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx\) |
\(\Big \downarrow \) 2057 |
\(\displaystyle \int \frac {x}{\sqrt {\frac {a c+a d x^n+b}{c+d x^n}}}dx\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {\sqrt {a c+a d x^n+b} \int \frac {x \sqrt {d x^n+c}}{\sqrt {a d x^n+b+a c}}dx}{\sqrt {c+d x^n} \sqrt {\frac {a c+a d x^n+b}{c+d x^n}}}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt {a c+a d x^n+b} \int \frac {x \sqrt {\frac {d x^n}{c}+1}}{\sqrt {a d x^n+b+a c}}dx}{\sqrt {\frac {d x^n}{c}+1} \sqrt {\frac {a c+a d x^n+b}{c+d x^n}}}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt {\frac {a d x^n}{a c+b}+1} \int \frac {x \sqrt {\frac {d x^n}{c}+1}}{\sqrt {\frac {a d x^n}{b+a c}+1}}dx}{\sqrt {\frac {d x^n}{c}+1} \sqrt {\frac {a c+a d x^n+b}{c+d x^n}}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {x^2 \sqrt {\frac {a d x^n}{a c+b}+1} \operatorname {AppellF1}\left (\frac {2}{n},-\frac {1}{2},\frac {1}{2},\frac {n+2}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )}{2 \sqrt {\frac {d x^n}{c}+1} \sqrt {\frac {a c+a d x^n+b}{c+d x^n}}}\) |
Input:
Int[x/Sqrt[a + b/(c + d*x^n)],x]
Output:
(x^2*Sqrt[1 + (a*d*x^n)/(b + a*c)]*AppellF1[2/n, -1/2, 1/2, (2 + n)/n, -(( d*x^n)/c), -((a*d*x^n)/(b + a*c))])/(2*Sqrt[(b + a*c + a*d*x^n)/(c + d*x^n )]*Sqrt[1 + (d*x^n)/c])
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[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
\[\int \frac {x}{\sqrt {a +\frac {b}{c +d \,x^{n}}}}d x\]
Input:
int(x/(a+b/(c+d*x^n))^(1/2),x)
Output:
int(x/(a+b/(c+d*x^n))^(1/2),x)
Exception generated. \[ \int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x/(a+b/(c+d*x^n))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx=\int \frac {x}{\sqrt {\frac {a c + a d x^{n} + b}{c + d x^{n}}}}\, dx \] Input:
integrate(x/(a+b/(c+d*x**n))**(1/2),x)
Output:
Integral(x/sqrt((a*c + a*d*x**n + b)/(c + d*x**n)), x)
\[ \int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx=\int { \frac {x}{\sqrt {a + \frac {b}{d x^{n} + c}}} \,d x } \] Input:
integrate(x/(a+b/(c+d*x^n))^(1/2),x, algorithm="maxima")
Output:
integrate(x/sqrt(a + b/(d*x^n + c)), x)
\[ \int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx=\int { \frac {x}{\sqrt {a + \frac {b}{d x^{n} + c}}} \,d x } \] Input:
integrate(x/(a+b/(c+d*x^n))^(1/2),x, algorithm="giac")
Output:
integrate(x/sqrt(a + b/(d*x^n + c)), x)
Timed out. \[ \int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx=\int \frac {x}{\sqrt {a+\frac {b}{c+d\,x^n}}} \,d x \] Input:
int(x/(a + b/(c + d*x^n))^(1/2),x)
Output:
int(x/(a + b/(c + d*x^n))^(1/2), x)
\[ \int \frac {x}{\sqrt {a+\frac {b}{c+d x^n}}} \, dx=\int \frac {\sqrt {x^{n} d +c}\, \sqrt {x^{n} a d +a c +b}\, x}{x^{n} a d +a c +b}d x \] Input:
int(x/(a+b/(c+d*x^n))^(1/2),x)
Output:
int((sqrt(x**n*d + c)*sqrt(x**n*a*d + a*c + b)*x)/(x**n*a*d + a*c + b),x)