Integrand size = 23, antiderivative size = 82 \[ \int \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^m \, dx=\frac {\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2 (1+m),3+2 m,-\frac {b}{a \sqrt {\frac {c}{x}}}\right )}{d (1+m) \sqrt {1+\frac {b}{a \sqrt {\frac {c}{x}}}}} \] Output:
(a+b/(c/x)^(1/2))^(1/2)*(d*x)^(1+m)*hypergeom([-1/2, 2+2*m],[3+2*m],-b/a/( c/x)^(1/2))/d/(1+m)/(1+b/a/(c/x)^(1/2))^(1/2)
Time = 2.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04 \[ \int \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^m \, dx=\frac {4 \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} x (d x)^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {5}{2}-2 m,-\frac {3}{2}-2 m,-\frac {a \sqrt {\frac {c}{x}}}{b}\right )}{(5+4 m) \sqrt {1+\frac {a \sqrt {\frac {c}{x}}}{b}}} \] Input:
Integrate[Sqrt[a + b/Sqrt[c/x]]*(d*x)^m,x]
Output:
(4*Sqrt[a + b/Sqrt[c/x]]*x*(d*x)^m*Hypergeometric2F1[-1/2, -5/2 - 2*m, -3/ 2 - 2*m, -((a*Sqrt[c/x])/b)])/((5 + 4*m)*Sqrt[1 + (a*Sqrt[c/x])/b])
Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.40, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {893, 866, 864, 77, 75}
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.
\(\displaystyle \int (d x)^m \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} \, dx\) |
\(\Big \downarrow \) 893 |
\(\displaystyle \int (d x)^m \sqrt {a+\frac {b x \sqrt {\frac {c}{x}}}{c}}dx\) |
\(\Big \downarrow \) 866 |
\(\displaystyle x^{-m} (d x)^m \int x^m \sqrt {a+\frac {b \sqrt {\frac {c}{x}} x}{c}}dx\) |
\(\Big \downarrow \) 864 |
\(\displaystyle 2 x^{-m} (d x)^m \int \left (\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )^{2 m+1} \sqrt {a+\frac {b \sqrt {\frac {c}{x}} x}{c}}d\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\) |
\(\Big \downarrow \) 77 |
\(\displaystyle -\frac {2 a \sqrt {c} x^{-m} \left (\frac {x \sqrt {\frac {c}{x}}}{\sqrt {c}}\right )^{2 m} (d x)^m \left (-\frac {b x \sqrt {\frac {c}{x}}}{a c}\right )^{-2 m} \int \left (-\frac {b \sqrt {\frac {c}{x}} x}{a c}\right )^{2 m+1} \sqrt {a+\frac {b \sqrt {\frac {c}{x}} x}{c}}d\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}}{b}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {4 a c x^{-m} \left (\frac {x \sqrt {\frac {c}{x}}}{\sqrt {c}}\right )^{2 m} (d x)^m \left (a+\frac {b x \sqrt {\frac {c}{x}}}{c}\right )^{3/2} \left (-\frac {b x \sqrt {\frac {c}{x}}}{a c}\right )^{-2 m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-2 m-1,\frac {5}{2},\frac {b \sqrt {\frac {c}{x}} x}{a c}+1\right )}{3 b^2}\) |
Input:
Int[Sqrt[a + b/Sqrt[c/x]]*(d*x)^m,x]
Output:
(-4*a*c*(d*x)^m*((Sqrt[c/x]*x)/Sqrt[c])^(2*m)*(a + (b*Sqrt[c/x]*x)/c)^(3/2 )*Hypergeometric2F1[3/2, -1 - 2*m, 5/2, 1 + (b*Sqrt[c/x]*x)/(a*c)])/(3*b^2 *x^m*(-((b*Sqrt[c/x]*x)/(a*c)))^(2*m))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((-b)*(c/ d))^IntPart[m]*((b*x)^FracPart[m]/((-d)*(x/c))^FracPart[m]) Int[((-d)*(x/ c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0] && !GtQ[-d/(b*c), 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^Int Part[m]*((c*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a + b*x^n)^p, x], x] / ; FreeQ[{a, b, c, m, p}, x] && FractionQ[n]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> With[{k = Denominator[n]}, Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x ], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, d, m, p, q}, x] && FractionQ[n]
\[\int \sqrt {a +\frac {b}{\sqrt {\frac {c}{x}}}}\, \left (d x \right )^{m}d x\]
Input:
int((a+b/(c/x)^(1/2))^(1/2)*(d*x)^m,x)
Output:
int((a+b/(c/x)^(1/2))^(1/2)*(d*x)^m,x)
Exception generated. \[ \int \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^m \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b/(c/x)^(1/2))^(1/2)*(d*x)^m,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: algl ogextint: unimplemented
\[ \int \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^m \, dx=\int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {\frac {c}{x}}}}\, dx \] Input:
integrate((a+b/(c/x)**(1/2))**(1/2)*(d*x)**m,x)
Output:
Integral((d*x)**m*sqrt(a + b/sqrt(c/x)), x)
\[ \int \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^m \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {\frac {c}{x}}}} \,d x } \] Input:
integrate((a+b/(c/x)^(1/2))^(1/2)*(d*x)^m,x, algorithm="maxima")
Output:
integrate((d*x)^m*sqrt(a + b/sqrt(c/x)), x)
\[ \int \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^m \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {\frac {c}{x}}}} \,d x } \] Input:
integrate((a+b/(c/x)^(1/2))^(1/2)*(d*x)^m,x, algorithm="giac")
Output:
integrate((d*x)^m*sqrt(a + b/sqrt(c/x)), x)
Timed out. \[ \int \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^m \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} \,d x \] Input:
int((d*x)^m*(a + b/(c/x)^(1/2))^(1/2),x)
Output:
int((d*x)^m*(a + b/(c/x)^(1/2))^(1/2), x)
\[ \int \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^m \, dx=\frac {4 d^{m} c^{\frac {1}{4}} \left (x^{m +\frac {1}{2}} \sqrt {c}\, \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}\, a b -4 x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}\, a^{2} c m -2 x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}\, a^{2} c +4 x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}\, b^{2} m x +3 x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}\, b^{2} x +4 \left (\int \frac {x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}}{x}d x \right ) a^{2} c \,m^{2}+2 \left (\int \frac {x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}}{x}d x \right ) a^{2} c m \right )}{\sqrt {c}\, b^{2} \left (16 m^{2}+32 m +15\right )} \] Input:
int((a+b/(c/x)^(1/2))^(1/2)*(d*x)^m,x)
Output:
(4*d**m*c**(1/4)*(x**((2*m + 1)/2)*sqrt(c)*sqrt(sqrt(c)*a + sqrt(x)*b)*a*b - 4*x**m*sqrt(sqrt(c)*a + sqrt(x)*b)*a**2*c*m - 2*x**m*sqrt(sqrt(c)*a + s qrt(x)*b)*a**2*c + 4*x**m*sqrt(sqrt(c)*a + sqrt(x)*b)*b**2*m*x + 3*x**m*sq rt(sqrt(c)*a + sqrt(x)*b)*b**2*x + 4*int((x**m*sqrt(sqrt(c)*a + sqrt(x)*b) )/x,x)*a**2*c*m**2 + 2*int((x**m*sqrt(sqrt(c)*a + sqrt(x)*b))/x,x)*a**2*c* m))/(sqrt(c)*b**2*(16*m**2 + 32*m + 15))