\(\int \frac {(d x)^m}{\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}}} \, dx\) [99]

Optimal result

Integrand size = 23, antiderivative size = 82 \[ \int \frac {(d x)^m}{\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}}} \, dx=\frac {\sqrt {1+\frac {b}{a \sqrt {\frac {c}{x}}}} (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},2 (1+m),3+2 m,-\frac {b}{a \sqrt {\frac {c}{x}}}\right )}{d (1+m) \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}}} \] Output:

(1+b/a/(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)/(a+b/(c/x)^(1/2))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 2.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.41 \[ \int \frac {(d x)^m}{\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}}} \, dx=\frac {a^2 c \left (\frac {a \sqrt {\frac {c}{x}}}{b+a \sqrt {\frac {c}{x}}}\right )^{-\frac {1}{2}+2 m} (d x)^m \operatorname {Hypergeometric2F1}\left (2+2 m,\frac {5}{2}+2 m,3+2 m,\frac {b}{b+a \sqrt {\frac {c}{x}}}\right )}{(1+m) \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} \left (b+a \sqrt {\frac {c}{x}}\right )^2} \] Input:

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

Output:

(a^2*c*((a*Sqrt[c/x])/(b + a*Sqrt[c/x]))^(-1/2 + 2*m)*(d*x)^m*Hypergeometr 
ic2F1[2 + 2*m, 5/2 + 2*m, 3 + 2*m, b/(b + a*Sqrt[c/x])])/((1 + m)*Sqrt[a + 
 b/Sqrt[c/x]]*(b + a*Sqrt[c/x])^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.38, 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.

\(\displaystyle \int \frac {(d x)^m}{\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}}} \, dx\)

\(\Big \downarrow \) 893

\(\displaystyle \int \frac {(d x)^m}{\sqrt {a+\frac {b x \sqrt {\frac {c}{x}}}{c}}}dx\)

\(\Big \downarrow \) 866

\(\displaystyle x^{-m} (d x)^m \int \frac {x^m}{\sqrt {a+\frac {b \sqrt {\frac {c}{x}} x}{c}}}dx\)

\(\Big \downarrow \) 864

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

\(\Big \downarrow \) 77

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

\(\Big \downarrow \) 75

\(\displaystyle -\frac {4 a c x^{-m} \left (\frac {x \sqrt {\frac {c}{x}}}{\sqrt {c}}\right )^{2 m} (d x)^m \sqrt {a+\frac {b x \sqrt {\frac {c}{x}}}{c}} \left (-\frac {b x \sqrt {\frac {c}{x}}}{a c}\right )^{-2 m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-2 m-1,\frac {3}{2},\frac {b \sqrt {\frac {c}{x}} x}{a c}+1\right )}{b^2}\)

Input:

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

Output:

(-4*a*c*(d*x)^m*((Sqrt[c/x]*x)/Sqrt[c])^(2*m)*Sqrt[a + (b*Sqrt[c/x]*x)/c]* 
Hypergeometric2F1[1/2, -1 - 2*m, 3/2, 1 + (b*Sqrt[c/x]*x)/(a*c)])/(b^2*x^m 
*(-((b*Sqrt[c/x]*x)/(a*c)))^(2*m))
 

Defintions of rubi rules used

rule 75

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])
 

rule 77

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]
 

rule 864

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]
 

rule 866

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]
 

rule 893

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]
 

Maple [F]

\[\int \frac {\left (d x \right )^{m}}{\sqrt {a +\frac {b}{\sqrt {\frac {c}{x}}}}}d x\]

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)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {(d x)^m}{\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   algl 
ogextint: unimplemented
 

Sympy [F]

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

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

Output:

Integral((d*x)**m/sqrt(a + b/sqrt(c/x)), x)
 

Maxima [F]

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

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

Output:

integrate((d*x)^m/sqrt(a + b/sqrt(c/x)), x)
 

Giac [F]

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

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

Output:

integrate((d*x)^m/sqrt(a + b/sqrt(c/x)), x)
 

Mupad [F(-1)]

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

Reduce [F]

\[ \int \frac {(d x)^m}{\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}}} \, dx=\frac {4 d^{m} c^{\frac {1}{4}} \left (x^{m +\frac {1}{2}} \sqrt {c}\, \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}\, b -4 x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}\, a c m -2 x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}\, a c +4 \left (\int \frac {x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}}{x}d x \right ) a c \,m^{2}+2 \left (\int \frac {x^{m} \sqrt {\sqrt {c}\, a +\sqrt {x}\, b}}{x}d x \right ) a c m \right )}{\sqrt {c}\, b^{2} \left (4 m +3\right )} \] Input:

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

Output:

(4*d**m*c**(1/4)*(x**((2*m + 1)/2)*sqrt(c)*sqrt(sqrt(c)*a + sqrt(x)*b)*b - 
 4*x**m*sqrt(sqrt(c)*a + sqrt(x)*b)*a*c*m - 2*x**m*sqrt(sqrt(c)*a + sqrt(x 
)*b)*a*c + 4*int((x**m*sqrt(sqrt(c)*a + sqrt(x)*b))/x,x)*a*c*m**2 + 2*int( 
(x**m*sqrt(sqrt(c)*a + sqrt(x)*b))/x,x)*a*c*m))/(sqrt(c)*b**2*(4*m + 3))