\(\int \frac {(d x)^m}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx\) [98]

Optimal result

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

(1+b*(c/x)^(1/2)/a)^(1/2)*(d*x)^(1+m)*hypergeom([1/2, -2-2*m],[-1-2*m],-b* 
(c/x)^(1/2)/a)/d/(1+m)/(a+b*(c/x)^(1/2))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.50 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.17 \[ \int \frac {(d x)^m}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\frac {4 b^2 c \left (1-\frac {a}{a+b \sqrt {\frac {c}{x}}}\right )^{2 m} (d x)^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2}+2 m,3+2 m,\frac {7}{2}+2 m,\frac {a}{a+b \sqrt {\frac {c}{x}}}\right )}{(5+4 m) \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}} \] Input:

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

Output:

(4*b^2*c*(1 - a/(a + b*Sqrt[c/x]))^(2*m)*(d*x)^m*Hypergeometric2F1[5/2 + 2 
*m, 3 + 2*m, 7/2 + 2*m, a/(a + b*Sqrt[c/x])])/((5 + 4*m)*(a + b*Sqrt[c/x]) 
^(5/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {893, 866, 864, 862, 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.

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

\(\Big \downarrow \) 893

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

\(\Big \downarrow \) 866

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

\(\Big \downarrow \) 864

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

\(\Big \downarrow \) 862

\(\displaystyle -2 x^{-m} \left (\frac {\sqrt {c}}{x \sqrt {\frac {c}{x}}}\right )^{2 m} \left (\frac {x \sqrt {\frac {c}{x}}}{\sqrt {c}}\right )^{2 m} (d x)^m \int \frac {\left (\frac {\sqrt {c}}{\sqrt {\frac {c}{x}} x}\right )^{-2 m-3}}{\sqrt {a+b \sqrt {\frac {c}{x}} x}}d\frac {\sqrt {c}}{\sqrt {\frac {c}{x}} x}\)

\(\Big \downarrow \) 77

\(\displaystyle \frac {2 b^3 c^{3/2} x^{-m} \left (\frac {x \sqrt {\frac {c}{x}}}{\sqrt {c}}\right )^{2 m} (d x)^m \left (-\frac {b x \sqrt {\frac {c}{x}}}{a}\right )^{2 m} \int \frac {\left (-\frac {b \sqrt {\frac {c}{x}} x}{a}\right )^{-2 m-3}}{\sqrt {a+b \sqrt {\frac {c}{x}} x}}d\frac {\sqrt {c}}{\sqrt {\frac {c}{x}} x}}{a^3}\)

\(\Big \downarrow \) 75

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

Input:

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

Output:

(4*b^2*c*(d*x)^m*(-((b*Sqrt[c/x]*x)/a))^(2*m)*((Sqrt[c/x]*x)/Sqrt[c])^(2*m 
)*Sqrt[a + b*Sqrt[c/x]*x]*Hypergeometric2F1[1/2, 3 + 2*m, 3/2, 1 + (b*Sqrt 
[c/x]*x)/a])/(a^3*x^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 862

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-c^ 
(-1))*(c*x)^(m + 1)*(1/x)^(m + 1)   Subst[Int[(a + b/x^n)^p/x^(m + 2), x], 
x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m]
 

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]

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

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

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

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

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

Output:

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

Giac [F]

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

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

Output:

sage0*x
 

Mupad [F(-1)]

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

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

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

Output:

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