\(\int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^m \, dx\) [166]

Optimal result

Integrand size = 26, antiderivative size = 230 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^m \, dx=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^{1+m} \operatorname {AppellF1}\left (-2 (1+m),-\frac {1}{2},-\frac {1}{2},-1-2 m,-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {-4 a c+b^2 d}\right )},-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}+\sqrt {-4 a c+b^2 d}\right )}\right )}{(1+m) \sqrt {1+\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {-4 a c+b^2 d}\right )}} \sqrt {1+\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}+\sqrt {-4 a c+b^2 d}\right )}}} \] Output:

(a+b*(d/x)^(1/2)+c/x)^(1/2)*x^(1+m)*AppellF1(-2-2*m,-1/2,-1/2,-1-2*m,-2*c* 
(d/x)^(1/2)/d^(1/2)/(b*d^(1/2)-(b^2*d-4*a*c)^(1/2)),-2*c*(d/x)^(1/2)/d^(1/ 
2)/(b*d^(1/2)+(b^2*d-4*a*c)^(1/2)))/(1+m)/(1+2*c*(d/x)^(1/2)/d^(1/2)/(b*d^ 
(1/2)-(b^2*d-4*a*c)^(1/2)))^(1/2)/(1+2*c*(d/x)^(1/2)/d^(1/2)/(b*d^(1/2)+(b 
^2*d-4*a*c)^(1/2)))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.90 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2067, 1715, 1179, 150}

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.

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

\(\Big \downarrow \) 2067

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

\(\Big \downarrow \) 1715

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

\(\Big \downarrow \) 1179

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {x^{m+1} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} \operatorname {AppellF1}\left (-2 (m+1),-\frac {1}{2},-\frac {1}{2},-2 m-1,-\frac {2 c \sqrt {d}}{\left (b \sqrt {d}-\sqrt {b^2 d-4 a c}\right ) x},-\frac {2 c \sqrt {d}}{\left (\sqrt {d} b+\sqrt {b^2 d-4 a c}\right ) x}\right )}{(m+1) \sqrt {\frac {2 c \sqrt {d}}{x \left (b \sqrt {d}-\sqrt {b^2 d-4 a c}\right )}+1} \sqrt {\frac {2 c \sqrt {d}}{x \left (\sqrt {b^2 d-4 a c}+b \sqrt {d}\right )}+1}}\)

Input:

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

Output:

(Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^(1 + m)*AppellF1[-2*(1 + m), -1/2, -1 
/2, -1 - 2*m, (-2*c*Sqrt[d])/((b*Sqrt[d] - Sqrt[-4*a*c + b^2*d])*x), (-2*c 
*Sqrt[d])/((b*Sqrt[d] + Sqrt[-4*a*c + b^2*d])*x)])/((1 + m)*Sqrt[1 + (2*c* 
Sqrt[d])/((b*Sqrt[d] - Sqrt[-4*a*c + b^2*d])*x)]*Sqrt[1 + (2*c*Sqrt[d])/(( 
b*Sqrt[d] + Sqrt[-4*a*c + b^2*d])*x)])
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1179

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( 
d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) 
^p)   Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d 
- e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m 
, p}, x]
 

rule 1715

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] 
 :> With[{k = Denominator[n]}, Simp[k   Subst[Int[x^(k*(m + 1) - 1)*(a + b* 
x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, m, p}, x] 
 && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
 

rule 2067

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^ 
(p_), x_Symbol] :> Simp[(-d)*(e*x)^m*(d/x)^m   Subst[Int[(a + b*x^n + (c/d^ 
(2*n))*x^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, e, n, p} 
, x] && EqQ[n2, -2*n] &&  !IntegerQ[m] && IntegerQ[2*n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^m \, dx=\int x^m\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^m \, dx=\int \frac {x^{m} \sqrt {\sqrt {d}\, b x +\sqrt {x}\, a x +\sqrt {x}\, c}}{x^{\frac {3}{4}}}d x \] Input:

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

Output:

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