\(\int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx\) [158]

Optimal result

Integrand size = 24, antiderivative size = 248 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{64 a^{9/2}} \] Output:

-7/12*b*d^2*(a+b*(d/x)^(1/2)+c/x)^(1/2)/a^2/(d/x)^(3/2)+5/96*b*d*(-21*b^2* 
d+44*a*c)*(a+b*(d/x)^(1/2)+c/x)^(1/2)/a^4/(d/x)^(1/2)-1/48*(-35*b^2*d+36*a 
*c)*(a+b*(d/x)^(1/2)+c/x)^(1/2)*x/a^3+1/2*(a+b*(d/x)^(1/2)+c/x)^(1/2)*x^2/ 
a+1/64*(35*b^4*d^2-120*a*b^2*c*d+48*a^2*c^2)*arctanh(1/2*(2*a+b*(d/x)^(1/2 
))/a^(1/2)/(a+b*(d/x)^(1/2)+c/x)^(1/2))/a^(9/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.12 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\frac {\sqrt {a} d \left (-105 b^3 d \left (b d+c \sqrt {\frac {d}{x}}\right )+48 a^4 x^2-8 a^3 x \left (3 c+b \sqrt {\frac {d}{x}} x\right )+a^2 \left (-72 c^2+14 b^2 d x+92 b c \sqrt {\frac {d}{x}} x\right )-5 a b \left (-58 b c d-44 c^2 \sqrt {\frac {d}{x}}+7 b^2 d \sqrt {\frac {d}{x}} x\right )\right )-3 \sqrt {d} \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{96 a^{9/2} d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \] Input:

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

Output:

(Sqrt[a]*d*(-105*b^3*d*(b*d + c*Sqrt[d/x]) + 48*a^4*x^2 - 8*a^3*x*(3*c + b 
*Sqrt[d/x]*x) + a^2*(-72*c^2 + 14*b^2*d*x + 92*b*c*Sqrt[d/x]*x) - 5*a*b*(- 
58*b*c*d - 44*c^2*Sqrt[d/x] + 7*b^2*d*Sqrt[d/x]*x)) - 3*Sqrt[d]*(48*a^2*c^ 
2 - 120*a*b^2*c*d + 35*b^4*d^2)*Sqrt[(d*(c + (a + b*Sqrt[d/x])*x))/x]*ArcT 
anh[(Sqrt[c]*Sqrt[d/x] - Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x))/x])/(Sqrt[a]*S 
qrt[d])])/(96*a^(9/2)*d*Sqrt[a + b*Sqrt[d/x] + c/x])
 

Rubi [A] (warning: unable to verify)

Time = 0.93 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {2066, 1693, 1167, 27, 1237, 27, 1237, 27, 1228, 1154, 219}

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.

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

\(\Big \downarrow \) 2066

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

\(\Big \downarrow \) 1693

\(\displaystyle -2 d^2 \int \frac {x^5}{d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}\)

\(\Big \downarrow \) 1167

\(\displaystyle -2 d^2 \left (-\frac {\int \frac {\left (6 \sqrt {\frac {d}{x}} c+7 b d\right ) x^4}{2 d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{4 a}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 d^2 \left (-\frac {\int \frac {\left (6 \sqrt {\frac {d}{x}} c+7 b d\right ) x^4}{d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle -2 d^2 \left (-\frac {-\frac {\int -\frac {\left (-35 d b^2-28 c \sqrt {\frac {d}{x}} b+36 a c\right ) x^3}{2 d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{3 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 d^2 \left (-\frac {\frac {\int \frac {\left (-35 d b^2-28 c \sqrt {\frac {d}{x}} b+36 a c\right ) x^3}{d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {\int \frac {\left (2 c \sqrt {\frac {d}{x}} \left (36 a c-35 b^2 d\right )+5 b d \left (44 a c-21 b^2 d\right )\right ) x^2}{2 d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{2 a}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {\int \frac {\left (2 c \sqrt {\frac {d}{x}} \left (36 a c-35 b^2 d\right )+5 b d \left (44 a c-21 b^2 d\right )\right ) x^2}{d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{4 a d}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\)

\(\Big \downarrow \) 1228

\(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {\frac {3 \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \int \frac {x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{2 a}-\frac {5 b x \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a}}{4 a d}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {-\frac {3 \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \int \frac {1}{4 a-\frac {d^2}{x^2}}d\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{a}-\frac {5 b x \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a}}{4 a d}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle -2 d^2 \left (-\frac {\frac {-\frac {-\frac {3 \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}\right )}{2 a^{3/2}}-\frac {5 b x \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a}}{4 a d}-\frac {x^2 \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{6 a}-\frac {7 b x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 a d^2}}{8 a d}-\frac {x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}\right )\)

Input:

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

Output:

-2*d^2*(-1/4*(Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^4)/(a*d^4) - ((-7*b*Sqrt 
[a + b*Sqrt[d/x] + (c*d)/x^2]*x^3)/(3*a*d^2) + (-1/2*((36*a*c - 35*b^2*d)* 
Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^2)/(a*d^2) - ((-5*b*(44*a*c - 21*b^2*d 
)*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x)/a - (3*(48*a^2*c^2 - 120*a*b^2*c*d 
+ 35*b^4*d^2)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] 
+ (c*d)/x^2])])/(2*a^(3/2)))/(4*a*d))/(6*a))/(8*a*d))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1167

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d 
^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[ 
(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1693

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

rule 2066

Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_), x 
_Symbol] :> Simp[-d^(m + 1)   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, n, p}, x] && EqQ[n2, -2*n 
] && IntegerQ[2*n] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.60

method	result	size
default	\(-\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \left (210 a^{\frac {3}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}+112 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} \sqrt {\frac {d}{x}}\, x^{\frac {3}{2}} b -96 x^{\frac {3}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {9}{2}}-140 d \,a^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {x}\, b^{2}-105 d^{2} \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a \,b^{4}-440 a^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, \sqrt {x}\, b c +144 a^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c \sqrt {x}+360 d \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{2} b^{2} c -144 \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{3} c^{2}\right )}{192 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {11}{2}}}\)	\(398\)

Input:

int(x/(a+b*(d/x)^(1/2)+c/x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/192*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(210*a^(3/2)*(b*(d/x)^(1/ 
2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^(3/2)*b^3+112*(b*(d/x)^(1/2)*x+a*x+c)^(1/2 
)*a^(7/2)*(d/x)^(1/2)*x^(3/2)*b-96*x^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a 
^(9/2)-140*d*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^(1/2)*b^2-105*d^2*ln( 
1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^( 
1/2))/a^(1/2))*a*b^4-440*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2) 
*x^(1/2)*b*c+144*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c*x^(1/2)+360*d*ln( 
1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^( 
1/2))/a^(1/2))*a^2*b^2*c-144*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2 
)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^3*c^2)/(b*(d/x)^(1/2)*x+a 
*x+c)^(1/2)/a^(11/2)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.20 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=-\frac {2 \, \sqrt {a d x + \sqrt {d x} b d + c d} {\left (2 \, \sqrt {d x} {\left (4 \, \sqrt {d x} {\left (\frac {7 \, b d}{a^{2}} - \frac {6 \, \sqrt {d x}}{a}\right )} - \frac {35 \, a b^{2} d^{2} - 36 \, a^{2} c d}{a^{4}}\right )} + \frac {5 \, {\left (21 \, b^{3} d^{3} - 44 \, a b c d^{2}\right )}}{a^{4}}\right )} + \frac {3 \, {\left (35 \, b^{4} d^{4} - 120 \, a b^{2} c d^{3} + 48 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | b d + 2 \, {\left (\sqrt {d x} \sqrt {a} - \sqrt {a d x + \sqrt {d x} b d + c d}\right )} \sqrt {a} \right |}\right )}{a^{\frac {9}{2}}} - \frac {105 \, b^{4} d^{4} \log \left ({\left | b d - 2 \, \sqrt {c d} \sqrt {a} \right |}\right ) - 360 \, a b^{2} c d^{3} \log \left ({\left | b d - 2 \, \sqrt {c d} \sqrt {a} \right |}\right ) + 210 \, \sqrt {c d} \sqrt {a} b^{3} d^{3} + 144 \, a^{2} c^{2} d^{2} \log \left ({\left | b d - 2 \, \sqrt {c d} \sqrt {a} \right |}\right ) - 440 \, \sqrt {c d} a^{\frac {3}{2}} b c d^{2}}{a^{\frac {9}{2}}}}{192 \, d^{2} \mathrm {sgn}\left (x\right )} \] Input:

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

Output:

-1/192*(2*sqrt(a*d*x + sqrt(d*x)*b*d + c*d)*(2*sqrt(d*x)*(4*sqrt(d*x)*(7*b 
*d/a^2 - 6*sqrt(d*x)/a) - (35*a*b^2*d^2 - 36*a^2*c*d)/a^4) + 5*(21*b^3*d^3 
 - 44*a*b*c*d^2)/a^4) + 3*(35*b^4*d^4 - 120*a*b^2*c*d^3 + 48*a^2*c^2*d^2)* 
log(abs(b*d + 2*(sqrt(d*x)*sqrt(a) - sqrt(a*d*x + sqrt(d*x)*b*d + c*d))*sq 
rt(a)))/a^(9/2) - (105*b^4*d^4*log(abs(b*d - 2*sqrt(c*d)*sqrt(a))) - 360*a 
*b^2*c*d^3*log(abs(b*d - 2*sqrt(c*d)*sqrt(a))) + 210*sqrt(c*d)*sqrt(a)*b^3 
*d^3 + 144*a^2*c^2*d^2*log(abs(b*d - 2*sqrt(c*d)*sqrt(a))) - 440*sqrt(c*d) 
*a^(3/2)*b*c*d^2)/a^(9/2))/(d^2*sgn(x))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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