\(\int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx\) [155]

Optimal result

Integrand size = 26, antiderivative size = 233 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=-\frac {b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{64 c^4}+\frac {\left (32 a c-35 b^2 d+42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{120 c^3}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{5 c x}-\frac {b \sqrt {d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \text {arctanh}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{128 c^{9/2}} \] Output:

-1/64*b*(-7*b^2*d+12*a*c)*(b*d+2*c*(d/x)^(1/2))*(a+b*(d/x)^(1/2)+c/x)^(1/2 
)/c^4+1/120*(32*a*c-35*b^2*d+42*b*c*(d/x)^(1/2))*(a+b*(d/x)^(1/2)+c/x)^(3/ 
2)/c^3-2/5*(a+b*(d/x)^(1/2)+c/x)^(3/2)/c/x-1/128*b*d^(1/2)*(-7*b^2*d+12*a* 
c)*(-b^2*d+4*a*c)*arctanh(1/2*(b*d+2*c*(d/x)^(1/2))/c^(1/2)/d^(1/2)/(a+b*( 
d/x)^(1/2)+c/x)^(1/2))/c^(9/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {2 \sqrt {c} \left (-384 c^4-16 c^3 \left (8 a+3 b \sqrt {\frac {d}{x}}\right ) x+105 b^4 d^2 x^2-10 b^2 c d \left (46 a+7 b \sqrt {\frac {d}{x}}\right ) x^2+8 c^2 x \left (7 b^2 d+32 a^2 x+29 a b \sqrt {\frac {d}{x}} x\right )\right )}{x^2}+\frac {15 b d \left (48 a^2 c^2-40 a b^2 c d+7 b^4 d^2\right ) \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{1920 c^{9/2}} \] Input:

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]
 

Output:

(Sqrt[a + b*Sqrt[d/x] + c/x]*((2*Sqrt[c]*(-384*c^4 - 16*c^3*(8*a + 3*b*Sqr 
t[d/x])*x + 105*b^4*d^2*x^2 - 10*b^2*c*d*(46*a + 7*b*Sqrt[d/x])*x^2 + 8*c^ 
2*x*(7*b^2*d + 32*a^2*x + 29*a*b*Sqrt[d/x]*x)))/x^2 + (15*b*d*(48*a^2*c^2 
- 40*a*b^2*c*d + 7*b^4*d^2)*Log[b*d + 2*c*Sqrt[d/x] - 2*Sqrt[c]*Sqrt[(d*(c 
 + a*x + b*Sqrt[d/x]*x))/x]])/Sqrt[(d*(c + (a + b*Sqrt[d/x])*x))/x]))/(192 
0*c^(9/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.70 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2066, 1693, 1166, 27, 1225, 1087, 1092, 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.

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

\(\Big \downarrow \) 2066

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

\(\Big \downarrow \) 1693

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

\(\Big \downarrow \) 1166

\(\displaystyle -\frac {2 \left (\frac {d \int -\frac {1}{2} \sqrt {a+\frac {b d}{x}+\frac {c d}{x^2}} \left (4 a+\frac {7 b d}{x}\right ) \sqrt {\frac {d}{x}}d\sqrt {\frac {d}{x}}}{5 c}+\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \int \sqrt {a+\frac {b d}{x}+\frac {c d}{x^2}} \left (4 a+\frac {7 b d}{x}\right ) \sqrt {\frac {d}{x}}d\sqrt {\frac {d}{x}}}{10 c}\right )}{d^2}\)

\(\Big \downarrow \) 1225

\(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \left (-\frac {5 b d \left (12 a c-7 b^2 d\right ) \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}d\sqrt {\frac {d}{x}}}{16 c^2}-\frac {d \left (d \left (35 b^2-\frac {32 a c}{d}\right )-42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}\right )}{d^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \left (-\frac {5 b d \left (12 a c-7 b^2 d\right ) \left (\frac {\left (4 a c-b^2 d\right ) \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 c}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{16 c^2}-\frac {d \left (d \left (35 b^2-\frac {32 a c}{d}\right )-42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}\right )}{d^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \left (-\frac {5 b d \left (12 a c-7 b^2 d\right ) \left (\frac {\left (4 a c-b^2 d\right ) \int \frac {1}{\frac {4 c}{d}-\frac {d^2}{x^2}}d\frac {2 \sqrt {\frac {d}{x}} c+b d}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{4 c}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{16 c^2}-\frac {d \left (d \left (35 b^2-\frac {32 a c}{d}\right )-42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}\right )}{d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \left (-\frac {5 b d \left (12 a c-7 b^2 d\right ) \left (\frac {\sqrt {d} \left (4 a c-b^2 d\right ) \text {arctanh}\left (\frac {d^{3/2}}{2 \sqrt {c} x}\right )}{8 c^{3/2}}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{16 c^2}-\frac {d \left (d \left (35 b^2-\frac {32 a c}{d}\right )-42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}\right )}{d^2}\)

Input:

Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]
 

Output:

(-2*((d^3*(a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2))/(5*c*x^2) - (d*(-1/24*(d*(( 
35*b^2 - (32*a*c)/d)*d - 42*b*c*Sqrt[d/x])*(a + b*Sqrt[d/x] + (c*d)/x^2)^( 
3/2))/c^2 - (5*b*d*(12*a*c - 7*b^2*d)*(((b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*S 
qrt[d/x] + (c*d)/x^2])/(4*c) + (Sqrt[d]*(4*a*c - b^2*d)*ArcTanh[d^(3/2)/(2 
*Sqrt[c]*x)])/(8*c^(3/2))))/(16*c^2)))/(10*c)))/d^2
 

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1166

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)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1225

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(614\) vs. \(2(193)=386\).

Time = 0.09 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.64

method	result	size
default	\(-\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \left (105 \sqrt {c}\, \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{5} b^{5}-210 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{5} b^{5}-600 a \,c^{\frac {3}{2}} \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{4} b^{3}+780 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{4} b^{3} c -420 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{3} b^{3} c +720 a^{2} c^{\frac {5}{2}} \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \sqrt {\frac {d}{x}}\, x^{3} b +210 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d^{2} x^{2} b^{4}-210 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a \,d^{2} x^{3} b^{4}-720 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{2} \sqrt {\frac {d}{x}}\, x^{3} b \,c^{2}+720 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a \sqrt {\frac {d}{x}}\, x^{2} b \,c^{2}-360 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a d \,x^{2} b^{2} c +360 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{2} d \,x^{3} b^{2} c +560 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d x \,b^{2} c^{2}-672 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, x b \,c^{3}-512 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a \,c^{3} x +768 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} c^{4}\right )}{1920 x^{2} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{5}}\)	\(615\)

Input:

int((a+b*(d/x)^(1/2)+c/x)^(1/2)/x^3,x,method=_RETURNVERBOSE)
 

Output:

-1/1920*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*(105*c^(1/2)*ln((2*c+b*(d/x)^(1/ 
2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*(d/x)^(5/2)*x^5*b^5 
-210*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(5/2)*x^5*b^5-600*a*c^(3/2)*ln((2 
*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*(d/x) 
^(3/2)*x^4*b^3+780*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a*(d/x)^(3/2)*x^4*b^3*c-4 
20*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(3/2)*x^3*b^3*c+720*a^2*c^(5/2)*ln( 
(2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*(d/ 
x)^(1/2)*x^3*b+210*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d^2*x^2*b^4-210*(b*(d/x)^ 
(1/2)*x+a*x+c)^(1/2)*a*d^2*x^3*b^4-720*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^2*( 
d/x)^(1/2)*x^3*b*c^2+720*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a*(d/x)^(1/2)*x^2*b 
*c^2-360*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a*d*x^2*b^2*c+360*(b*(d/x)^(1/2)*x+ 
a*x+c)^(1/2)*a^2*d*x^3*b^2*c+560*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x*b^2*c^2 
-672*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x*b*c^3-512*(b*(d/x)^(1/2)* 
x+a*x+c)^(3/2)*a*c^3*x+768*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c^4)/x^2/(b*(d/x) 
^(1/2)*x+a*x+c)^(1/2)/c^5
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\text {Timed out} \] Input:

integrate((a+b*(d/x)^(1/2)+c/x)^(1/2)/x^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((a+b*(d/x)**(1/2)+c/x)**(1/2)/x**3,x)
 

Output:

Integral(sqrt(a + b*sqrt(d/x) + c/x)/x**3, x)
 

Maxima [F]

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

integrate((a+b*(d/x)^(1/2)+c/x)^(1/2)/x^3,x, algorithm="maxima")
 

Output:

integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^3, x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\text {Timed out} \] Input:

integrate((a+b*(d/x)^(1/2)+c/x)^(1/2)/x^3,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int((a + c/x + b*(d/x)^(1/2))^(1/2)/x^3,x)
                                                                                    
                                                                                    
 

Output:

int((a + c/x + b*(d/x)^(1/2))^(1/2)/x^3, x)
 

Reduce [B] (verification not implemented)

Time = 1.50 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\frac {464 \sqrt {x}\, \sqrt {d}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, a b \,c^{3} x -140 \sqrt {x}\, \sqrt {d}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, b^{3} c^{2} d x -96 \sqrt {x}\, \sqrt {d}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, b \,c^{4}+512 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, a^{2} c^{3} x^{2}-920 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, a \,b^{2} c^{2} d \,x^{2}-256 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, a \,c^{4} x +210 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, b^{4} c \,d^{2} x^{2}+112 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, b^{2} c^{3} d x -768 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, c^{5}+720 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}-\sqrt {x}\, \sqrt {d}\, b -2 c \right ) a^{2} b \,c^{2} x^{2}-600 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}-\sqrt {x}\, \sqrt {d}\, b -2 c \right ) a \,b^{3} c d \,x^{2}+105 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}-\sqrt {x}\, \sqrt {d}\, b -2 c \right ) b^{5} d^{2} x^{2}-720 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\sqrt {x}\right ) a^{2} b \,c^{2} x^{2}+600 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\sqrt {x}\right ) a \,b^{3} c d \,x^{2}-105 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\sqrt {x}\right ) b^{5} d^{2} x^{2}}{1920 \sqrt {x}\, c^{5} x^{2}} \] Input:

int((a+b*(d/x)^(1/2)+c/x)^(1/2)/x^3,x)
 

Output:

(464*sqrt(x)*sqrt(d)*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*a*b*c**3*x - 140*sq 
rt(x)*sqrt(d)*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*b**3*c**2*d*x - 96*sqrt(x) 
*sqrt(d)*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*b*c**4 + 512*sqrt(sqrt(x)*sqrt( 
d)*b + a*x + c)*a**2*c**3*x**2 - 920*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*a*b 
**2*c**2*d*x**2 - 256*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*a*c**4*x + 210*sqr 
t(sqrt(x)*sqrt(d)*b + a*x + c)*b**4*c*d**2*x**2 + 112*sqrt(sqrt(x)*sqrt(d) 
*b + a*x + c)*b**2*c**3*d*x - 768*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*c**5 + 
 720*sqrt(x)*sqrt(d)*sqrt(c)*log(2*sqrt(c)*sqrt(sqrt(x)*sqrt(d)*b + a*x + 
c) - sqrt(x)*sqrt(d)*b - 2*c)*a**2*b*c**2*x**2 - 600*sqrt(x)*sqrt(d)*sqrt( 
c)*log(2*sqrt(c)*sqrt(sqrt(x)*sqrt(d)*b + a*x + c) - sqrt(x)*sqrt(d)*b - 2 
*c)*a*b**3*c*d*x**2 + 105*sqrt(x)*sqrt(d)*sqrt(c)*log(2*sqrt(c)*sqrt(sqrt( 
x)*sqrt(d)*b + a*x + c) - sqrt(x)*sqrt(d)*b - 2*c)*b**5*d**2*x**2 - 720*sq 
rt(x)*sqrt(d)*sqrt(c)*log(sqrt(x))*a**2*b*c**2*x**2 + 600*sqrt(x)*sqrt(d)* 
sqrt(c)*log(sqrt(x))*a*b**3*c*d*x**2 - 105*sqrt(x)*sqrt(d)*sqrt(c)*log(sqr 
t(x))*b**5*d**2*x**2)/(1920*sqrt(x)*c**5*x**2)