\(\int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx\) [163]

Optimal result

Integrand size = 26, antiderivative size = 289 \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx=-\frac {\left (1024 a^2 c^2-2940 a b^2 c d+945 b^4 d^2+14 b c \left (92 a c-45 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {b \sqrt {d} \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) \text {arctanh}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{128 c^{11/2}} \] Output:

-1/960*(1024*c^2*a^2-2940*a*b^2*c*d+945*b^4*d^2+14*b*c*(-45*b^2*d+92*a*c)* 
(d/x)^(1/2))*(a+b*(d/x)^(1/2)+c/x)^(1/2)/c^5+9/20*b*(a+b*(d/x)^(1/2)+c/x)^ 
(1/2)*(d/x)^(3/2)/c^2/d-2/5*(a+b*(d/x)^(1/2)+c/x)^(1/2)/c/x^2+1/120*(-63*b 
^2*d+64*a*c)*(a+b*(d/x)^(1/2)+c/x)^(1/2)/c^3/x+1/128*b*d^(1/2)*(63*b^4*d^2 
-280*a*b^2*c*d+240*a^2*c^2)*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^(11/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.48 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx=\frac {-2 \sqrt {c} \left (384 c^5-16 c^4 \left (8 a+3 b \sqrt {\frac {d}{x}}\right ) x+945 b^4 d^2 \left (a+b \sqrt {\frac {d}{x}}\right ) x^3-105 b^2 c d x^2 \left (-3 b^2 d+28 a^2 x+34 a b \sqrt {\frac {d}{x}} x\right )+8 c^3 x \left (9 b^2 d+64 a^2 x+43 a b \sqrt {\frac {d}{x}} x\right )+2 c^2 x^2 \left (-574 a b^2 d-63 b^3 d \sqrt {\frac {d}{x}}+512 a^3 x+1156 a^2 b \sqrt {\frac {d}{x}} x\right )\right )-15 b \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) x^3 \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )}{1920 c^{11/2} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3} \] Input:

Integrate[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x^4),x]
 

Output:

(-2*Sqrt[c]*(384*c^5 - 16*c^4*(8*a + 3*b*Sqrt[d/x])*x + 945*b^4*d^2*(a + b 
*Sqrt[d/x])*x^3 - 105*b^2*c*d*x^2*(-3*b^2*d + 28*a^2*x + 34*a*b*Sqrt[d/x]* 
x) + 8*c^3*x*(9*b^2*d + 64*a^2*x + 43*a*b*Sqrt[d/x]*x) + 2*c^2*x^2*(-574*a 
*b^2*d - 63*b^3*d*Sqrt[d/x] + 512*a^3*x + 1156*a^2*b*Sqrt[d/x]*x)) - 15*b* 
(240*a^2*c^2 - 280*a*b^2*c*d + 63*b^4*d^2)*x^3*Sqrt[(d*(c + (a + b*Sqrt[d/ 
x])*x))/x]*Log[b*d + 2*c*Sqrt[d/x] - 2*Sqrt[c]*Sqrt[(d*(c + a*x + b*Sqrt[d 
/x]*x))/x]])/(1920*c^(11/2)*Sqrt[a + b*Sqrt[d/x] + c/x]*x^3)
 

Rubi [A] (warning: unable to verify)

Time = 1.03 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2066, 1693, 1166, 27, 1236, 27, 1236, 27, 1225, 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.

\(\displaystyle \int \frac {1}{x^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx\)

\(\Big \downarrow \) 2066

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

\(\Big \downarrow \) 1693

\(\displaystyle -\frac {2 \int \frac {d^5}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^5}d\sqrt {\frac {d}{x}}}{d^3}\)

\(\Big \downarrow \) 1166

\(\displaystyle -\frac {2 \left (\frac {d \int -\frac {d^3 \left (8 a+9 b \sqrt {\frac {d}{x}}\right )}{2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^3}d\sqrt {\frac {d}{x}}}{5 c}+\frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 c x^4}\right )}{d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 c x^4}-\frac {d \int \frac {d^3 \left (8 a+9 b \sqrt {\frac {d}{x}}\right )}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^3}d\sqrt {\frac {d}{x}}}{10 c}\right )}{d^3}\)

\(\Big \downarrow \) 1236

\(\displaystyle -\frac {2 \left (\frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 c x^4}-\frac {d \left (\frac {d \int -\frac {d \left (54 a b d-\left (64 a c-63 b^2 d\right ) \sqrt {\frac {d}{x}}\right )}{2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^2}d\sqrt {\frac {d}{x}}}{4 c}+\frac {9 b d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c x^3}\right )}{10 c}\right )}{d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 c x^4}-\frac {d \left (\frac {9 b d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c x^3}-\frac {\int \frac {d^2 \left (54 a b d-\left (64 a c-63 b^2 d\right ) \sqrt {\frac {d}{x}}\right )}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^2}d\sqrt {\frac {d}{x}}}{8 c}\right )}{10 c}\right )}{d^3}\)

\(\Big \downarrow \) 1236

\(\displaystyle -\frac {2 \left (\frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 c x^4}-\frac {d \left (\frac {9 b d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c x^3}-\frac {\frac {d \int \frac {\left (4 a \left (64 a c-63 b^2 d\right )+\frac {7 b d \left (92 a c-45 b^2 d\right )}{x}\right ) \sqrt {\frac {d}{x}}}{2 \sqrt {a+\frac {b d}{x}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{3 c}-\frac {d^3 \left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 c x^2}}{8 c}\right )}{10 c}\right )}{d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 c x^4}-\frac {d \left (\frac {9 b d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c x^3}-\frac {\frac {d \int \frac {\left (4 a \left (64 a c-63 b^2 d\right )+\frac {7 b d \left (92 a c-45 b^2 d\right )}{x}\right ) \sqrt {\frac {d}{x}}}{\sqrt {a+\frac {b d}{x}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{6 c}-\frac {d^3 \left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 c x^2}}{8 c}\right )}{10 c}\right )}{d^3}\)

\(\Big \downarrow \) 1225

\(\displaystyle -\frac {2 \left (\frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 c x^4}-\frac {d \left (\frac {9 b d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c x^3}-\frac {\frac {d \left (-\frac {15 b d \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 c^2}-\frac {d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} \left (d \left (-\frac {1024 a^2 c^2}{d}+2940 a b^2 c-945 b^4 d\right )-14 b c \sqrt {\frac {d}{x}} \left (92 a c-45 b^2 d\right )\right )}{4 c^2}\right )}{6 c}-\frac {d^3 \left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 c x^2}}{8 c}\right )}{10 c}\right )}{d^3}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {2 \left (\frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 c x^4}-\frac {d \left (\frac {9 b d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c x^3}-\frac {\frac {d \left (-\frac {15 b d \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) \int \frac {1}{\frac {4 c}{d}-\frac {d^2}{x^2}}d\frac {2 \sqrt {\frac {d}{x}} c+b d}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{4 c^2}-\frac {d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} \left (d \left (-\frac {1024 a^2 c^2}{d}+2940 a b^2 c-945 b^4 d\right )-14 b c \sqrt {\frac {d}{x}} \left (92 a c-45 b^2 d\right )\right )}{4 c^2}\right )}{6 c}-\frac {d^3 \left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 c x^2}}{8 c}\right )}{10 c}\right )}{d^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \left (\frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 c x^4}-\frac {d \left (\frac {9 b d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c x^3}-\frac {\frac {d \left (-\frac {15 b d^{3/2} \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) \text {arctanh}\left (\frac {d^{3/2}}{2 \sqrt {c} x}\right )}{8 c^{5/2}}-\frac {d \left (d \left (-\frac {1024 a^2 c^2}{d}+2940 a b^2 c-945 b^4 d\right )-14 b c \sqrt {\frac {d}{x}} \left (92 a c-45 b^2 d\right )\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c^2}\right )}{6 c}-\frac {d^3 \left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{3 c x^2}}{8 c}\right )}{10 c}\right )}{d^3}\)

Input:

Int[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x^4),x]
 

Output:

(-2*((d^5*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2])/(5*c*x^4) - (d*((9*b*d^4*Sqrt 
[a + b*Sqrt[d/x] + (c*d)/x^2])/(4*c*x^3) - (-1/3*(d^3*(64*a*c - 63*b^2*d)* 
Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2])/(c*x^2) + (d*(-1/4*(d*(d*(2940*a*b^2*c 
- (1024*a^2*c^2)/d - 945*b^4*d) - 14*b*c*(92*a*c - 45*b^2*d)*Sqrt[d/x])*Sq 
rt[a + b*Sqrt[d/x] + (c*d)/x^2])/c^2 - (15*b*d^(3/2)*(240*a^2*c^2 - 280*a* 
b^2*c*d + 63*b^4*d^2)*ArcTanh[d^(3/2)/(2*Sqrt[c]*x)])/(8*c^(5/2))))/(6*c)) 
/(8*c)))/(10*c)))/d^3
 

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 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 1236

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

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 = 487, normalized size of antiderivative = 1.69

method	result	size
default	\(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \left (945 \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{5} b^{5} c -1890 d^{2} c^{\frac {3}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x^{2} b^{4}-4200 \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{4} a \,b^{3} c^{2}+1260 c^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{3} b^{3}+5880 d \,c^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x^{2} a \,b^{2}-1008 d \,c^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x \,b^{2}+3600 \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \sqrt {\frac {d}{x}}\, x^{3} a^{2} b \,c^{3}-2576 c^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x^{2} a b -2048 c^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{2} x^{2}+864 c^{\frac {9}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x b +1024 c^{\frac {9}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a x -768 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{\frac {11}{2}}\right )}{1920 x^{2} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{\frac {13}{2}}}\)	\(487\)

Input:

int(1/(a+b*(d/x)^(1/2)+c/x)^(1/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

1/1920*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)/x^2*(945*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*c-18 
90*d^2*c^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^2*b^4-4200*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* 
a*b^3*c^2+1260*c^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^3*b^3+5 
880*d*c^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^2*a*b^2-1008*d*c^(7/2)*(b*(d 
/x)^(1/2)*x+a*x+c)^(1/2)*x*b^2+3600*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*a^2*b*c^3-2576*c^(7/2) 
*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^2*a*b-2048*c^(7/2)*(b*(d/x)^( 
1/2)*x+a*x+c)^(1/2)*a^2*x^2+864*c^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x 
)^(1/2)*x*b+1024*c^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a*x-768*(b*(d/x)^(1 
/2)*x+a*x+c)^(1/2)*c^(11/2))/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^(13/2)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*(d/x)^(1/2)+c/x)^(1/2)/x^4,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(1/(a+b*(d/x)**(1/2)+c/x)**(1/2)/x**4,x)
 

Output:

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

Maxima [F]

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

integrate(1/(a+b*(d/x)^(1/2)+c/x)^(1/2)/x^4,x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x^4), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*(d/x)^(1/2)+c/x)^(1/2)/x^4,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx=\int \frac {1}{x^4\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \] Input:

int(1/(x^4*(a + c/x + b*(d/x)^(1/2))^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/(x^4*(a + c/x + b*(d/x)^(1/2))^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 1.55 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx=\frac {-2576 \sqrt {x}\, \sqrt {d}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, a b \,c^{3} x +1260 \sqrt {x}\, \sqrt {d}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, b^{3} c^{2} d x +864 \sqrt {x}\, \sqrt {d}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, b \,c^{4}-2048 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, a^{2} c^{3} x^{2}+5880 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, a \,b^{2} c^{2} d \,x^{2}+1024 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, a \,c^{4} x -1890 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, b^{4} c \,d^{2} x^{2}-1008 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, b^{2} c^{3} d x -768 \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}\, c^{5}+3600 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}-\sqrt {x}\, \sqrt {d}\, b -2 c \right ) a^{2} b \,c^{2} x^{2}-4200 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}-\sqrt {x}\, \sqrt {d}\, b -2 c \right ) a \,b^{3} c d \,x^{2}+945 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {\sqrt {x}\, \sqrt {d}\, b +a x +c}-\sqrt {x}\, \sqrt {d}\, b -2 c \right ) b^{5} d^{2} x^{2}-3600 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\sqrt {x}\right ) a^{2} b \,c^{2} x^{2}+4200 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\sqrt {x}\right ) a \,b^{3} c d \,x^{2}-945 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\sqrt {x}\right ) b^{5} d^{2} x^{2}}{1920 \sqrt {x}\, c^{6} x^{2}} \] Input:

int(1/(a+b*(d/x)^(1/2)+c/x)^(1/2)/x^4,x)
 

Output:

( - 2576*sqrt(x)*sqrt(d)*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*a*b*c**3*x + 12 
60*sqrt(x)*sqrt(d)*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*b**3*c**2*d*x + 864*s 
qrt(x)*sqrt(d)*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*b*c**4 - 2048*sqrt(sqrt(x 
)*sqrt(d)*b + a*x + c)*a**2*c**3*x**2 + 5880*sqrt(sqrt(x)*sqrt(d)*b + a*x 
+ c)*a*b**2*c**2*d*x**2 + 1024*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*a*c**4*x 
- 1890*sqrt(sqrt(x)*sqrt(d)*b + a*x + c)*b**4*c*d**2*x**2 - 1008*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 + 3600*sqrt(x)*sqrt(d)*sqrt(c)*log( - 2*sqrt(c)*sqrt(sqrt(x)*sqr 
t(d)*b + a*x + c) - sqrt(x)*sqrt(d)*b - 2*c)*a**2*b*c**2*x**2 - 4200*sqrt( 
x)*sqrt(d)*sqrt(c)*log( - 2*sqrt(c)*sqrt(sqrt(x)*sqrt(d)*b + a*x + c) - sq 
rt(x)*sqrt(d)*b - 2*c)*a*b**3*c*d*x**2 + 945*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 - 3600*sqrt(x)*sqrt(d)*sqrt(c)*log(sqrt(x))*a**2*b*c**2*x**2 
 + 4200*sqrt(x)*sqrt(d)*sqrt(c)*log(sqrt(x))*a*b**3*c*d*x**2 - 945*sqrt(x) 
*sqrt(d)*sqrt(c)*log(sqrt(x))*b**5*d**2*x**2)/(1920*sqrt(x)*c**6*x**2)