\(\int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx\) [85]

Optimal result

Integrand size = 21, antiderivative size = 174 \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=\frac {4 a^5 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac {4 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}+\frac {40 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^6 c^3}+\frac {20 a \left (a+b \sqrt {\frac {c}{x}}\right )^{11/2}}{11 b^6 c^3}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{13/2}}{13 b^6 c^3} \] Output:

4/3*a^5*(a+b*(c/x)^(1/2))^(3/2)/b^6/c^3-4*a^4*(a+b*(c/x)^(1/2))^(5/2)/b^6/ 
c^3+40/7*a^3*(a+b*(c/x)^(1/2))^(7/2)/b^6/c^3-40/9*a^2*(a+b*(c/x)^(1/2))^(9 
/2)/b^6/c^3+20/11*a*(a+b*(c/x)^(1/2))^(11/2)/b^6/c^3-4/13*(a+b*(c/x)^(1/2) 
)^(13/2)/b^6/c^3
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2} \left (-256 a^5+384 a^4 b \sqrt {\frac {c}{x}}+560 a^2 b^3 \left (\frac {c}{x}\right )^{3/2}+693 b^5 \left (\frac {c}{x}\right )^{5/2}-\frac {630 a b^4 c^2}{x^2}-\frac {480 a^3 b^2 c}{x}\right )}{9009 b^6 c^3} \] Input:

Integrate[Sqrt[a + b*Sqrt[c/x]]/x^4,x]
 

Output:

(-4*(a + b*Sqrt[c/x])^(3/2)*(-256*a^5 + 384*a^4*b*Sqrt[c/x] + 560*a^2*b^3* 
(c/x)^(3/2) + 693*b^5*(c/x)^(5/2) - (630*a*b^4*c^2)/x^2 - (480*a^3*b^2*c)/ 
x))/(9009*b^6*c^3)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {893, 798, 53, 2009}

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.

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

\(\Big \downarrow \) 893

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

\(\Big \downarrow \) 798

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

\(\Big \downarrow \) 53

\(\displaystyle -2 \int \left (\frac {\left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{11/2}}{b^5 c^{5/2}}-\frac {5 a \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{9/2}}{b^5 c^{5/2}}+\frac {10 a^2 \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{7/2}}{b^5 c^{5/2}}-\frac {10 a^3 \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{5/2}}{b^5 c^{5/2}}+\frac {5 a^4 \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{3/2}}{b^5 c^{5/2}}-\frac {a^5 \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{b^5 c^{5/2}}\right )d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (-\frac {2 a^5 \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{3/2}}{3 b^6 c^3}+\frac {2 a^4 \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{5/2}}{b^6 c^3}-\frac {20 a^3 \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{7/2}}{7 b^6 c^3}+\frac {20 a^2 \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{9/2}}{9 b^6 c^3}+\frac {2 \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{13/2}}{13 b^6 c^3}-\frac {10 a \left (a+\frac {b \sqrt {c}}{\sqrt {x}}\right )^{11/2}}{11 b^6 c^3}\right )\)

Input:

Int[Sqrt[a + b*Sqrt[c/x]]/x^4,x]
 

Output:

-2*((-2*a^5*(a + (b*Sqrt[c])/Sqrt[x])^(3/2))/(3*b^6*c^3) + (2*a^4*(a + (b* 
Sqrt[c])/Sqrt[x])^(5/2))/(b^6*c^3) - (20*a^3*(a + (b*Sqrt[c])/Sqrt[x])^(7/ 
2))/(7*b^6*c^3) + (20*a^2*(a + (b*Sqrt[c])/Sqrt[x])^(9/2))/(9*b^6*c^3) - ( 
10*a*(a + (b*Sqrt[c])/Sqrt[x])^(11/2))/(11*b^6*c^3) + (2*(a + (b*Sqrt[c])/ 
Sqrt[x])^(13/2))/(13*b^6*c^3))
 

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 798

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.76

method	result	size
default	\(-\frac {4 \sqrt {a +b \sqrt {\frac {c}{x}}}\, \left (b x \sqrt {\frac {c}{x}}+x a \right )^{\frac {3}{2}} \left (693 x^{2} \left (\frac {c}{x}\right )^{\frac {5}{2}} b^{5}+560 x^{2} \left (\frac {c}{x}\right )^{\frac {3}{2}} a^{2} b^{3}+384 x^{2} \sqrt {\frac {c}{x}}\, a^{4} b -256 x^{2} a^{5}-480 c x \,a^{3} b^{2}-630 a \,b^{4} c^{2}\right )}{9009 c^{3} x^{3} \sqrt {x \left (a +b \sqrt {\frac {c}{x}}\right )}\, b^{6}}\)	\(133\)

Input:

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

Output:

-4/9009*(a+b*(c/x)^(1/2))^(1/2)*(b*x*(c/x)^(1/2)+x*a)^(3/2)*(693*x^2*(c/x) 
^(5/2)*b^5+560*x^2*(c/x)^(3/2)*a^2*b^3+384*x^2*(c/x)^(1/2)*a^4*b-256*x^2*a 
^5-480*c*x*a^3*b^2-630*a*b^4*c^2)/c^3/x^3/(x*(a+b*(c/x)^(1/2)))^(1/2)/b^6
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=-\frac {4 \, {\left (693 \, b^{6} c^{3} - 70 \, a^{2} b^{4} c^{2} x - 96 \, a^{4} b^{2} c x^{2} - 256 \, a^{6} x^{3} + {\left (63 \, a b^{5} c^{2} x + 80 \, a^{3} b^{3} c x^{2} + 128 \, a^{5} b x^{3}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{9009 \, b^{6} c^{3} x^{3}} \] Input:

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

Output:

-4/9009*(693*b^6*c^3 - 70*a^2*b^4*c^2*x - 96*a^4*b^2*c*x^2 - 256*a^6*x^3 + 
 (63*a*b^5*c^2*x + 80*a^3*b^3*c*x^2 + 128*a^5*b*x^3)*sqrt(c/x))*sqrt(b*sqr 
t(c/x) + a)/(b^6*c^3*x^3)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=-\frac {4 \, {\left (\frac {693 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {13}{2}}}{b^{6}} - \frac {4095 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {11}{2}} a}{b^{6}} + \frac {10010 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {9}{2}} a^{2}}{b^{6}} - \frac {12870 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} a^{3}}{b^{6}} + \frac {9009 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a^{4}}{b^{6}} - \frac {3003 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{5}}{b^{6}}\right )}}{9009 \, c^{3}} \] Input:

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

Output:

-4/9009*(693*(b*sqrt(c/x) + a)^(13/2)/b^6 - 4095*(b*sqrt(c/x) + a)^(11/2)* 
a/b^6 + 10010*(b*sqrt(c/x) + a)^(9/2)*a^2/b^6 - 12870*(b*sqrt(c/x) + a)^(7 
/2)*a^3/b^6 + 9009*(b*sqrt(c/x) + a)^(5/2)*a^4/b^6 - 3003*(b*sqrt(c/x) + a 
)^(3/2)*a^5/b^6)/c^3
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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