\(\int \frac {\sqrt {a+\frac {b}{x}}}{(c+\frac {d}{x})^{3/2}} \, dx\) [48]

Optimal result

Integrand size = 23, antiderivative size = 122 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=-\frac {(b c-3 a d) \sqrt {a+\frac {b}{x}}}{a c^2 \sqrt {c+\frac {d}{x}}}+\frac {\left (a+\frac {b}{x}\right )^{3/2} x}{a c \sqrt {c+\frac {d}{x}}}+\frac {(b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{5/2}} \] Output:

-(-3*a*d+b*c)*(a+b/x)^(1/2)/a/c^2/(c+d/x)^(1/2)+(a+b/x)^(3/2)*x/a/c/(c+d/x 
)^(1/2)+(-3*a*d+b*c)*arctanh(c^(1/2)*(a+b/x)^(1/2)/a^(1/2)/(c+d/x)^(1/2))/ 
a^(1/2)/c^(5/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x \left (\sqrt {a} \sqrt {c} \sqrt {b+a x} (3 d+c x)+(b c-3 a d) \sqrt {d+c x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {b+a x}}{\sqrt {a} \sqrt {d+c x}}\right )\right )}{\sqrt {a} c^{5/2} \sqrt {b+a x} (d+c x)} \] Input:

Integrate[Sqrt[a + b/x]/(c + d/x)^(3/2),x]
 

Output:

(Sqrt[a + b/x]*Sqrt[c + d/x]*x*(Sqrt[a]*Sqrt[c]*Sqrt[b + a*x]*(3*d + c*x) 
+ (b*c - 3*a*d)*Sqrt[d + c*x]*ArcTanh[(Sqrt[c]*Sqrt[b + a*x])/(Sqrt[a]*Sqr 
t[d + c*x])]))/(Sqrt[a]*c^(5/2)*Sqrt[b + a*x]*(d + c*x))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {899, 107, 105, 104, 221}

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.

\(\displaystyle \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {x \left (a+\frac {b}{x}\right )^{3/2}}{a c \sqrt {c+\frac {d}{x}}}-\frac {(b c-3 a d) \left (\frac {2 \sqrt {a+\frac {b}{x}}}{c \sqrt {c+\frac {d}{x}}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{c^{3/2}}\right )}{2 a c}\)

Input:

Int[Sqrt[a + b/x]/(c + d/x)^(3/2),x]
 

Output:

((a + b/x)^(3/2)*x)/(a*c*Sqrt[c + d/x]) - ((b*c - 3*a*d)*((2*Sqrt[a + b/x] 
)/(c*Sqrt[c + d/x]) - (2*Sqrt[a]*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]* 
Sqrt[c + d/x])])/c^(3/2)))/(2*a*c)
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

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

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

rule 221

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

rule 899

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol 
] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, 
 b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(279\) vs. \(2(102)=204\).

Time = 0.31 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.30

method	result	size
default	\(\frac {\sqrt {\frac {a x +b}{x}}\, x \sqrt {\frac {c x +d}{x}}\, \left (-3 \ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a c d x +\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b \,c^{2} x +2 c x \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}-3 \ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a \,d^{2}+\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b c d +6 d \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}\right )}{2 \sqrt {a c}\, \left (c x +d \right ) \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, c^{2}}\)	\(280\)

Input:

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

Output:

1/2*((a*x+b)/x)^(1/2)*x*((c*x+d)/x)^(1/2)*(-3*ln(1/2*(2*a*c*x+2*((a*x+b)*( 
c*x+d))^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*a*c*d*x+ln(1/2*(2*a*c*x+2* 
((a*x+b)*(c*x+d))^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*b*c^2*x+2*c*x*(( 
a*x+b)*(c*x+d))^(1/2)*(a*c)^(1/2)-3*ln(1/2*(2*a*c*x+2*((a*x+b)*(c*x+d))^(1 
/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*a*d^2+ln(1/2*(2*a*c*x+2*((a*x+b)*(c* 
x+d))^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*b*c*d+6*d*((a*x+b)*(c*x+d))^ 
(1/2)*(a*c)^(1/2))/(a*c)^(1/2)/(c*x+d)/((a*x+b)*(c*x+d))^(1/2)/c^2
 

Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.61 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\left [-\frac {{\left (b c d - 3 \, a d^{2} + {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {a c} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} + 4 \, {\left (2 \, a c x^{2} + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - 8 \, {\left (a b c^{2} + a^{2} c d\right )} x\right ) - 4 \, {\left (a c^{2} x^{2} + 3 \, a c d x\right )} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{4 \, {\left (a c^{4} x + a c^{3} d\right )}}, -\frac {{\left (b c d - 3 \, a d^{2} + {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {2 \, \sqrt {-a c} x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, a c x + b c + a d}\right ) - 2 \, {\left (a c^{2} x^{2} + 3 \, a c d x\right )} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, {\left (a c^{4} x + a c^{3} d\right )}}\right ] \] Input:

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

Output:

[-1/4*((b*c*d - 3*a*d^2 + (b*c^2 - 3*a*c*d)*x)*sqrt(a*c)*log(-8*a^2*c^2*x^ 
2 - b^2*c^2 - 6*a*b*c*d - a^2*d^2 + 4*(2*a*c*x^2 + (b*c + a*d)*x)*sqrt(a*c 
)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - 8*(a*b*c^2 + a^2*c*d)*x) - 4*(a*c^ 
2*x^2 + 3*a*c*d*x)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x))/(a*c^4*x + a*c^3*d 
), -1/2*((b*c*d - 3*a*d^2 + (b*c^2 - 3*a*c*d)*x)*sqrt(-a*c)*arctan(2*sqrt( 
-a*c)*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)/(2*a*c*x + b*c + a*d)) - 2*(a* 
c^2*x^2 + 3*a*c*d*x)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x))/(a*c^4*x + a*c^3 
*d)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\int \frac {\sqrt {a + \frac {b}{x}}}{\left (c + \frac {d}{x}\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

Integral(sqrt(a + b/x)/(c + d/x)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{{\left (c + \frac {d}{x}\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

integrate(sqrt(a + b/x)/(c + d/x)^(3/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{{\left (c + \frac {d}{x}\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

integrate(sqrt(a + b/x)/(c + d/x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\int \frac {\sqrt {a+\frac {b}{x}}}{{\left (c+\frac {d}{x}\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.13 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\frac {4 \sqrt {c x +d}\, \sqrt {a x +b}\, a \,c^{2} x +12 \sqrt {c x +d}\, \sqrt {a x +b}\, a c d -12 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}}{\sqrt {a d -b c}}\right ) a c d x -12 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}}{\sqrt {a d -b c}}\right ) a \,d^{2}+4 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}}{\sqrt {a d -b c}}\right ) b \,c^{2} x +4 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}}{\sqrt {a d -b c}}\right ) b c d +9 \sqrt {c}\, \sqrt {a}\, a c d x +9 \sqrt {c}\, \sqrt {a}\, a \,d^{2}-\sqrt {c}\, \sqrt {a}\, b \,c^{2} x -\sqrt {c}\, \sqrt {a}\, b c d}{4 a \,c^{3} \left (c x +d \right )} \] Input:

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

Output:

(4*sqrt(c*x + d)*sqrt(a*x + b)*a*c**2*x + 12*sqrt(c*x + d)*sqrt(a*x + b)*a 
*c*d - 12*sqrt(c)*sqrt(a)*log((sqrt(c)*sqrt(a*x + b) + sqrt(a)*sqrt(c*x + 
d))/sqrt(a*d - b*c))*a*c*d*x - 12*sqrt(c)*sqrt(a)*log((sqrt(c)*sqrt(a*x + 
b) + sqrt(a)*sqrt(c*x + d))/sqrt(a*d - b*c))*a*d**2 + 4*sqrt(c)*sqrt(a)*lo 
g((sqrt(c)*sqrt(a*x + b) + sqrt(a)*sqrt(c*x + d))/sqrt(a*d - b*c))*b*c**2* 
x + 4*sqrt(c)*sqrt(a)*log((sqrt(c)*sqrt(a*x + b) + sqrt(a)*sqrt(c*x + d))/ 
sqrt(a*d - b*c))*b*c*d + 9*sqrt(c)*sqrt(a)*a*c*d*x + 9*sqrt(c)*sqrt(a)*a*d 
**2 - sqrt(c)*sqrt(a)*b*c**2*x - sqrt(c)*sqrt(a)*b*c*d)/(4*a*c**3*(c*x + d 
))