\(\int \frac {\sqrt {a+\frac {b}{x}}}{(c+\frac {d}{x})^{3/2}} \, dx\) [268]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {382, 98, 96, 95, 214} \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\frac {(b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{5/2}}-\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{a c^2 \sqrt {c+\frac {d}{x}}}+\frac {x \left (a+\frac {b}{x}\right )^{3/2}}{a c \sqrt {c+\frac {d}{x}}} \]

[In]

Int[Sqrt[a + b/x]/(c + d/x)^(3/2),x]

[Out]

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

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[(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] || Sum
SimplerQ[m, 1])

Rule 214

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 382

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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\left (a+\frac {b}{x}\right )^{3/2} x}{a c \sqrt {c+\frac {d}{x}}}+\frac {\left (-\frac {b c}{2}+\frac {3 a d}{2}\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = -\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 {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\frac {1}{x}\right )}{2 c^2} \\ & = -\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 {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}}\right )}{c^2} \\ & = -\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) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} c^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x \left (\sqrt {a} \sqrt {c} \sqrt {b+a x} (3 d+c x)+(b c-3 a d) \sqrt {d+c x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {b+a x}}{\sqrt {a} \sqrt {d+c x}}\right )\right )}{\sqrt {a} c^{5/2} \sqrt {b+a x} (d+c x)} \]

[In]

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

[Out]

(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]*ArcTan
h[(Sqrt[c]*Sqrt[b + a*x])/(Sqrt[a]*Sqrt[d + c*x])]))/(Sqrt[a]*c^(5/2)*Sqrt[b + a*x]*(d + c*x))

Maple [B] (verified)

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

Time = 0.16 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.30

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

[In]

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

[Out]

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)

none

Time = 0.39 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.61 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^{3/2}} \, dx=\left [-\frac {{\left (b c d - 3 \, a d^{2} + {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {a c} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} + 4 \, {\left (2 \, a c x^{2} + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - 8 \, {\left (a b c^{2} + a^{2} c d\right )} x\right ) - 4 \, {\left (a c^{2} x^{2} + 3 \, a c d x\right )} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{4 \, {\left (a c^{4} x + a c^{3} d\right )}}, -\frac {{\left (b c d - 3 \, a d^{2} + {\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {2 \, \sqrt {-a c} x \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, a c x + b c + a d}\right ) - 2 \, {\left (a c^{2} x^{2} + 3 \, a c d x\right )} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, {\left (a c^{4} x + a c^{3} d\right )}}\right ] \]

[In]

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

[Out]

[-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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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