3.229 \(\int \frac {\sqrt {a+\frac {b}{x}}}{(c+\frac {d}{x})^2} \, dx\)

Optimal. Leaf size=147 \[ \frac {\sqrt {d} (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 \sqrt {b c-a d}}+\frac {(b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^3}+\frac {2 d \sqrt {a+\frac {b}{x}}}{c^2 \left (c+\frac {d}{x}\right )}+\frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )} \]

[Out]

(-4*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/c^3/a^(1/2)+(-4*a*d+3*b*c)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)
^(1/2))*d^(1/2)/c^3/(-a*d+b*c)^(1/2)+2*d*(a+b/x)^(1/2)/c^2/(c+d/x)+x*(a+b/x)^(1/2)/c/(c+d/x)

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Rubi [A]  time = 0.21, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {375, 99, 151, 156, 63, 208, 205} \[ \frac {2 d \sqrt {a+\frac {b}{x}}}{c^2 \left (c+\frac {d}{x}\right )}+\frac {\sqrt {d} (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 \sqrt {b c-a d}}+\frac {(b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^3}+\frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]/(c + d/x)^2,x]

[Out]

(2*d*Sqrt[a + b/x])/(c^2*(c + d/x)) + (Sqrt[a + b/x]*x)/(c*(c + d/x)) + (Sqrt[d]*(3*b*c - 4*a*d)*ArcTan[(Sqrt[
d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c^3*Sqrt[b*c - a*d]) + ((b*c - 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(Sq
rt[a]*c^3)

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 99

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 205

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

Rule 208

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

Rule 375

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*} \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^2} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2 (c+d x)^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (b c-4 a d)-\frac {3 b d x}{2}}{x \sqrt {a+b x} (c+d x)^2} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {2 d \sqrt {a+\frac {b}{x}}}{c^2 \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} (b c-4 a d) (b c-a d)+b d (b c-a d) x}{x \sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{c^2 (b c-a d)}\\ &=\frac {2 d \sqrt {a+\frac {b}{x}}}{c^2 \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )}-\frac {(b c-4 a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 c^3}+\frac {(d (3 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{2 c^3}\\ &=\frac {2 d \sqrt {a+\frac {b}{x}}}{c^2 \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )}-\frac {(b c-4 a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b c^3}+\frac {(d (3 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{c-\frac {a d}{b}+\frac {d x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b c^3}\\ &=\frac {2 d \sqrt {a+\frac {b}{x}}}{c^2 \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )}+\frac {\sqrt {d} (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 \sqrt {b c-a d}}+\frac {(b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^3}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 122, normalized size = 0.83 \[ \frac {\frac {c x \sqrt {a+\frac {b}{x}} (c x+2 d)}{c x+d}+\frac {\sqrt {d} (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}+\frac {(b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*Sqrt[a + b/x]*x*(2*d + c*x))/(d + c*x) + (Sqrt[d]*(3*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c
- a*d]])/Sqrt[b*c - a*d] + ((b*c - 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a])/c^3

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fricas [A]  time = 1.36, size = 801, normalized size = 5.45 \[ \left [-\frac {{\left (b c d - 4 \, a d^{2} + {\left (b c^{2} - 4 \, a c d\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \, {\left (a c^{2} x^{2} + 2 \, a c d x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a c^{4} x + a c^{3} d\right )}}, -\frac {2 \, {\left (b c d - 4 \, a d^{2} + {\left (b c^{2} - 4 \, a c d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {d}{b c - a d}} \log \left (-\frac {2 \, {\left (b c - a d\right )} x \sqrt {-\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}} - b d + {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \, {\left (a c^{2} x^{2} + 2 \, a c d x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a c^{4} x + a c^{3} d\right )}}, \frac {2 \, {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x\right )} \sqrt {\frac {d}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} x \sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) - {\left (b c d - 4 \, a d^{2} + {\left (b c^{2} - 4 \, a c d\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (a c^{2} x^{2} + 2 \, a c d x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a c^{4} x + a c^{3} d\right )}}, \frac {{\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x\right )} \sqrt {\frac {d}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} x \sqrt {\frac {d}{b c - a d}} \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) - {\left (b c d - 4 \, a d^{2} + {\left (b c^{2} - 4 \, a c d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a c^{2} x^{2} + 2 \, a c d x\right )} \sqrt {\frac {a x + b}{x}}}{a c^{4} x + a c^{3} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Unable to divide, perhaps due to rounding error%%%{%%{[-2,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,6,4,0]%%%}+%%%{%%
%{8,[1]%%%},[3,5,4,1]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1]%%%}]%%},[2,5,5,1]%%%}+%%%{%%{[%%%{-8,[1]%%%},0]:[1,0,%
%%{-1,[1]%%%}]%%},[2,4,4,2]%%%}+%%%{%%%{8,[1]%%%},[1,4,5,2]%%%}+%%%{%%{[-2,0]:[1,0,%%%{-1,[1]%%%}]%%},[0,4,6,2
]%%%} / %%%{%%%{1,[1]%%%},[4,2,0,0]%%%}+%%%{%%{poly1[%%%{-4,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,0,1]%%%}+%
%%{%%%{2,[1]%%%},[2,1,1,1]%%%}+%%%{%%%{4,[2]%%%},[2,0,0,2]%%%}+%%%{%%{poly1[%%%{-4,[1]%%%},0]:[1,0,%%%{-1,[1]%
%%}]%%},[1,0,1,2]%%%}+%%%{%%%{1,[1]%%%},[0,0,2,2]%%%} Error: Bad Argument Value

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maple [B]  time = 0.07, size = 943, normalized size = 6.41 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (4 a^{\frac {7}{2}} c \,d^{3} x \ln \left (\frac {-2 a d x +b c x -b d +2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {\left (a x +b \right ) x}\, c}{c x +d}\right )-7 a^{\frac {5}{2}} b \,c^{2} d^{2} x \ln \left (\frac {-2 a d x +b c x -b d +2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {\left (a x +b \right ) x}\, c}{c x +d}\right )+3 a^{\frac {3}{2}} b^{2} c^{3} d x \ln \left (\frac {-2 a d x +b c x -b d +2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {\left (a x +b \right ) x}\, c}{c x +d}\right )+4 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{3} c^{2} d^{2} x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-5 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{2} b \,c^{3} d x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+\sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a \,b^{2} c^{4} x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+4 a^{\frac {7}{2}} d^{4} \ln \left (\frac {-2 a d x +b c x -b d +2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {\left (a x +b \right ) x}\, c}{c x +d}\right )-7 a^{\frac {5}{2}} b c \,d^{3} \ln \left (\frac {-2 a d x +b c x -b d +2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {\left (a x +b \right ) x}\, c}{c x +d}\right )+2 \sqrt {\left (a x +b \right ) x}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{\frac {5}{2}} c^{4} x^{2}+3 a^{\frac {3}{2}} b^{2} c^{2} d^{2} \ln \left (\frac {-2 a d x +b c x -b d +2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {\left (a x +b \right ) x}\, c}{c x +d}\right )+4 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{3} c \,d^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-5 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{2} b \,c^{2} d^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+\sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a \,b^{2} c^{3} d \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-2 \sqrt {\left (a x +b \right ) x}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{\frac {5}{2}} c^{3} d x +4 \sqrt {\left (a x +b \right ) x}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{\frac {3}{2}} b \,c^{4} x -4 \sqrt {\left (a x +b \right ) x}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{\frac {5}{2}} c^{2} d^{2}+4 \sqrt {\left (a x +b \right ) x}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{\frac {3}{2}} b \,c^{3} d -2 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{\frac {3}{2}} c^{4}\right ) x}{2 \sqrt {\left (a x +b \right ) x}\, \left (a d -b c \right ) \left (c x +d \right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{\frac {3}{2}} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a*x+b)/x)^(1/2)*x*(4*a^(7/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x
+d))*x*c*d^3+2*a^(5/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*c^4+4*a^(7/2)*ln((-2*a*d*x+b*c*x-b*d+2*((
a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*d^4-2*a^(5/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x*
c^3*d-7*a^(5/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*x*b*c^2*d^2-4*a
^(5/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*c^2*d^2-7*a^(5/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^
(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*b*c*d^3-2*c^4*((a*x+b)*x)^(3/2)*a^(3/2)*((a*d-b*c)/c^2*d)^(1/2)+4*a^(3/2)*
((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b*c^4+3*a^(3/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*(
(a*x+b)*x)^(1/2)*c)/(c*x+d))*x*b^2*c^3*d+4*a^(3/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*b*c^3*d+3*a^(3/2)
*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*b^2*c^2*d^2+4*a^3*ln(1/2*(2*a*
x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x*c^2*d^2-5*a^2*ln(1/2*(2*a*x+b+2*((a*x+b)*x
)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x*b*c^3*d+a*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(
1/2))*((a*d-b*c)/c^2*d)^(1/2)*x*b^2*c^4+4*a^3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)
/c^2*d)^(1/2)*c*d^3-5*a^2*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*b*c^2*
d^2+a*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*b^2*c^3*d)/c^4/((a*x+b)*x)
^(1/2)/(a*d-b*c)/(c*x+d)/a^(3/2)/((a*d-b*c)/c^2*d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + \frac {b}{x}}}{{\left (c + \frac {d}{x}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.26, size = 1195, normalized size = 8.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(1/2)/(c + d/x)^2,x)

[Out]

- ((2*b*d*(a + b/x)^(3/2))/c^2 - (b*(a + b/x)^(1/2)*(2*a*d - b*c))/c^2)/((a + b/x)*(2*a*d - b*c) - d*(a + b/x)
^2 - a^2*d + a*b*c) - (atanh((8*b^5*d^3*(a + b/x)^(1/2))/(a^(1/2)*(8*b^5*d^3 - (2*b^6*c*d^2)/a)) + (2*b^6*d^2*
(a + b/x)^(1/2))/(a^(3/2)*((2*b^6*d^2)/a - (8*b^5*d^3)/c)))*(4*a*d - b*c))/(a^(1/2)*c^3) - (atan((((d*(a*d - b
*c))^(1/2)*((4*(a + b/x)^(1/2)*(16*a^2*b^2*d^5 + 5*b^4*c^2*d^3 - 16*a*b^3*c*d^4))/c^4 - (((2*(2*b^4*c^7*d^2 -
4*a*b^3*c^6*d^3))/c^6 - (2*(2*b^3*c^7*d^2 - 4*a*b^2*c^6*d^3)*(a + b/x)^(1/2)*(d*(a*d - b*c))^(1/2)*(4*a*d - 3*
b*c))/(c^4*(b*c^4 - a*c^3*d)))*(d*(a*d - b*c))^(1/2)*(4*a*d - 3*b*c))/(2*(b*c^4 - a*c^3*d)))*(4*a*d - 3*b*c)*1
i)/(2*(b*c^4 - a*c^3*d)) + ((d*(a*d - b*c))^(1/2)*((4*(a + b/x)^(1/2)*(16*a^2*b^2*d^5 + 5*b^4*c^2*d^3 - 16*a*b
^3*c*d^4))/c^4 + (((2*(2*b^4*c^7*d^2 - 4*a*b^3*c^6*d^3))/c^6 + (2*(2*b^3*c^7*d^2 - 4*a*b^2*c^6*d^3)*(a + b/x)^
(1/2)*(d*(a*d - b*c))^(1/2)*(4*a*d - 3*b*c))/(c^4*(b*c^4 - a*c^3*d)))*(d*(a*d - b*c))^(1/2)*(4*a*d - 3*b*c))/(
2*(b*c^4 - a*c^3*d)))*(4*a*d - 3*b*c)*1i)/(2*(b*c^4 - a*c^3*d)))/((4*(16*a^2*b^3*d^5 + 3*b^5*c^2*d^3 - 16*a*b^
4*c*d^4))/c^6 - ((d*(a*d - b*c))^(1/2)*((4*(a + b/x)^(1/2)*(16*a^2*b^2*d^5 + 5*b^4*c^2*d^3 - 16*a*b^3*c*d^4))/
c^4 - (((2*(2*b^4*c^7*d^2 - 4*a*b^3*c^6*d^3))/c^6 - (2*(2*b^3*c^7*d^2 - 4*a*b^2*c^6*d^3)*(a + b/x)^(1/2)*(d*(a
*d - b*c))^(1/2)*(4*a*d - 3*b*c))/(c^4*(b*c^4 - a*c^3*d)))*(d*(a*d - b*c))^(1/2)*(4*a*d - 3*b*c))/(2*(b*c^4 -
a*c^3*d)))*(4*a*d - 3*b*c))/(2*(b*c^4 - a*c^3*d)) + ((d*(a*d - b*c))^(1/2)*((4*(a + b/x)^(1/2)*(16*a^2*b^2*d^5
+ 5*b^4*c^2*d^3 - 16*a*b^3*c*d^4))/c^4 + (((2*(2*b^4*c^7*d^2 - 4*a*b^3*c^6*d^3))/c^6 + (2*(2*b^3*c^7*d^2 - 4*
a*b^2*c^6*d^3)*(a + b/x)^(1/2)*(d*(a*d - b*c))^(1/2)*(4*a*d - 3*b*c))/(c^4*(b*c^4 - a*c^3*d)))*(d*(a*d - b*c))
^(1/2)*(4*a*d - 3*b*c))/(2*(b*c^4 - a*c^3*d)))*(4*a*d - 3*b*c))/(2*(b*c^4 - a*c^3*d))))*(d*(a*d - b*c))^(1/2)*
(4*a*d - 3*b*c)*1i)/(b*c^4 - a*c^3*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {a + \frac {b}{x}}}{\left (c x + d\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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