3.151 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2} \, dx\)
Optimal. Leaf size=172 \[ -\frac{(4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2} c^3}-\frac{d^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 (b c-a d)^{3/2}}+\frac{d \sqrt{a+\frac{b}{x}} (b c-2 a d)}{a c^2 \left (c+\frac{d}{x}\right ) (b c-a d)}+\frac{x \sqrt{a+\frac{b}{x}}}{a c \left (c+\frac{d}{x}\right )} \]
[Out]
(d*(b*c - 2*a*d)*Sqrt[a + b/x])/(a*c^2*(b*c - a*d)*(c + d/x)) + (Sqrt[a + b/x]*x
)/(a*c*(c + d/x)) - (d^(3/2)*(5*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt
[b*c - a*d]])/(c^3*(b*c - a*d)^(3/2)) - ((b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqr
t[a]])/(a^(3/2)*c^3)
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Rubi [A] time = 0.650209, antiderivative size = 172, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333
\[ -\frac{(4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2} c^3}-\frac{d^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 (b c-a d)^{3/2}}+\frac{d \sqrt{a+\frac{b}{x}} (b c-2 a d)}{a c^2 \left (c+\frac{d}{x}\right ) (b c-a d)}+\frac{x \sqrt{a+\frac{b}{x}}}{a c \left (c+\frac{d}{x}\right )} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b/x]*(c + d/x)^2),x]
[Out]
(d*(b*c - 2*a*d)*Sqrt[a + b/x])/(a*c^2*(b*c - a*d)*(c + d/x)) + (Sqrt[a + b/x]*x
)/(a*c*(c + d/x)) - (d^(3/2)*(5*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt
[b*c - a*d]])/(c^3*(b*c - a*d)^(3/2)) - ((b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqr
t[a]])/(a^(3/2)*c^3)
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Rubi in Sympy [A] time = 68.0127, size = 141, normalized size = 0.82 \[ - \frac{d x \sqrt{a + \frac{b}{x}}}{c \left (c + \frac{d}{x}\right ) \left (a d - b c\right )} + \frac{d^{\frac{3}{2}} \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{3} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{x \sqrt{a + \frac{b}{x}} \left (2 a d - b c\right )}{a c^{2} \left (a d - b c\right )} - \frac{\left (4 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c+d/x)**2/(a+b/x)**(1/2),x)
[Out]
-d*x*sqrt(a + b/x)/(c*(c + d/x)*(a*d - b*c)) + d**(3/2)*(4*a*d - 5*b*c)*atanh(sq
rt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(c**3*(a*d - b*c)**(3/2)) + x*sqrt(a + b/x)
*(2*a*d - b*c)/(a*c**2*(a*d - b*c)) - (4*a*d + b*c)*atanh(sqrt(a + b/x)/sqrt(a))
/(a**(3/2)*c**3)
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Mathematica [C] time = 0.84081, size = 224, normalized size = 1.3 \[ -\frac{\frac{(4 a d+b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{3/2}}+\frac{i d^{3/2} (5 b c-4 a d) \log \left (\frac{2 c^4 \sqrt{b c-a d} \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 i a d x-i b (d-c x)\right )}{d^{5/2} (c x+d) (5 b c-4 a d)}\right )}{(b c-a d)^{3/2}}+\frac{2 c x \sqrt{a+\frac{b}{x}} (b c (c x+d)-a d (c x+2 d))}{a (c x+d) (a d-b c)}}{2 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b/x]*(c + d/x)^2),x]
[Out]
-((2*c*Sqrt[a + b/x]*x*(b*c*(d + c*x) - a*d*(2*d + c*x)))/(a*(-(b*c) + a*d)*(d +
c*x)) + ((b*c + 4*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/a^(3/2) + (I
*d^(3/2)*(5*b*c - 4*a*d)*Log[(2*c^4*Sqrt[b*c - a*d]*((-2*I)*a*d*x + 2*Sqrt[d]*Sq
rt[b*c - a*d]*Sqrt[a + b/x]*x - I*b*(d - c*x)))/(d^(5/2)*(5*b*c - 4*a*d)*(d + c*
x))])/(b*c - a*d)^(3/2))/(2*c^3)
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Maple [B] time = 0.029, size = 1137, normalized size = 6.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c+d/x)^2/(a+b/x)^(1/2),x)
[Out]
1/2*((a*x+b)/x)^(1/2)*x*(-4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2)
)*((a*d-b*c)*d/c^2)^(1/2)*a^4*x*c^2*d^3-2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1
/2)*a^(7/2)*x^2*c^4*d-4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((
a*d-b*c)*d/c^2)^(1/2)*a^4*c*d^4+7*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a
^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*a^3*x*b*c^3*d^2+2*d*c^4*(x*(a*x+b))^(3/2)*a^(5/2
)*((a*d-b*c)*d/c^2)^(1/2)+2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(7/2)*x*
c^3*d^2-6*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(5/2)*x*b*c^4*d+2*(x*(a*x+
b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(3/2)*x*b^2*c^5+7*ln(1/2*(2*(x*(a*x+b))^(1/2
)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*a^3*b*c^2*d^3-2*ln(1/2*(2*(x
*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*a^2*x*b^2*c^4*
d-4*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d)
)*a^(9/2)*x*c*d^4+9*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*
c*x-b*d)/(c*x+d))*a^(7/2)*x*b*c^2*d^3-5*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2
)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(5/2)*x*b^2*c^3*d^2+4*(x*(a*x+b))^(1/2)*
((a*d-b*c)*d/c^2)^(1/2)*a^(7/2)*c^2*d^3-6*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1
/2)*a^(5/2)*b*c^3*d^2+2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(3/2)*b^2*c^
4*d-2*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1
/2)*a^2*b^2*c^3*d^2-ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-
b*c)*d/c^2)^(1/2)*x*a*b^3*c^5-4*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*
c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(9/2)*d^5+9*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d
/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(7/2)*b*c*d^4-5*ln((2*(x*(a*x+b))^(1
/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(5/2)*b^2*c^2*d^3-ln
(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*a*b^
3*c^4*d)/(x*(a*x+b))^(1/2)/a^(5/2)/(a*d-b*c)^2/c^4/((a*d-b*c)*d/c^2)^(1/2)/(c*x+
d)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x)*(c + d/x)^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.359771, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x)*(c + d/x)^2),x, algorithm="fricas")
[Out]
[1/2*((5*a*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(a)*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*((b*c^3 - a*c^2*d)*x^2 + (b*c^2*d - 2*a*c*d^2)
*x)*sqrt(a)*sqrt((a*x + b)/x) + (b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3
+ 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(
a)))/((a*b*c^4*d - a^2*c^3*d^2 + (a*b*c^5 - a^2*c^4*d)*x)*sqrt(a)), -1/2*(2*(5*a
*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(a)*sqrt(d/(b*c - a*d)
)*arctan(-(b*c - a*d)*sqrt(d/(b*c - a*d))/(d*sqrt((a*x + b)/x))) - 2*((b*c^3 - a
*c^2*d)*x^2 + (b*c^2*d - 2*a*c*d^2)*x)*sqrt(a)*sqrt((a*x + b)/x) - (b^2*c^2*d +
3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*log(-2*a*x*sq
rt((a*x + b)/x) + (2*a*x + b)*sqrt(a)))/((a*b*c^4*d - a^2*c^3*d^2 + (a*b*c^5 - a
^2*c^4*d)*x)*sqrt(a)), 1/2*((5*a*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^
2)*x)*sqrt(-a)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*s
qrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) + 2*((b*c^3 - a*c^2*d)*x^2
+ (b*c^2*d - 2*a*c*d^2)*x)*sqrt(-a)*sqrt((a*x + b)/x) + 2*(b^2*c^2*d + 3*a*b*c*d
^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*arctan(a/(sqrt(-a)*sqr
t((a*x + b)/x))))/((a*b*c^4*d - a^2*c^3*d^2 + (a*b*c^5 - a^2*c^4*d)*x)*sqrt(-a))
, -((5*a*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(-a)*sqrt(d/(b
*c - a*d))*arctan(-(b*c - a*d)*sqrt(d/(b*c - a*d))/(d*sqrt((a*x + b)/x))) - ((b*
c^3 - a*c^2*d)*x^2 + (b*c^2*d - 2*a*c*d^2)*x)*sqrt(-a)*sqrt((a*x + b)/x) - (b^2*
c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*arcta
n(a/(sqrt(-a)*sqrt((a*x + b)/x))))/((a*b*c^4*d - a^2*c^3*d^2 + (a*b*c^5 - a^2*c^
4*d)*x)*sqrt(-a))]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c+d/x)**2/(a+b/x)**(1/2),x)
[Out]
Exception raised: ValueError
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GIAC/XCAS [A] time = 0.25707, size = 393, normalized size = 2.28 \[ -b{\left (\frac{{\left (5 \, b c d^{2} - 4 \, a d^{3}\right )} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{2} c^{4} - a b c^{3} d\right )} \sqrt{b c d - a d^{2}}} + \frac{b^{2} c^{2} \sqrt{\frac{a x + b}{x}} - 2 \, a b c d \sqrt{\frac{a x + b}{x}} + 2 \, a^{2} d^{2} \sqrt{\frac{a x + b}{x}} + \frac{{\left (a x + b\right )} b c d \sqrt{\frac{a x + b}{x}}}{x} - \frac{2 \,{\left (a x + b\right )} a d^{2} \sqrt{\frac{a x + b}{x}}}{x}}{{\left (a b c^{3} - a^{2} c^{2} d\right )}{\left (a b c - a^{2} d - \frac{{\left (a x + b\right )} b c}{x} + \frac{2 \,{\left (a x + b\right )} a d}{x} - \frac{{\left (a x + b\right )}^{2} d}{x^{2}}\right )}} - \frac{{\left (b c + 4 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b c^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x)*(c + d/x)^2),x, algorithm="giac")
[Out]
-b*((5*b*c*d^2 - 4*a*d^3)*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/((b^2*
c^4 - a*b*c^3*d)*sqrt(b*c*d - a*d^2)) + (b^2*c^2*sqrt((a*x + b)/x) - 2*a*b*c*d*s
qrt((a*x + b)/x) + 2*a^2*d^2*sqrt((a*x + b)/x) + (a*x + b)*b*c*d*sqrt((a*x + b)/
x)/x - 2*(a*x + b)*a*d^2*sqrt((a*x + b)/x)/x)/((a*b*c^3 - a^2*c^2*d)*(a*b*c - a^
2*d - (a*x + b)*b*c/x + 2*(a*x + b)*a*d/x - (a*x + b)^2*d/x^2)) - (b*c + 4*a*d)*
arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a*b*c^3))