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

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]

(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]])/(Sqrt[a]*c^3)

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Rubi [A]  time = 0.606933, antiderivative size = 147, 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 \[ \frac{\sqrt{d} (3 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 \sqrt{b c-a d}}+\frac{(b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a} c^3}+\frac{2 d \sqrt{a+\frac{b}{x}}}{c^2 \left (c+\frac{d}{x}\right )}+\frac{x \sqrt{a+\frac{b}{x}}}{c \left (c+\frac{d}{x}\right )} \]

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]])/(Sqrt[a]*c^3)

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Rubi in Sympy [A]  time = 67.7229, size = 122, normalized size = 0.83 \[ \frac{x \sqrt{a + \frac{b}{x}}}{c \left (c + \frac{d}{x}\right )} + \frac{2 d \sqrt{a + \frac{b}{x}}}{c^{2} \left (c + \frac{d}{x}\right )} + \frac{\sqrt{d} \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{3} \sqrt{a d - b c}} - \frac{\left (4 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{\sqrt{a} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(1/2)/(c+d/x)**2,x)

[Out]

x*sqrt(a + b/x)/(c*(c + d/x)) + 2*d*sqrt(a + b/x)/(c**2*(c + d/x)) + sqrt(d)*(4*
a*d - 3*b*c)*atanh(sqrt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(c**3*sqrt(a*d - b*c))
- (4*a*d - b*c)*atanh(sqrt(a + b/x)/sqrt(a))/(sqrt(a)*c**3)

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Mathematica [C]  time = 0.599008, size = 197, normalized size = 1.34 \[ \frac{\frac{i \sqrt{d} (3 b c-4 a d) \log \left (-\frac{2 i c^4 \left (-2 i \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 a d x+b c x-b d\right )}{d^{3/2} (c x+d) (3 b c-4 a d) \sqrt{b c-a d}}\right )}{\sqrt{b c-a d}}+\frac{2 c x \sqrt{a+\frac{b}{x}} (c x+2 d)}{c x+d}+\frac{(b c-4 a d) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{\sqrt{a}}}{2 c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.028, size = 939, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/(c+d/x)^2,x)

[Out]

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^2*((a*d-b*c)*d/c^2)^(1/2)*x*c^2*d^2-2*(x*(a*x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d
/c^2)^(1/2)*x^2*c^4-4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*
((a*d-b*c)*d/c^2)^(1/2)*c*d^3+5*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(
1/2))*a*((a*d-b*c)*d/c^2)^(1/2)*x*b*c^3*d+2*c^4*(x*(a*x+b))^(3/2)*a^(1/2)*((a*d-
b*c)*d/c^2)^(1/2)+2*(x*(a*x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x*c^3*d-4*
(x*(a*x+b))^(1/2)*a^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b*c^4+5*ln(1/2*(2*(x*(a*x+b)
)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*((a*d-b*c)*d/c^2)^(1/2)*b*c^2*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*b^2*c^4-4*
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^
(5/2)*x*c*d^3+7*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^(3/2)*x*b*c^2*d^2-3*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^(1/2)*x*b^2*c^3*d+4*(x*(a*x+b))^(1/2)*a^(3/2
)*((a*d-b*c)*d/c^2)^(1/2)*c^2*d^2-4*(x*(a*x+b))^(1/2)*a^(1/2)*((a*d-b*c)*d/c^2)^
(1/2)*b*c^3*d-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)*b^2*c^3*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^(5/2)*d^4+7*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^(3/2)*b*c*d^3-3*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^(1/2)*b^2*c^2*d^2)/(x*(a*x+b
))^(1/2)/c^4/a^(1/2)/((a*d-b*c)*d/c^2)^(1/2)/(a*d-b*c)/(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(sqrt(a + b/x)/(c + d/x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300437, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/(c + d/x)^2,x, algorithm="fricas")

[Out]

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

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

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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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/(c + d/x)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError