3.150 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )} \, dx\)

Optimal. Leaf size=108 \[ -\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2} c^2}-\frac{2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 \sqrt{b c-a d}}+\frac{x \sqrt{a+\frac{b}{x}}}{a c} \]

[Out]

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

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Rubi [A]  time = 0.316674, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2} c^2}-\frac{2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 \sqrt{b c-a d}}+\frac{x \sqrt{a+\frac{b}{x}}}{a c} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x]*(c + d/x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 40.7588, size = 92, normalized size = 0.85 \[ \frac{2 d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{2} \sqrt{a d - b c}} + \frac{x \sqrt{a + \frac{b}{x}}}{a c} - \frac{2 \left (a d + \frac{b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.467142, size = 157, normalized size = 1.45 \[ \frac{-\frac{(2 a d+b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{3/2}}+\frac{2 d^{3/2} \log (c x+d)}{\sqrt{a d-b c}}-\frac{2 d^{3/2} \log \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{a d-b c}-2 a d x+b c x-b d\right )}{\sqrt{a d-b c}}+\frac{2 c x \sqrt{a+\frac{b}{x}}}{a}}{2 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x]*(c + d/x)),x]

[Out]

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

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Maple [B]  time = 0.019, size = 389, normalized size = 3.6 \[{\frac{x}{2\,{c}^{3} \left ( ad-bc \right ) }\sqrt{{\frac{ax+b}{x}}} \left ( -2\,{d}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}c\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}+2\,d\sqrt{x \left ( ax+b \right ) }{c}^{2}{a}^{5/2}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}-2\,b\sqrt{x \left ( ax+b \right ) }{c}^{3}{a}^{3/2}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}-2\,{d}^{3}\ln \left ({\frac{1}{cx+d} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}c-2\,adx+bcx-bd \right ) } \right ){a}^{7/2}+2\,{d}^{2}\ln \left ({\frac{1}{cx+d} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}c-2\,adx+bcx-bd \right ) } \right ) bc{a}^{5/2}+d\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) b{c}^{2}{a}^{2}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}+{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){c}^{3}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}a \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+d/x)/(a+b/x)^(1/2),x)

[Out]

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

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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)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277234, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a^{\frac{3}{2}} d \sqrt{-\frac{d}{b c - a d}} \log \left (-\frac{2 \,{\left (b c - a d\right )} x \sqrt{-\frac{d}{b c - a d}} \sqrt{\frac{a x + b}{x}} - b d +{\left (b c - 2 \, a d\right )} x}{c x + d}\right ) + 2 \, \sqrt{a} c x \sqrt{\frac{a x + b}{x}} +{\left (b c + 2 \, a d\right )} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{2 \, a^{\frac{3}{2}} c^{2}}, \frac{\sqrt{-a} a d \sqrt{-\frac{d}{b c - a d}} \log \left (-\frac{2 \,{\left (b c - a d\right )} x \sqrt{-\frac{d}{b c - a d}} \sqrt{\frac{a x + b}{x}} - b d +{\left (b c - 2 \, a d\right )} x}{c x + d}\right ) + \sqrt{-a} c x \sqrt{\frac{a x + b}{x}} +{\left (b c + 2 \, a d\right )} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a} a c^{2}}, -\frac{4 \, a^{\frac{3}{2}} d \sqrt{\frac{d}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{\frac{d}{b c - a d}}}{d \sqrt{\frac{a x + b}{x}}}\right ) - 2 \, \sqrt{a} c x \sqrt{\frac{a x + b}{x}} -{\left (b c + 2 \, a d\right )} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{2 \, a^{\frac{3}{2}} c^{2}}, -\frac{2 \, \sqrt{-a} a d \sqrt{\frac{d}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{\frac{d}{b c - a d}}}{d \sqrt{\frac{a x + b}{x}}}\right ) - \sqrt{-a} c x \sqrt{\frac{a x + b}{x}} -{\left (b c + 2 \, a d\right )} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a} a c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*(c + d/x)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.252986, size = 174, normalized size = 1.61 \[ -b{\left (\frac{2 \, d^{2} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{\sqrt{b c d - a d^{2}} b c^{2}} + \frac{\sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a c} - \frac{{\left (b c + 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*(c + d/x)),x, algorithm="giac")

[Out]

-b*(2*d^2*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^2)*b
*c^2) + sqrt((a*x + b)/x)/((a - (a*x + b)/x)*a*c) - (b*c + 2*a*d)*arctan(sqrt((a
*x + b)/x)/sqrt(-a))/(sqrt(-a)*a*b*c^2))