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

Optimal. Leaf size=104 \[ \frac{2 \sqrt{d} \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2}+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a} c^2}+\frac{x \sqrt{a+\frac{b}{x}}}{c} \]

[Out]

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

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Rubi [A]  time = 0.354658, antiderivative size = 104, 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 \[ \frac{2 \sqrt{d} \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2}+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a} c^2}+\frac{x \sqrt{a+\frac{b}{x}}}{c} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/(c + d/x),x]

[Out]

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

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Rubi in Sympy [A]  time = 41.2538, size = 90, normalized size = 0.87 \[ \frac{x \sqrt{a + \frac{b}{x}}}{c} + \frac{2 \sqrt{d} \sqrt{a d - b c} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{2}} - \frac{2 \left (a d - \frac{b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{\sqrt{a} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.358711, size = 153, normalized size = 1.47 \[ \frac{2 \sqrt{d} \sqrt{a d-b c} \log (c x+d)+\frac{(b c-2 a d) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{\sqrt{a}}-2 \sqrt{d} \sqrt{a d-b c} \log \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{a d-b c}-2 a d x+b c x-b d\right )+2 c x \sqrt{a+\frac{b}{x}}}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.03, size = 286, normalized size = 2.8 \[{\frac{x}{2\,{c}^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 2\,\sqrt{x \left ( ax+b \right ) }{c}^{2}\sqrt{a}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}-2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) \sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}acd+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) \sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}b{c}^{2}-2\,\ln \left ({\frac{1}{cx+d} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}c-2\,adx+bcx-bd \right ) } \right ){a}^{3/2}{d}^{2}+2\,\ln \left ({\frac{1}{cx+d} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}c-2\,adx+bcx-bd \right ) } \right ) \sqrt{a}bcd \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/(c+d/x),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288149, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{a} c x \sqrt{\frac{a x + b}{x}} -{\left (b c - 2 \, a d\right )} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \, \sqrt{-b c d + a d^{2}} \sqrt{a} \log \left (\frac{b d -{\left (b c - 2 \, a d\right )} x + 2 \, \sqrt{-b c d + a d^{2}} x \sqrt{\frac{a x + b}{x}}}{c x + d}\right )}{2 \, \sqrt{a} c^{2}}, \frac{2 \, \sqrt{a} c x \sqrt{\frac{a x + b}{x}} - 4 \, \sqrt{b c d - a d^{2}} \sqrt{a} \arctan \left (\frac{\sqrt{b c d - a d^{2}}}{d \sqrt{\frac{a x + b}{x}}}\right ) -{\left (b c - 2 \, a d\right )} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{2 \, \sqrt{a} c^{2}}, \frac{\sqrt{-a} c x \sqrt{\frac{a x + b}{x}} -{\left (b c - 2 \, a d\right )} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) + \sqrt{-b c d + a d^{2}} \sqrt{-a} \log \left (\frac{b d -{\left (b c - 2 \, a d\right )} x + 2 \, \sqrt{-b c d + a d^{2}} x \sqrt{\frac{a x + b}{x}}}{c x + d}\right )}{\sqrt{-a} c^{2}}, \frac{\sqrt{-a} c x \sqrt{\frac{a x + b}{x}} -{\left (b c - 2 \, a d\right )} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) - 2 \, \sqrt{b c d - a d^{2}} \sqrt{-a} \arctan \left (\frac{\sqrt{b c d - a d^{2}}}{d \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(a + b/x)/(c*x + d), 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),x, algorithm="giac")

[Out]

Exception raised: TypeError