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

Optimal. Leaf size=51 \[ \frac{c x \sqrt{a+\frac{b}{x}}}{a}-\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.109843, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{c x \sqrt{a+\frac{b}{x}}}{a}-\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d/x)/Sqrt[a + b/x],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.44179, size = 42, normalized size = 0.82 \[ \frac{c x \sqrt{a + \frac{b}{x}}}{a} + \frac{2 \left (a d - \frac{b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0753147, size = 62, normalized size = 1.22 \[ \frac{(2 a d-b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{3/2}}+\frac{c x \sqrt{a+\frac{b}{x}}}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 179, normalized size = 3.5 \[ -{\frac{x}{2\,b}\sqrt{{\frac{ax+b}{x}}} \left ( 2\,\sqrt{x \left ( ax+b \right ) }d{a}^{5/2}-2\,\sqrt{x \left ( ax+b \right ) }cb{a}^{3/2}-2\,d\sqrt{a{x}^{2}+bx}{a}^{5/2}-d\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) b{a}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) db{a}^{2}+{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) ca \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)/(a+b/x)^(1/2),x)

[Out]

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

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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((c + d/x)/sqrt(a + b/x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249634, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{a} c x \sqrt{\frac{a x + b}{x}} -{\left (b c - 2 \, a d\right )} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{2 \, a^{\frac{3}{2}}}, \frac{\sqrt{-a} c x \sqrt{\frac{a x + b}{x}} +{\left (b c - 2 \, a d\right )} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c + d/x)/sqrt(a + b/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)))/a^(3/2), (sqrt(-a)*c*x*sqrt((a*x + b)/x) + (b*c - 2*a*
d)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))))/(sqrt(-a)*a)]

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Sympy [A]  time = 25.0676, size = 73, normalized size = 1.43 \[ \frac{\sqrt{b} c \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} + \frac{2 d \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a}} - \frac{b c \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*c*sqrt(x)*sqrt(a*x/b + 1)/a + 2*d*asinh(sqrt(a)*sqrt(x)/sqrt(b))/sqrt(a)
- b*c*asinh(sqrt(a)*sqrt(x)/sqrt(b))/a**(3/2)

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GIAC/XCAS [A]  time = 0.250741, size = 99, normalized size = 1.94 \[ -b{\left (\frac{c \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a} - \frac{{\left (b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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