3.169 \(\int \frac{\sqrt{a+\frac{b}{x}}}{\left (c+\frac{d}{x}\right )^{3/2}} \, dx\)

Optimal. Leaf size=122 \[ \frac{(b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{5/2}}-\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{a c^2 \sqrt{c+\frac{d}{x}}}+\frac{x \left (a+\frac{b}{x}\right )^{3/2}}{a c \sqrt{c+\frac{d}{x}}} \]

[Out]

-(((b*c - 3*a*d)*Sqrt[a + b/x])/(a*c^2*Sqrt[c + d/x])) + ((a + b/x)^(3/2)*x)/(a*
c*Sqrt[c + d/x]) + ((b*c - 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt[
c + d/x])])/(Sqrt[a]*c^(5/2))

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Rubi [A]  time = 0.300756, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{(b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{5/2}}-\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{a c^2 \sqrt{c+\frac{d}{x}}}+\frac{x \left (a+\frac{b}{x}\right )^{3/2}}{a c \sqrt{c+\frac{d}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/(c + d/x)^(3/2),x]

[Out]

-(((b*c - 3*a*d)*Sqrt[a + b/x])/(a*c^2*Sqrt[c + d/x])) + ((a + b/x)^(3/2)*x)/(a*
c*Sqrt[c + d/x]) + ((b*c - 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt[
c + d/x])])/(Sqrt[a]*c^(5/2))

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Rubi in Sympy [A]  time = 24.1821, size = 114, normalized size = 0.93 \[ - \frac{2 d x \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{c \sqrt{c + \frac{d}{x}} \left (a d - b c\right )} + \frac{x \sqrt{a + \frac{b}{x}} \sqrt{c + \frac{d}{x}} \left (3 a d - b c\right )}{c^{2} \left (a d - b c\right )} - \frac{\left (3 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + \frac{b}{x}}}{\sqrt{a} \sqrt{c + \frac{d}{x}}} \right )}}{\sqrt{a} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(1/2)/(c+d/x)**(3/2),x)

[Out]

-2*d*x*(a + b/x)**(3/2)/(c*sqrt(c + d/x)*(a*d - b*c)) + x*sqrt(a + b/x)*sqrt(c +
d/x)*(3*a*d - b*c)/(c**2*(a*d - b*c)) - (3*a*d - b*c)*atanh(sqrt(c)*sqrt(a + b/
x)/(sqrt(a)*sqrt(c + d/x)))/(sqrt(a)*c**(5/2))

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Mathematica [A]  time = 0.214013, size = 112, normalized size = 0.92 \[ \frac{(b c-3 a d) \log \left (2 \sqrt{a} \sqrt{c} x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+2 a c x+a d+b c\right )}{2 \sqrt{a} c^{5/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}} (c x+3 d)}{c^2 (c x+d)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]/(c + d/x)^(3/2),x]

[Out]

(Sqrt[a + b/x]*Sqrt[c + d/x]*x*(3*d + c*x))/(c^2*(d + c*x)) + ((b*c - 3*a*d)*Log
[b*c + a*d + 2*a*c*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b/x]*Sqrt[c + d/x]*x])/(2*Sqrt
[a]*c^(5/2))

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Maple [B]  time = 0.049, size = 280, normalized size = 2.3 \[{\frac{x}{ \left ( 2\,cx+2\,d \right ){c}^{2}}\sqrt{{\frac{ax+b}{x}}}\sqrt{{\frac{cx+d}{x}}} \left ( -3\,\ln \left ( 1/2\,{\frac{2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc}{\sqrt{ac}}} \right ) xacd+\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) xb{c}^{2}+2\,xc\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}-3\,\ln \left ( 1/2\,{\frac{2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc}{\sqrt{ac}}} \right ) a{d}^{2}+\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) bcd+6\,d\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/(c+d/x)^(3/2),x)

[Out]

1/2*((a*x+b)/x)^(1/2)*x*((c*x+d)/x)^(1/2)*(-3*ln(1/2*(2*a*c*x+2*((c*x+d)*(a*x+b)
)^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*x*a*c*d+ln(1/2*(2*a*c*x+2*((c*x+d)*(a*
x+b))^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*x*b*c^2+2*x*c*((c*x+d)*(a*x+b))^(1
/2)*(a*c)^(1/2)-3*ln(1/2*(2*a*c*x+2*((c*x+d)*(a*x+b))^(1/2)*(a*c)^(1/2)+a*d+b*c)
/(a*c)^(1/2))*a*d^2+ln(1/2*(2*a*c*x+2*((c*x+d)*(a*x+b))^(1/2)*(a*c)^(1/2)+a*d+b*
c)/(a*c)^(1/2))*b*c*d+6*d*((c*x+d)*(a*x+b))^(1/2)*(a*c)^(1/2))/(c*x+d)/(a*c)^(1/
2)/((c*x+d)*(a*x+b))^(1/2)/c^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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.328971, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (c x^{2} + 3 \, d x\right )} \sqrt{a c} \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} -{\left (b c d - 3 \, a d^{2} +{\left (b c^{2} - 3 \, a c d\right )} x\right )} \log \left (4 \,{\left (2 \, a^{2} c^{2} x^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} -{\left (8 \, a^{2} c^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}\right )}{4 \,{\left (c^{3} x + c^{2} d\right )} \sqrt{a c}}, \frac{2 \,{\left (c x^{2} + 3 \, d x\right )} \sqrt{-a c} \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} +{\left (b c d - 3 \, a d^{2} +{\left (b c^{2} - 3 \, a c d\right )} x\right )} \arctan \left (\frac{2 \, \sqrt{-a c} x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}}}{2 \, a c x + b c + a d}\right )}{2 \,{\left (c^{3} x + c^{2} d\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/(c + d/x)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(4*(c*x^2 + 3*d*x)*sqrt(a*c)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - (b*c*d -
3*a*d^2 + (b*c^2 - 3*a*c*d)*x)*log(4*(2*a^2*c^2*x^2 + (a*b*c^2 + a^2*c*d)*x)*sq
rt((a*x + b)/x)*sqrt((c*x + d)/x) - (8*a^2*c^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d
^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c)))/((c^3*x + c^2*d)*sqrt(a*c)), 1/2*(2*(c
*x^2 + 3*d*x)*sqrt(-a*c)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + (b*c*d - 3*a*d^2
+ (b*c^2 - 3*a*c*d)*x)*arctan(2*sqrt(-a*c)*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)
/(2*a*c*x + b*c + a*d)))/((c^3*x + c^2*d)*sqrt(-a*c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + \frac{b}{x}}}{\left (c + \frac{d}{x}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(1/2)/(c+d/x)**(3/2),x)

[Out]

Integral(sqrt(a + b/x)/(c + d/x)**(3/2), 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)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError