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

Optimal. Leaf size=76 \[ \frac{(c+d x) \sqrt{\frac{a+b x}{c+d x}}}{d}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right )}{\sqrt{b} d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0879393, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{(c+d x) \sqrt{\frac{a+b x}{c+d x}}}{d}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right )}{\sqrt{b} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.92557, size = 76, normalized size = 1. \[ - \frac{\sqrt{\frac{a + b x}{c + d x}} \left (a d - b c\right )}{d \left (b - \frac{d \left (a + b x\right )}{c + d x}\right )} + \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{\frac{a + b x}{c + d x}}}{\sqrt{b}} \right )}}{\sqrt{b} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.112044, size = 127, normalized size = 1.67 \[ \frac{\sqrt{c+d x} (a d-b c) \sqrt{\frac{a+b x}{c+d x}} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 \sqrt{b} d^{3/2} \sqrt{a+b x}}+\frac{(c+d x) \sqrt{\frac{a+b x}{c+d x}}}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 152, normalized size = 2. \[{\frac{dx+c}{2\,d}\sqrt{{\frac{bx+a}{dx+c}}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) ad-\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) bc+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)/(d*x+c))^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291685, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b d}{\left (d x + c\right )} \sqrt{\frac{b x + a}{d x + c}} -{\left (b c - a d\right )} \log \left ({\left (2 \, b d x + b c + a d\right )} \sqrt{b d} + 2 \,{\left (b d^{2} x + b c d\right )} \sqrt{\frac{b x + a}{d x + c}}\right )}{2 \, \sqrt{b d} d}, \frac{\sqrt{-b d}{\left (d x + c\right )} \sqrt{\frac{b x + a}{d x + c}} +{\left (b c - a d\right )} \arctan \left (\frac{b}{\sqrt{-b d} \sqrt{\frac{b x + a}{d x + c}}}\right )}{\sqrt{-b d} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296043, size = 161, normalized size = 2.12 \[ \frac{\sqrt{b d x^{2} + b c x + a d x + a c}{\rm sign}\left (d x + c\right )}{d} + \frac{{\left (b c{\rm sign}\left (d x + c\right ) - a d{\rm sign}\left (d x + c\right )\right )} \sqrt{b d}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b d} x - \sqrt{b d x^{2} + b c x + a d x + a c}\right )} b d - \sqrt{b d} b c - \sqrt{b d} a d \right |}\right )}{2 \, b d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*d*x^2 + b*c*x + a*d*x + a*c)*sign(d*x + c)/d + 1/2*(b*c*sign(d*x + c) - a
*d*sign(d*x + c))*sqrt(b*d)*ln(abs(-2*(sqrt(b*d)*x - sqrt(b*d*x^2 + b*c*x + a*d*
x + a*c))*b*d - sqrt(b*d)*b*c - sqrt(b*d)*a*d))/(b*d^2)