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

Optimal. Leaf size=41 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.11072, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.63461, size = 36, normalized size = 0.88 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{\frac{a + b x}{c + d x}}}{\sqrt{b}} \right )}}{\sqrt{b} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.049, size = 89, normalized size = 2.17 \[ \frac{\sqrt{c+d x} \sqrt{\frac{a+b x}{c+d x}} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b} \sqrt{d} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)/(c + d*x)]*Sqrt[c + d*x]*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqr
t[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(Sqrt[b]*Sqrt[d]*Sqrt[a + b*x])

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Maple [B]  time = 0.034, size = 80, normalized size = 2. \[{(dx+c)\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) \sqrt{{\frac{bx+a}{dx+c}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279138, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left ({\left (2 \, b d x + b c + a d\right )} \sqrt{b d} + 2 \,{\left (b d^{2} x + b c d\right )} \sqrt{\frac{b x + a}{d x + c}}\right )}{\sqrt{b d}}, -\frac{2 \, \arctan \left (\frac{b}{\sqrt{-b d} \sqrt{\frac{b x + a}{d x + c}}}\right )}{\sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296455, size = 100, normalized size = 2.44 \[ -\frac{\sqrt{b d}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b d} x - \sqrt{b d x^{2} + b c x + a d x + a c}\right )} b d - \sqrt{b d} b c - \sqrt{b d} a d \right |}\right ){\rm sign}\left (d x + c\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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 - s
qrt(b*d)*b*c - sqrt(b*d)*a*d))*sign(d*x + c)/(b*d)