3.1464 \(\int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=72 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} \sqrt{d}}+\frac{\sqrt{a+b x} \sqrt{c+d x}}{b} \]

[Out]

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

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Rubi [A]  time = 0.0776794, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} \sqrt{d}}+\frac{\sqrt{a+b x} \sqrt{c+d x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.0075, size = 63, normalized size = 0.88 \[ \frac{\sqrt{a + b x} \sqrt{c + d x}}{b} - \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{3}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0627237, size = 88, normalized size = 1.22 \[ \frac{(b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{3/2} \sqrt{d}}+\frac{\sqrt{a+b x} \sqrt{c+d x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 107, normalized size = 1.5 \[{\frac{1}{b}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{ad-bc}{2\,b}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262102, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c - a d\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right ) - 4 \, \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c}}{4 \, \sqrt{b d} b}, \frac{{\left (b c - a d\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right ) + 2 \, \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \, \sqrt{-b d} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c + d*x)/sqrt(a + b*x), x)

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GIAC/XCAS [A]  time = 0.232225, size = 126, normalized size = 1.75 \[ -\frac{{\left (\frac{{\left (b^{2} c - a b d\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d}} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}\right )}{\left | b \right |}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-((b^2*c - a*b*d)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d -
a*b*d)))/sqrt(b*d) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a))*abs(b)/
b^3