3.1522 \(\int \frac{1}{\sqrt{a+b x} \sqrt{4+a+b x}} \, dx\)

Optimal. Leaf size=19 \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{a+b x}\right )}{b} \]

[Out]

(2*ArcSinh[Sqrt[a + b*x]/2])/b

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Rubi [A]  time = 0.0270635, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*x]*Sqrt[4 + a + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[a + b*x]/2])/b

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Rubi in Sympy [A]  time = 6.96682, size = 14, normalized size = 0.74 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a + b x}}{2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/2)/(b*x+a+4)**(1/2),x)

[Out]

2*asinh(sqrt(a + b*x)/2)/b

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Mathematica [A]  time = 0.0114375, size = 19, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b*x]*Sqrt[4 + a + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[a + b*x]/2])/b

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Maple [B]  time = 0.012, size = 86, normalized size = 4.5 \[{1\sqrt{ \left ( bx+a \right ) \left ( bx+a+4 \right ) }\ln \left ({1 \left ({\frac{ab}{2}}+{\frac{b \left ( a+4 \right ) }{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( ab+b \left ( a+4 \right ) \right ) x+a \left ( a+4 \right ) } \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{bx+a+4}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/2)/(b*x+a+4)^(1/2),x)

[Out]

((b*x+a)*(b*x+a+4))^(1/2)/(b*x+a)^(1/2)/(b*x+a+4)^(1/2)*ln((1/2*a*b+1/2*b*(a+4)+
b^2*x)/(b^2)^(1/2)+(b^2*x^2+(a*b+b*(a+4))*x+a*(a+4))^(1/2))/(b^2)^(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(1/(sqrt(b*x + a + 4)*sqrt(b*x + a)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226534, size = 42, normalized size = 2.21 \[ -\frac{\log \left (-b x + \sqrt{b x + a + 4} \sqrt{b x + a} - a - 2\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a + 4)*sqrt(b*x + a)),x, algorithm="fricas")

[Out]

-log(-b*x + sqrt(b*x + a + 4)*sqrt(b*x + a) - a - 2)/b

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Sympy [A]  time = 3.24125, size = 19, normalized size = 1. \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{\frac{a}{b} + x}}{2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(1/2)/(b*x+a+4)**(1/2),x)

[Out]

2*asinh(sqrt(b)*sqrt(a/b + x)/2)/b

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GIAC/XCAS [A]  time = 0.331838, size = 34, normalized size = 1.79 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + a + 4} + \sqrt{b x + a} \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a + 4)*sqrt(b*x + a)),x, algorithm="giac")

[Out]

-2*ln(abs(-sqrt(b*x + a + 4) + sqrt(b*x + a)))/b