3.2798 \(\int \frac{1}{\sqrt{c (a+b x)^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{c (a+b x)^2}} \]

[Out]

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

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Rubi [A]  time = 0.0288452, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{c (a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[c*(a + b*x)^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.03479, size = 37, normalized size = 1.32 \[ \frac{\left (a + b x\right ) \log{\left (a + b x \right )}}{b \sqrt{a^{2} c + 2 a b c x + b^{2} c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)*log(a + b*x)/(b*sqrt(a**2*c + 2*a*b*c*x + b**2*c*x**2))

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Mathematica [A]  time = 0.0108394, size = 28, normalized size = 1. \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{c (a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 27, normalized size = 1. \[{\frac{ \left ( bx+a \right ) \ln \left ( bx+a \right ) }{b}{\frac{1}{\sqrt{c \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*(b*x+a)^2)^(1/2),x)

[Out]

(b*x+a)*ln(b*x+a)/b/(c*(b*x+a)^2)^(1/2)

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Maxima [A]  time = 1.35333, size = 24, normalized size = 0.86 \[ \sqrt{\frac{1}{b^{2} c}} \log \left (x + \frac{a}{b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((b*x + a)^2*c),x, algorithm="maxima")

[Out]

sqrt(1/(b^2*c))*log(x + a/b)

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Fricas [A]  time = 0.213904, size = 57, normalized size = 2.04 \[ \frac{\sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c} \log \left (b x + a\right )}{b^{2} c x + a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((b*x + a)^2*c),x, algorithm="fricas")

[Out]

sqrt(b^2*c*x^2 + 2*a*b*c*x + a^2*c)*log(b*x + a)/(b^2*c*x + a*b*c)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c \left (a + b x\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(c*(a + b*x)**2), x)

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GIAC/XCAS [A]  time = 0.218425, size = 45, normalized size = 1.61 \[ \frac{{\rm ln}\left (\sqrt{c}{\left | b x + a \right |}{\left |{\rm sign}\left (b x + a\right ) \right |}\right )}{b \sqrt{c}{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((b*x + a)^2*c),x, algorithm="giac")

[Out]

ln(sqrt(c)*abs(b*x + a)*abs(sign(b*x + a)))/(b*sqrt(c)*sign(b*x + a))