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

Optimal. Leaf size=20 \[ \frac{x \log (a+b x)}{b \sqrt{c x^2}} \]

[Out]

(x*Log[a + b*x])/(b*Sqrt[c*x^2])

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Rubi [A]  time = 0.0104734, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x \log (a+b x)}{b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Log[a + b*x])/(b*Sqrt[c*x^2])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(sqrt(c*x**2)*(a + b*x)), x)

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Mathematica [A]  time = 0.00359213, size = 20, normalized size = 1. \[ \frac{x \log (a+b x)}{b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Log[a + b*x])/(b*Sqrt[c*x^2])

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Maple [A]  time = 0.004, size = 19, normalized size = 1. \[{\frac{x\ln \left ( bx+a \right ) }{b}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x+a)/(c*x^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.209833, size = 31, normalized size = 1.55 \[ \frac{\sqrt{c x^{2}} \log \left (b x + a\right )}{b c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*log(b*x + a)/(b*c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(sqrt(c*x**2)*(a + b*x)), x)

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GIAC/XCAS [A]  time = 0.212909, size = 47, normalized size = 2.35 \[ -\frac{{\rm ln}\left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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