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