Optimal. Leaf size=38 \[ \frac{\sqrt{c x^2}}{b}-\frac{a \sqrt{c x^2} \log (a+b x)}{b^2 x} \]
[Out]
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Rubi [A] time = 0.0312646, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\sqrt{c x^2}}{b}-\frac{a \sqrt{c x^2} \log (a+b x)}{b^2 x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*x^2]/(a + b*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{2} x} + \frac{\sqrt{c x^{2}} \int \frac{1}{b}\, dx}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2)**(1/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0109076, size = 28, normalized size = 0.74 \[ \frac{c x (b x-a \log (a+b x))}{b^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*x^2]/(a + b*x),x]
[Out]
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Maple [A] time = 0.005, size = 29, normalized size = 0.8 \[ -{\frac{a\ln \left ( bx+a \right ) -bx}{{b}^{2}x}\sqrt{c{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2)^(1/2)/(b*x+a),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(sqrt(c*x^2)/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204184, size = 36, normalized size = 0.95 \[ \frac{\sqrt{c x^{2}}{\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2)**(1/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.207965, size = 50, normalized size = 1.32 \[ \sqrt{c}{\left (\frac{x{\rm sign}\left (x\right )}{b} - \frac{a{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{2}} + \frac{a{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)/(b*x + a),x, algorithm="giac")
[Out]