Optimal. Leaf size=65 \[ -\frac{a^2 \sqrt{c x^2}}{b^3 x (a+b x)}-\frac{2 a \sqrt{c x^2} \log (a+b x)}{b^3 x}+\frac{\sqrt{c x^2}}{b^2} \]
[Out]
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Rubi [A] time = 0.056326, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{a^2 \sqrt{c x^2}}{b^3 x (a+b x)}-\frac{2 a \sqrt{c x^2} \log (a+b x)}{b^3 x}+\frac{\sqrt{c x^2}}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(x*Sqrt[c*x^2])/(a + b*x)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sqrt{c x^{2}}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(c*x**2)**(1/2)/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.0299677, size = 53, normalized size = 0.82 \[ \frac{c x \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(x*Sqrt[c*x^2])/(a + b*x)^2,x]
[Out]
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Maple [A] time = 0.007, size = 62, normalized size = 1. \[ -{\frac{2\,\ln \left ( bx+a \right ) xab-{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -abx+{a}^{2}}{x \left ( bx+a \right ){b}^{3}}\sqrt{c{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c*x^2)^(1/2)/(b*x+a)^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(sqrt(c*x^2)*x/(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209758, size = 77, normalized size = 1.18 \[ \frac{{\left (b^{2} x^{2} + a b x - a^{2} - 2 \,{\left (a b x + a^{2}\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{b^{4} x^{2} + a b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*x/(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sqrt{c x^{2}}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c*x**2)**(1/2)/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.206885, size = 78, normalized size = 1.2 \[ \sqrt{c}{\left (\frac{x{\rm sign}\left (x\right )}{b^{2}} - \frac{2 \, a{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{3}} + \frac{{\left (2 \, a{\rm ln}\left ({\left | a \right |}\right ) + a\right )}{\rm sign}\left (x\right )}{b^{3}} - \frac{a^{2}{\rm sign}\left (x\right )}{{\left (b x + a\right )} b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*x/(b*x + a)^2,x, algorithm="giac")
[Out]