Optimal. Leaf size=76 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}-\frac{3 b \sqrt{b x+c x^2}}{4 c^2}+\frac{x \sqrt{b x+c x^2}}{2 c} \]
[Out]
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Rubi [A] time = 0.0877493, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}-\frac{3 b \sqrt{b x+c x^2}}{4 c^2}+\frac{x \sqrt{b x+c x^2}}{2 c} \]
Antiderivative was successfully verified.
[In] Int[x^2/Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 9.83411, size = 68, normalized size = 0.89 \[ \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 c^{\frac{5}{2}}} - \frac{3 b \sqrt{b x + c x^{2}}}{4 c^{2}} + \frac{x \sqrt{b x + c x^{2}}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0552627, size = 90, normalized size = 1.18 \[ \frac{\sqrt{c} x \left (-3 b^2-b c x+2 c^2 x^2\right )+3 b^2 \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{4 c^{5/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/Sqrt[b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.007, size = 68, normalized size = 0.9 \[{\frac{x}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,b}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(c*x^2+b*x)^(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^2/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226823, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x - 3 \, b\right )} \sqrt{c}}{8 \, c^{\frac{5}{2}}}, \frac{3 \, b^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x}{\left (2 \, c x - 3 \, b\right )} \sqrt{-c}}{4 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(c*x^2 + b*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227192, size = 88, normalized size = 1.16 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (\frac{2 \, x}{c} - \frac{3 \, b}{c^{2}}\right )} - \frac{3 \, b^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]