Optimal. Leaf size=119 \[ \frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2} \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.131767, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2} \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a*x^2 + b*x^3 + c*x^4]/x,x]
[Out]
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Rubi in Sympy [A] time = 19.825, size = 107, normalized size = 0.9 \[ \frac{\left (b + 2 c x\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{4 c x} - \frac{x \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**3+a*x**2)**(1/2)/x,x)
[Out]
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Mathematica [A] time = 0.236237, size = 100, normalized size = 0.84 \[ \frac{x \left (2 \sqrt{c} (b+2 c x) (a+x (b+c x))-\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{8 c^{3/2} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x,x]
[Out]
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Maple [A] time = 0.007, size = 146, normalized size = 1.2 \[{\frac{1}{8\,x}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 4\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x+2\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}b+4\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{c}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b \right ){\frac{1}{\sqrt{c}}}} \right ){b}^{2}c \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^3+a*x^2)^(1/2)/x,x)
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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^4 + b*x^3 + a*x^2)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285538, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{c} x \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} +{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )}}{16 \, c^{2} x}, \frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) + 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )}}{8 \, c^{2} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^3 + a*x^2)/x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (a + b x + c x^{2}\right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+b*x**3+a*x**2)**(1/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.294393, size = 169, normalized size = 1.42 \[ \frac{1}{8} \,{\left (2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, x + \frac{b}{c}\right )} + \frac{{\left (b^{2} - 4 \, a c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}}\right )}{\rm sign}\left (x\right ) - \frac{{\left (b^{2}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 4 \, a c{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 2 \, \sqrt{a} b \sqrt{c}\right )}{\rm sign}\left (x\right )}{8 \, c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^3 + a*x^2)/x,x, algorithm="giac")
[Out]