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 = 0.0777735, 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, Rules used = {1918, 1914, 621, 206} \[ \frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2} \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1918
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^2+b x^3+c x^4}}{x} \, dx &=\frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{\left (b^2-4 a c\right ) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{8 c}\\ &=\frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{\left (\left (b^2-4 a c\right ) x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 c \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{\left (\left (b^2-4 a c\right ) x \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{4 c \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{\left (b^2-4 a c\right ) x \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}}\\ \end{align*}
Mathematica [A] time = 0.14577, 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.
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Maple [A] time = 0.004, size = 146, normalized size = 1.2 \begin{align*}{\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66874, size = 502, normalized size = 4.22 \begin{align*} \left [-\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{c} x \log \left (-\frac{8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{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{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} \left (a + b x + c x^{2}\right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16641, size = 169, normalized size = 1.42 \begin{align*} \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 )} \log \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 )} \mathrm{sgn}\left (x\right ) - \frac{{\left (b^{2} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 4 \, a c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 2 \, \sqrt{a} b \sqrt{c}\right )} \mathrm{sgn}\left (x\right )}{8 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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