Optimal. Leaf size=163 \[ \frac{b \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt{a+b x+c x^2}}-\frac{b (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{8 c^2 x}+\frac{\left (a+b x+c x^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3 c x} \]
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Rubi [A] time = 0.0576031, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1903, 640, 612, 621, 206} \[ \frac{b \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt{a+b x+c x^2}}-\frac{b (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{8 c^2 x}+\frac{\left (a+b x+c x^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3 c x} \]
Antiderivative was successfully verified.
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Rule 1903
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a x^2+b x^3+c x^4} \, dx &=\frac{\sqrt{a x^2+b x^3+c x^4} \int x \sqrt{a+b x+c x^2} \, dx}{x \sqrt{a+b x+c x^2}}\\ &=\frac{\left (a+b x+c x^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3 c x}-\frac{\left (b \sqrt{a x^2+b x^3+c x^4}\right ) \int \sqrt{a+b x+c x^2} \, dx}{2 c x \sqrt{a+b x+c x^2}}\\ &=-\frac{b (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{8 c^2 x}+\frac{\left (a+b x+c x^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3 c x}+\frac{\left (b \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16 c^2 x \sqrt{a+b x+c x^2}}\\ &=-\frac{b (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{8 c^2 x}+\frac{\left (a+b x+c x^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3 c x}+\frac{\left (b \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}\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 )}{8 c^2 x \sqrt{a+b x+c x^2}}\\ &=-\frac{b (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{8 c^2 x}+\frac{\left (a+b x+c x^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3 c x}+\frac{b \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.219505, size = 119, normalized size = 0.73 \[ \frac{2 \sqrt{c} x (a+x (b+c x)) \left (8 c \left (a+c x^2\right )-3 b^2+2 b c x\right )+3 b x \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{48 c^{5/2} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 167, normalized size = 1. \begin{align*}{\frac{1}{48\,x}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 16\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}-12\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}xb-6\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}{b}^{2}-12\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) ab{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{3}c \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{c}^{-{\frac{7}{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 \sqrt{c x^{4} + b x^{3} + a x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61506, size = 591, normalized size = 3.63 \begin{align*} \left [-\frac{3 \,{\left (b^{3} - 4 \, a b 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 \,{\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{96 \, c^{3} x}, -\frac{3 \,{\left (b^{3} - 4 \, a b 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 \,{\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{48 \, c^{3} 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 \sqrt{a x^{2} + b x^{3} + c x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14418, size = 224, normalized size = 1.37 \begin{align*} \frac{1}{24} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \, x \mathrm{sgn}\left (x\right ) + \frac{b \mathrm{sgn}\left (x\right )}{c}\right )} x - \frac{3 \, b^{2} \mathrm{sgn}\left (x\right ) - 8 \, a c \mathrm{sgn}\left (x\right )}{c^{2}}\right )} - \frac{{\left (b^{3} \mathrm{sgn}\left (x\right ) - 4 \, a b c \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{5}{2}}} + \frac{{\left (3 \, b^{3} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 12 \, a b c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 6 \, \sqrt{a} b^{2} \sqrt{c} - 16 \, a^{\frac{3}{2}} c^{\frac{3}{2}}\right )} \mathrm{sgn}\left (x\right )}{48 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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