Optimal. Leaf size=205 \[ \frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}-\frac{x \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c} \]
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Rubi [A] time = 0.370274, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1919, 1949, 12, 1914, 621, 206} \[ \frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}-\frac{x \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c} \]
Antiderivative was successfully verified.
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Rule 1919
Rule 1949
Rule 12
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x \sqrt{a x^2+b x^3+c x^4} \, dx &=\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c}+\frac{\int \frac{x^2 \left (-2 a b-\frac{1}{2} \left (5 b^2-12 a c\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{24 c}\\ &=-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c}-\frac{\int \frac{x \left (-\frac{1}{2} a \left (5 b^2-12 a c\right )-\frac{1}{4} b \left (15 b^2-52 a c\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{48 c^2}\\ &=-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c}+\frac{\int -\frac{3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) x}{8 \sqrt{a x^2+b x^3+c x^4}} \, dx}{48 c^3}\\ &=-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c}-\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{128 c^3}\\ &=-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c}-\frac{\left (\left (b^2-4 a c\right ) \left (5 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}{128 c^3 \sqrt{a x^2+b x^3+c x^4}}\\ &=-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c}-\frac{\left (\left (b^2-4 a c\right ) \left (5 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 )}{64 c^3 \sqrt{a x^2+b x^3+c x^4}}\\ &=-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c}-\frac{\left (b^2-4 a c\right ) \left (5 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 )}{128 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.192789, size = 150, normalized size = 0.73 \[ \frac{2 \sqrt{c} x (a+x (b+c x)) \left (b \left (8 c^2 x^2-52 a c\right )+24 c^2 x \left (a+2 c x^2\right )-10 b^2 c x+15 b^3\right )-3 x \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{384 c^{7/2} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 265, normalized size = 1.3 \begin{align*}{\frac{1}{384\,x}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 96\,x \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}-80\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}b-48\,{c}^{7/2}\sqrt{c{x}^{2}+bx+a}xa+60\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}x{b}^{2}-24\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}ab+30\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}{b}^{3}-48\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{c}^{3}+72\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{2}{c}^{2}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{4}c \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{c}^{-{\frac{9}{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}} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64592, size = 737, normalized size = 3.6 \begin{align*} \left [\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (48 \, c^{4} x^{3} + 8 \, b c^{3} x^{2} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \,{\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{768 \, c^{4} x}, \frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (48 \, c^{4} x^{3} + 8 \, b c^{3} x^{2} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \,{\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{384 \, c^{4} 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 x \sqrt{x^{2} \left (a + b x + c x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21113, size = 311, normalized size = 1.52 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, x \mathrm{sgn}\left (x\right ) + \frac{b \mathrm{sgn}\left (x\right )}{c}\right )} x - \frac{5 \, b^{2} c \mathrm{sgn}\left (x\right ) - 12 \, a c^{2} \mathrm{sgn}\left (x\right )}{c^{3}}\right )} x + \frac{15 \, b^{3} \mathrm{sgn}\left (x\right ) - 52 \, a b c \mathrm{sgn}\left (x\right )}{c^{3}}\right )} + \frac{{\left (5 \, b^{4} \mathrm{sgn}\left (x\right ) - 24 \, a b^{2} c \mathrm{sgn}\left (x\right ) + 16 \, a^{2} c^{2} \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 )}{128 \, c^{\frac{7}{2}}} - \frac{{\left (15 \, b^{4} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 72 \, a b^{2} c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 48 \, a^{2} c^{2} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 30 \, \sqrt{a} b^{3} \sqrt{c} - 104 \, a^{\frac{3}{2}} b c^{\frac{3}{2}}\right )} \mathrm{sgn}\left (x\right )}{384 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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