Optimal. Leaf size=143 \[ \frac{x \left (3 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c} \]
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Rubi [A] time = 0.174007, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1928, 1949, 12, 1914, 621, 206} \[ \frac{x \left (3 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c} \]
Antiderivative was successfully verified.
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Rule 1928
Rule 1949
Rule 12
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a x^2+b x^3+c x^4}} \, dx &=\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c}-\frac{\int \frac{x \left (a+\frac{3 b x}{2}\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{2 c}\\ &=\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\int \frac{\left (3 b^2-4 a c\right ) x}{4 \sqrt{a x^2+b x^3+c x^4}} \, dx}{2 c^2}\\ &=\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\left (3 b^2-4 a c\right ) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{8 c^2}\\ &=\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\left (\left (3 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^2 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\left (\left (3 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^2 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\left (3 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^{5/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.136673, size = 105, normalized size = 0.73 \[ \frac{x \left (\left (3 b^2-4 a c\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )+2 \sqrt{c} (2 c x-3 b) (a+x (b+c x))\right )}{8 c^{5/2} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 144, normalized size = 1. \begin{align*}{\frac{x}{8}\sqrt{c{x}^{2}+bx+a} \left ( 4\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x-6\,\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}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{2}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}}}}{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 \frac{x^{3}}{\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 = 1.56927, size = 514, normalized size = 3.59 \begin{align*} \left [-\frac{{\left (3 \, 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 - 3 \, b c\right )}}{16 \, c^{3} x}, -\frac{{\left (3 \, 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 - 3 \, b c\right )}}{8 \, 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 \frac{x^{3}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\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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