Optimal. Leaf size=144 \[ -\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{a^2 x^2 \left (b^2-4 a c\right )}+\frac{3 b \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{2 a^{5/2}}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.163171, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1924, 1951, 12, 1904, 206} \[ -\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{a^2 x^2 \left (b^2-4 a c\right )}+\frac{3 b \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{2 a^{5/2}}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1924
Rule 1951
Rule 12
Rule 1904
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 \int \frac{-\frac{3 b^2}{2}+4 a c-b c x}{x \sqrt{a x^2+b x^3+c x^4}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{a^2 \left (b^2-4 a c\right ) x^2}+\frac{2 \int -\frac{3 b \left (b^2-4 a c\right )}{4 \sqrt{a x^2+b x^3+c x^4}} \, dx}{a^2 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{a^2 \left (b^2-4 a c\right ) x^2}-\frac{(3 b) \int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{2 a^2}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{a^2 \left (b^2-4 a c\right ) x^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{x (2 a+b x)}{\sqrt{a x^2+b x^3+c x^4}}\right )}{a^2}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{a^2 \left (b^2-4 a c\right ) x^2}+\frac{3 b \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{2 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.105634, size = 138, normalized size = 0.96 \[ \frac{2 \sqrt{a} \left (-4 a^2 c+a \left (b^2-10 b c x-8 c^2 x^2\right )+3 b^2 x (b+c x)\right )-3 b x \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{2 a^{5/2} \left (4 a c-b^2\right ) \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 201, normalized size = 1.4 \begin{align*} -{\frac{{x}^{2} \left ( c{x}^{2}+bx+a \right ) }{8\,ac-2\,{b}^{2}} \left ( 16\,{a}^{5/2}{x}^{2}{c}^{2}-6\,{a}^{3/2}{x}^{2}{b}^{2}c+20\,{a}^{5/2}xbc-6\,{a}^{3/2}x{b}^{3}-12\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}x{a}^{2}bc+3\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}xa{b}^{3}+8\,{a}^{7/2}c-2\,{a}^{5/2}{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{2}}}{a}^{-{\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}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87612, size = 1041, normalized size = 7.23 \begin{align*} \left [\frac{3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{3} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt{a} \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a^{2} b^{2} - 4 \, a^{3} c +{\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} +{\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x\right )}}{4 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{4} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{3} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2}\right )}}, -\frac{3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{3} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a^{2} b^{2} - 4 \, a^{3} c +{\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} +{\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x\right )}}{2 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{4} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{3} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2}\right )}}\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}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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