Optimal. Leaf size=201 \[ \frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{c^2 x \left (b^2-4 a c\right )}+\frac{2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}-\frac{3 b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2} \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.304659, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1923, 1949, 12, 1914, 621, 206} \[ \frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{c^2 x \left (b^2-4 a c\right )}+\frac{2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}-\frac{3 b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2} \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1923
Rule 1949
Rule 12
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac{2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 \int \frac{x^2 (4 a+2 b x)}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{b^2-4 a c}\\ &=\frac{2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\int \frac{x \left (2 a b+\left (3 b^2-8 a c\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac{2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac{\int \frac{3 b \left (b^2-4 a c\right ) x}{2 \sqrt{a x^2+b x^3+c x^4}} \, dx}{c^2 \left (b^2-4 a c\right )}\\ &=\frac{2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac{(3 b) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{2 c^2}\\ &=\frac{2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac{\left (3 b x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 c^2 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac{\left (3 b 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 )}{c^2 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac{3 b 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 )}{2 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.15881, size = 141, normalized size = 0.7 \[ \frac{x \left (2 \sqrt{c} \left (8 a^2 c+a \left (-3 b^2+10 b c x+4 c^2 x^2\right )-b^2 x (3 b+c x)\right )+3 b \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{2 c^{5/2} \left (4 a c-b^2\right ) \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 199, normalized size = 1. \begin{align*}{\frac{{x}^{3} \left ( c{x}^{2}+bx+a \right ) }{8\,ac-2\,{b}^{2}} \left ( 8\,{c}^{7/2}{x}^{2}a-2\,{c}^{5/2}{x}^{2}{b}^{2}+20\,{c}^{5/2}xab-6\,{c}^{3/2}x{b}^{3}+16\,{c}^{5/2}{a}^{2}-6\,{c}^{3/2}a{b}^{2}-12\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) \sqrt{c{x}^{2}+bx+a}ab{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) \sqrt{c{x}^{2}+bx+a}{b}^{3}c \right ){c}^{-{\frac{7}{2}}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{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^{6}}{{\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 = 2.09024, size = 1026, normalized size = 5.1 \begin{align*} \left [\frac{3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{2} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt{c} \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 (3 \, a b^{2} c - 8 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x\right )}}{4 \,{\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{3} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2} +{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} x\right )}}, \frac{3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{2} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt{-c} \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 (3 \, a b^{2} c - 8 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x\right )}}{2 \,{\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{3} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2} +{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} x\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^{6}}{\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 [A] time = 1.16984, size = 263, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (\frac{b^{3} c^{2} - 3 \, a b c^{3}}{b^{2} c^{4} - 4 \, a c^{5}} + \frac{a b^{2} c^{2} - 2 \, a^{2} c^{3}}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x}\right )}}{\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}}} + \frac{3 \, b \arctan \left (\frac{\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}} - \frac{\sqrt{a}}{x}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{b{\left (\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}} - \frac{\sqrt{a}}{x}\right )} - 2 \, \sqrt{a} c}{{\left ({\left (\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}} - \frac{\sqrt{a}}{x}\right )}^{2} - c\right )} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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