Optimal. Leaf size=120 \[ \frac{2 x^3 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3 c^{3/2}} \]
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Rubi [A] time = 0.0895328, antiderivative size = 120, 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 = {1357, 738, 640, 621, 206} \[ \frac{2 x^3 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3 c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 738
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=\frac{2 x^3 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 \operatorname{Subst}\left (\int \frac{2 a+b x}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{3 \left (b^2-4 a c\right )}\\ &=\frac{2 x^3 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{3 c}\\ &=\frac{2 x^3 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^3}{\sqrt{a+b x^3+c x^6}}\right )}{3 c}\\ &=\frac{2 x^3 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.109485, size = 107, normalized size = 0.89 \[ \frac{\frac{2 \sqrt{c} \left (a \left (b-2 c x^3\right )+b^2 x^3\right )}{\sqrt{a+b x^3+c x^6}}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3 c^{3/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{{x}^{8} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82198, size = 833, normalized size = 6.94 \begin{align*} \left [\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} x^{3} + a b c\right )}}{6 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} + a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{3}\right )}}, -\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \, \sqrt{c x^{6} + b x^{3} + a}{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} x^{3} + a b c\right )}}{3 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} + a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{3}\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^{8}}{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, 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^{8}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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