Optimal. Leaf size=104 \[ \frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{24 c^{5/2}}-\frac{b \sqrt{a+b x^3+c x^6}}{4 c^2}+\frac{x^3 \sqrt{a+b x^3+c x^6}}{6 c} \]
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Rubi [A] time = 0.0886356, antiderivative size = 104, 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, 742, 640, 621, 206} \[ \frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{24 c^{5/2}}-\frac{b \sqrt{a+b x^3+c x^6}}{4 c^2}+\frac{x^3 \sqrt{a+b x^3+c x^6}}{6 c} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 742
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt{a+b x^3+c x^6}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )\\ &=\frac{x^3 \sqrt{a+b x^3+c x^6}}{6 c}+\frac{\operatorname{Subst}\left (\int \frac{-a-\frac{3 b x}{2}}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{6 c}\\ &=-\frac{b \sqrt{a+b x^3+c x^6}}{4 c^2}+\frac{x^3 \sqrt{a+b x^3+c x^6}}{6 c}+\frac{\left (3 b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{24 c^2}\\ &=-\frac{b \sqrt{a+b x^3+c x^6}}{4 c^2}+\frac{x^3 \sqrt{a+b x^3+c x^6}}{6 c}+\frac{\left (3 b^2-4 a c\right ) \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 )}{12 c^2}\\ &=-\frac{b \sqrt{a+b x^3+c x^6}}{4 c^2}+\frac{x^3 \sqrt{a+b x^3+c x^6}}{6 c}+\frac{\left (3 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 )}{24 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0333565, size = 88, normalized size = 0.85 \[ \frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )+2 \sqrt{c} \left (2 c x^3-3 b\right ) \sqrt{a+b x^3+c x^6}}{24 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{{x}^{8}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, 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 = 1.61118, size = 475, normalized size = 4.57 \begin{align*} \left [-\frac{{\left (3 \, b^{2} - 4 \, a 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 (2 \, c^{2} x^{3} - 3 \, b c\right )}}{48 \, c^{3}}, -\frac{{\left (3 \, b^{2} - 4 \, a 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 (2 \, c^{2} x^{3} - 3 \, b c\right )}}{24 \, c^{3}}\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}}{\sqrt{a + b x^{3} + c x^{6}}}\, 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}}{\sqrt{c x^{6} + b x^{3} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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