Optimal. Leaf size=121 \[ \frac{\left (-16 a c+15 b^2-10 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{72 c^3}-\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{7/2}}+\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c} \]
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Rubi [A] time = 0.105398, antiderivative size = 121, 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, 779, 621, 206} \[ \frac{\left (-16 a c+15 b^2-10 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{72 c^3}-\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{7/2}}+\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 742
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt{a+b x^3+c x^6}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )\\ &=\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c}+\frac{\operatorname{Subst}\left (\int \frac{x \left (-2 a-\frac{5 b x}{2}\right )}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{9 c}\\ &=\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c}+\frac{\left (15 b^2-16 a c-10 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{72 c^3}-\frac{\left (b \left (5 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{48 c^3}\\ &=\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c}+\frac{\left (15 b^2-16 a c-10 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{72 c^3}-\frac{\left (b \left (5 b^2-12 a c\right )\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 )}{24 c^3}\\ &=\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c}+\frac{\left (15 b^2-16 a c-10 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{72 c^3}-\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.053277, size = 104, normalized size = 0.86 \[ \frac{2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (8 c \left (c x^6-2 a\right )+15 b^2-10 b c x^3\right )+\left (36 a b c-15 b^3\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{144 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{{x}^{11}{\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 = 1.58784, size = 568, normalized size = 4.69 \begin{align*} \left [-\frac{3 \,{\left (5 \, b^{3} - 12 \, a b 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 \,{\left (8 \, c^{3} x^{6} - 10 \, b c^{2} x^{3} + 15 \, b^{2} c - 16 \, a c^{2}\right )} \sqrt{c x^{6} + b x^{3} + a}}{288 \, c^{4}}, \frac{3 \,{\left (5 \, b^{3} - 12 \, a b 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 \,{\left (8 \, c^{3} x^{6} - 10 \, b c^{2} x^{3} + 15 \, b^{2} c - 16 \, a c^{2}\right )} \sqrt{c x^{6} + b x^{3} + a}}{144 \, c^{4}}\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^{11}}{\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^{11}}{\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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