Optimal. Leaf size=145 \[ -\frac{\left (15 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{72 a^3 x^3}+\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{7/2}}+\frac{5 b \sqrt{a+b x^3+c x^6}}{36 a^2 x^6}-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9} \]
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Rubi [A] time = 0.157619, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1357, 744, 834, 806, 724, 206} \[ -\frac{\left (15 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{72 a^3 x^3}+\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{7/2}}+\frac{5 b \sqrt{a+b x^3+c x^6}}{36 a^2 x^6}-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 744
Rule 834
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^{10} \sqrt{a+b x^3+c x^6}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9}-\frac{\operatorname{Subst}\left (\int \frac{\frac{5 b}{2}+2 c x}{x^3 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{9 a}\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9}+\frac{5 b \sqrt{a+b x^3+c x^6}}{36 a^2 x^6}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (15 b^2-16 a c\right )+\frac{5 b c x}{2}}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{18 a^2}\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9}+\frac{5 b \sqrt{a+b x^3+c x^6}}{36 a^2 x^6}-\frac{\left (15 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{72 a^3 x^3}-\frac{\left (b \left (5 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{48 a^3}\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9}+\frac{5 b \sqrt{a+b x^3+c x^6}}{36 a^2 x^6}-\frac{\left (15 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{72 a^3 x^3}+\frac{\left (b \left (5 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^3}{\sqrt{a+b x^3+c x^6}}\right )}{24 a^3}\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9}+\frac{5 b \sqrt{a+b x^3+c x^6}}{36 a^2 x^6}-\frac{\left (15 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{72 a^3 x^3}+\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0793621, size = 112, normalized size = 0.77 \[ \frac{\sqrt{a+b x^3+c x^6} \left (-8 a^2+2 a \left (5 b x^3+8 c x^6\right )-15 b^2 x^6\right )}{72 a^3 x^9}+\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{7/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{10}}{\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 = 1.85863, size = 613, normalized size = 4.23 \begin{align*} \left [-\frac{3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} \sqrt{a} x^{9} \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \,{\left ({\left (15 \, a b^{2} - 16 \, a^{2} c\right )} x^{6} - 10 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{288 \, a^{4} x^{9}}, -\frac{3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} \sqrt{-a} x^{9} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \,{\left ({\left (15 \, a b^{2} - 16 \, a^{2} c\right )} x^{6} - 10 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{144 \, a^{4} x^{9}}\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{1}{x^{10} \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{1}{\sqrt{c x^{6} + b x^{3} + a} x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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