Optimal. Leaf size=93 \[ \frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^3 \sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]
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Rubi [A] time = 0.09026, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {465, 471, 12, 377, 205} \[ \frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^3 \sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 465
Rule 471
Rule 12
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right )^2 \sqrt{c+d x^2}} \, dx,x,x^3\right )\\ &=-\frac{x^3 \sqrt{c+d x^6}}{6 (b c-a d) \left (a+b x^6\right )}+\frac{\operatorname{Subst}\left (\int \frac{c}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^3\right )}{6 (b c-a d)}\\ &=-\frac{x^3 \sqrt{c+d x^6}}{6 (b c-a d) \left (a+b x^6\right )}+\frac{c \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^3\right )}{6 (b c-a d)}\\ &=-\frac{x^3 \sqrt{c+d x^6}}{6 (b c-a d) \left (a+b x^6\right )}+\frac{c \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x^3}{\sqrt{c+d x^6}}\right )}{6 (b c-a d)}\\ &=-\frac{x^3 \sqrt{c+d x^6}}{6 (b c-a d) \left (a+b x^6\right )}+\frac{c \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^3}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 \sqrt{a} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.443684, size = 124, normalized size = 1.33 \[ \frac{\sqrt{c+d x^6} \left (-\frac{x^6 (b c-a d)}{a+b x^6}-\frac{c \sqrt{x^6 \left (\frac{d}{c}-\frac{b}{a}\right )} \tanh ^{-1}\left (\frac{\sqrt{x^6 \left (\frac{d}{c}-\frac{b}{a}\right )}}{\sqrt{\frac{d x^6}{c}+1}}\right )}{\sqrt{\frac{d x^6}{c}+1}}\right )}{6 x^3 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{8}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \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^{8}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.44086, size = 887, normalized size = 9.54 \begin{align*} \left [-\frac{4 \, \sqrt{d x^{6} + c}{\left (a b c - a^{2} d\right )} x^{3} -{\left (b c x^{6} + a c\right )} \sqrt{-a b c + a^{2} d} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} + 4 \,{\left ({\left (b c - 2 \, a d\right )} x^{9} - a c x^{3}\right )} \sqrt{d x^{6} + c} \sqrt{-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right )}{24 \,{\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{6} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}, -\frac{2 \, \sqrt{d x^{6} + c}{\left (a b c - a^{2} d\right )} x^{3} -{\left (b c x^{6} + a c\right )} \sqrt{a b c - a^{2} d} \arctan \left (\frac{{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d}}{2 \,{\left ({\left (a b c d - a^{2} d^{2}\right )} x^{9} +{\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )}}\right )}{12 \,{\left ({\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{6} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27726, size = 248, normalized size = 2.67 \begin{align*} -\frac{1}{6} \, c{\left (\frac{\arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d}{\left (b c \mathrm{sgn}\left (x\right ) - a d \mathrm{sgn}\left (x\right )\right )}} + \frac{\sqrt{d + \frac{c}{x^{6}}}}{{\left (b c \mathrm{sgn}\left (x\right ) - a d \mathrm{sgn}\left (x\right )\right )}{\left (b c + a{\left (d + \frac{c}{x^{6}}\right )} - a d\right )}}\right )} + \frac{{\left (b c \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) + \sqrt{a b c - a^{2} d} \sqrt{d}\right )} \mathrm{sgn}\left (x\right )}{6 \,{\left (\sqrt{a b c - a^{2} d} b^{2} c - \sqrt{a b c - a^{2} d} a b d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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