Optimal. Leaf size=54 \[ \frac{\tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 \sqrt{a} \sqrt{b c-a d}} \]
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Rubi [A] time = 0.0546701, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {465, 377, 205} \[ \frac{\tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 \sqrt{a} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
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Rule 465
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x^4}{\sqrt{c+d x^8}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b c-a d} x^4}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 \sqrt{a} \sqrt{b c-a d}}\\ \end{align*}
Mathematica [A] time = 0.0660526, size = 95, normalized size = 1.76 \[ \frac{x^4 \sqrt{\frac{d x^8}{c}+1} \tanh ^{-1}\left (\frac{\sqrt{\frac{d x^8}{c}-\frac{b x^8}{a}}}{\sqrt{\frac{d x^8}{c}+1}}\right )}{4 a \sqrt{c+d x^8} \sqrt{\frac{d x^8}{c}-\frac{b x^8}{a}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+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^{3}}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.28894, size = 525, normalized size = 9.72 \begin{align*} \left [-\frac{\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^{16} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \,{\left ({\left (b c - 2 \, a d\right )} x^{12} - a c x^{4}\right )} \sqrt{d x^{8} + c} \sqrt{-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{16 \,{\left (a b c - a^{2} d\right )}}, \frac{\arctan \left (\frac{{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d}}{2 \,{\left ({\left (a b c d - a^{2} d^{2}\right )} x^{12} +{\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )}}\right )}{8 \, \sqrt{a b c - a^{2} d}}\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^{3}}{\left (a + b x^{8}\right ) \sqrt{c + d x^{8}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20488, size = 97, normalized size = 1.8 \begin{align*} -\frac{\sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x^{4} - \sqrt{d x^{8} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{4 \, \sqrt{a b c d - a^{2} d^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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