Optimal. Leaf size=104 \[ \frac{(b c-2 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{3/2} (b c-a d)^{3/2}}+\frac{b x^4 \sqrt{c+d x^8}}{8 a \left (a+b x^8\right ) (b c-a d)} \]
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Rubi [A] time = 0.090251, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {465, 382, 377, 205} \[ \frac{(b c-2 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{3/2} (b c-a d)^{3/2}}+\frac{b x^4 \sqrt{c+d x^8}}{8 a \left (a+b x^8\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 465
Rule 382
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^2 \sqrt{c+d x^2}} \, dx,x,x^4\right )\\ &=\frac{b x^4 \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac{(b c-2 a d) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )}{8 a (b c-a d)}\\ &=\frac{b x^4 \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac{(b c-2 a d) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x^4}{\sqrt{c+d x^8}}\right )}{8 a (b c-a d)}\\ &=\frac{b x^4 \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac{(b c-2 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^4}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{3/2} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.778745, size = 407, normalized size = 3.91 \[ \frac{x^4 \sqrt{c+d x^8} \left (-30 d x^8 \sqrt{\frac{a x^8 \left (c+d x^8\right ) (b c-a d)}{c^2 \left (a+b x^8\right )^2}}-45 c \sqrt{\frac{a x^8 \left (c+d x^8\right ) (b c-a d)}{c^2 \left (a+b x^8\right )^2}}+16 d x^8 \left (\frac{x^8 (b c-a d)}{c \left (a+b x^8\right )}\right )^{5/2} \sqrt{\frac{a \left (c+d x^8\right )}{c \left (a+b x^8\right )}} \, _2F_1\left (2,3;\frac{7}{2};\frac{(b c-a d) x^8}{c \left (b x^8+a\right )}\right )+16 c \left (\frac{x^8 (b c-a d)}{c \left (a+b x^8\right )}\right )^{5/2} \sqrt{\frac{a \left (c+d x^8\right )}{c \left (a+b x^8\right )}} \, _2F_1\left (2,3;\frac{7}{2};\frac{(b c-a d) x^8}{c \left (b x^8+a\right )}\right )+30 d x^8 \sin ^{-1}\left (\sqrt{\frac{x^8 (b c-a d)}{c \left (a+b x^8\right )}}\right )+45 c \sin ^{-1}\left (\sqrt{\frac{x^8 (b c-a d)}{c \left (a+b x^8\right )}}\right )\right )}{120 c^2 \left (a+b x^8\right )^2 \left (\frac{x^8 (b c-a d)}{c \left (a+b x^8\right )}\right )^{3/2} \sqrt{\frac{a \left (c+d x^8\right )}{c \left (a+b x^8\right )}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\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 )}^{2} \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 = 2.26653, size = 973, normalized size = 9.36 \begin{align*} \left [\frac{4 \, \sqrt{d x^{8} + c}{\left (a b^{2} c - a^{2} b d\right )} x^{4} -{\left ({\left (b^{2} c - 2 \, a b d\right )} x^{8} + a b c - 2 \, a^{2} d\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^{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 )}{32 \,{\left ({\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{8} + a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )}}, \frac{2 \, \sqrt{d x^{8} + c}{\left (a b^{2} c - a^{2} b d\right )} x^{4} +{\left ({\left (b^{2} c - 2 \, a b d\right )} x^{8} + a b c - 2 \, a^{2} d\right )} \sqrt{a b c - a^{2} d} \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 )}{16 \,{\left ({\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{8} + a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} 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 = 1.21005, size = 320, normalized size = 3.08 \begin{align*} -\frac{1}{8} \, d^{\frac{3}{2}}{\left (\frac{{\left (b c - 2 \, a d\right )} \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 )}{{\left (a b c d - a^{2} d^{2}\right )}^{\frac{3}{2}}} + \frac{2 \,{\left ({\left (\sqrt{d} x^{4} - \sqrt{d x^{8} + c}\right )}^{2} b c - 2 \,{\left (\sqrt{d} x^{4} - \sqrt{d x^{8} + c}\right )}^{2} a d - b c^{2}\right )}}{{\left ({\left (\sqrt{d} x^{4} - \sqrt{d x^{8} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x^{4} - \sqrt{d x^{8} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x^{4} - \sqrt{d x^{8} + c}\right )}^{2} a d + b c^{2}\right )}{\left (a b c d - a^{2} d^{2}\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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