Optimal. Leaf size=104 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^8} (a d+b c)}{4 b^2 d^2}+\frac{\left (c+d x^8\right )^{3/2}}{12 b d^2} \]
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Rubi [A] time = 0.0985776, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {446, 88, 63, 208} \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^8} (a d+b c)}{4 b^2 d^2}+\frac{\left (c+d x^8\right )^{3/2}}{12 b d^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^{23}}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x) \sqrt{c+d x}} \, dx,x,x^8\right )\\ &=\frac{1}{8} \operatorname{Subst}\left (\int \left (\frac{-b c-a d}{b^2 d \sqrt{c+d x}}+\frac{a^2}{b^2 (a+b x) \sqrt{c+d x}}+\frac{\sqrt{c+d x}}{b d}\right ) \, dx,x,x^8\right )\\ &=-\frac{(b c+a d) \sqrt{c+d x^8}}{4 b^2 d^2}+\frac{\left (c+d x^8\right )^{3/2}}{12 b d^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^8\right )}{8 b^2}\\ &=-\frac{(b c+a d) \sqrt{c+d x^8}}{4 b^2 d^2}+\frac{\left (c+d x^8\right )^{3/2}}{12 b d^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^8}\right )}{4 b^2 d}\\ &=-\frac{(b c+a d) \sqrt{c+d x^8}}{4 b^2 d^2}+\frac{\left (c+d x^8\right )^{3/2}}{12 b d^2}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}\\ \end{align*}
Mathematica [A] time = 0.17129, size = 91, normalized size = 0.88 \[ \frac{\sqrt{c+d x^8} \left (-3 a d-2 b c+b d x^8\right )}{12 b^2 d^2}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{23}}{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(-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 = 1.36578, size = 595, normalized size = 5.72 \begin{align*} \left [\frac{3 \, \sqrt{b^{2} c - a b d} a^{2} d^{2} \log \left (\frac{b d x^{8} + 2 \, b c - a d - 2 \, \sqrt{d x^{8} + c} \sqrt{b^{2} c - a b d}}{b x^{8} + a}\right ) + 2 \,{\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{8} - 2 \, b^{3} c^{2} - a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \sqrt{d x^{8} + c}}{24 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}, \frac{3 \, \sqrt{-b^{2} c + a b d} a^{2} d^{2} \arctan \left (\frac{\sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d}}{b d x^{8} + b c}\right ) +{\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{8} - 2 \, b^{3} c^{2} - a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \sqrt{d x^{8} + c}}{12 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\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 [A] time = 1.17604, size = 143, normalized size = 1.38 \begin{align*} \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \, \sqrt{-b^{2} c + a b d} b^{2}} + \frac{{\left (d x^{8} + c\right )}^{\frac{3}{2}} b^{2} d^{4} - 3 \, \sqrt{d x^{8} + c} b^{2} c d^{4} - 3 \, \sqrt{d x^{8} + c} a b d^{5}}{12 \, b^{3} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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