Optimal. Leaf size=51 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} \sqrt{b c-a d}} \]
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Rubi [A] time = 0.0468158, antiderivative size = 51, 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 = {444, 63, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
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Rule 444
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^8\right )\\ &=\frac{\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 d}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} \sqrt{b c-a d}}\\ \end{align*}
Mathematica [A] time = 0.0147228, size = 51, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{7}}{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 = 1.30495, size = 288, normalized size = 5.65 \begin{align*} \left [\frac{\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 )}{8 \, \sqrt{b^{2} c - a b d}}, \frac{\sqrt{-b^{2} c + a b d} \arctan \left (\frac{\sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d}}{b d x^{8} + b c}\right )}{4 \,{\left (b^{2} c - a b d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 37.5565, size = 37, normalized size = 0.73 \begin{align*} \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x^{8}}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{4 b \sqrt{\frac{a d - b c}{b}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11318, size = 54, normalized size = 1.06 \begin{align*} \frac{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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