Optimal. Leaf size=132 \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{4 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^8}}{8 a \left (a+b x^8\right ) (b c-a d)} \]
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Rubi [A] time = 0.137263, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {446, 103, 156, 63, 208} \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{4 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^8}}{8 a \left (a+b x^8\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 446
Rule 103
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2 \sqrt{c+d x}} \, dx,x,x^8\right )\\ &=\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac{\operatorname{Subst}\left (\int \frac{b c-a d+\frac{b d x}{2}}{x (a+b x) \sqrt{c+d x}} \, dx,x,x^8\right )}{8 a (b c-a d)}\\ &=\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^8\right )}{8 a^2}-\frac{(b (2 b c-3 a d)) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^8\right )}{16 a^2 (b c-a d)}\\ &=\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^8}\right )}{4 a^2 d}-\frac{(b (2 b c-3 a d)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^8}\right )}{8 a^2 d (b c-a d)}\\ &=\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{4 a^2 \sqrt{c}}+\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 a^2 (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.241851, size = 123, normalized size = 0.93 \[ \frac{\frac{a b \sqrt{c+d x^8}}{\left (a+b x^8\right ) (b c-a d)}+\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{\sqrt{c}}}{8 a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65469, size = 1833, normalized size = 13.89 \begin{align*} \text{result too large to display} \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 = 1.61945, size = 207, normalized size = 1.57 \begin{align*} -\frac{1}{8} \, d^{2}{\left (\frac{{\left (2 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{\sqrt{d x^{8} + c} b}{{\left (a b c d - a^{2} d^{2}\right )}{\left ({\left (d x^{8} + c\right )} b - b c + a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d x^{8} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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