Optimal. Leaf size=141 \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 b^2 (b c-a d)^{3/2}}+\frac{a x^4 \sqrt{c+d x^8}}{8 b \left (a+b x^8\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right )}{4 b^2 \sqrt{d}} \]
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Rubi [A] time = 0.159961, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {465, 470, 523, 217, 206, 377, 205} \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 b^2 (b c-a d)^{3/2}}+\frac{a x^4 \sqrt{c+d x^8}}{8 b \left (a+b x^8\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right )}{4 b^2 \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 465
Rule 470
Rule 523
Rule 217
Rule 206
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{19}}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^2\right )^2 \sqrt{c+d x^2}} \, dx,x,x^4\right )\\ &=\frac{a x^4 \sqrt{c+d x^8}}{8 b (b c-a d) \left (a+b x^8\right )}-\frac{\operatorname{Subst}\left (\int \frac{a c-2 (b c-a d) x^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )}{8 b (b c-a d)}\\ &=\frac{a x^4 \sqrt{c+d x^8}}{8 b (b c-a d) \left (a+b x^8\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^2}} \, dx,x,x^4\right )}{4 b^2}-\frac{(a (3 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 b^2 (b c-a d)}\\ &=\frac{a x^4 \sqrt{c+d x^8}}{8 b (b c-a d) \left (a+b x^8\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x^4}{\sqrt{c+d x^8}}\right )}{4 b^2}-\frac{(a (3 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 b^2 (b c-a d)}\\ &=\frac{a x^4 \sqrt{c+d x^8}}{8 b (b c-a d) \left (a+b x^8\right )}-\frac{\sqrt{a} (3 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 b^2 (b c-a d)^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right )}{4 b^2 \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.210382, size = 135, normalized size = 0.96 \[ \frac{\frac{a b x^4 \sqrt{c+d x^8}}{\left (a+b x^8\right ) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{(b c-a d)^{3/2}}+\frac{2 \log \left (\sqrt{d} \sqrt{c+d x^8}+d x^4\right )}{\sqrt{d}}}{8 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{19}}{ \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^{19}}{{\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 [A] time = 3.75799, size = 2284, normalized size = 16.2 \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 = 1.74024, size = 205, normalized size = 1.45 \begin{align*} \frac{1}{8} \, c^{2}{\left (\frac{{\left (3 \, a b c - 2 \, a^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{{\left (b^{3} c^{3} - a b^{2} c^{2} d\right )} \sqrt{a b c - a^{2} d}} + \frac{a \sqrt{d + \frac{c}{x^{8}}}}{{\left (b^{2} c^{2} - a b c d\right )}{\left (b c + a{\left (d + \frac{c}{x^{8}}\right )} - a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d + \frac{c}{x^{8}}}}{\sqrt{-d}}\right )}{b^{2} c^{2} \sqrt{-d}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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