Optimal. Leaf size=115 \[ \frac{b^2 \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^8} (2 a d+3 b c)}{12 a^2 c^2 x^4}-\frac{\sqrt{c+d x^8}}{12 a c x^{12}} \]
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Rubi [A] time = 0.16105, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {465, 480, 583, 12, 377, 205} \[ \frac{b^2 \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^8} (2 a d+3 b c)}{12 a^2 c^2 x^4}-\frac{\sqrt{c+d x^8}}{12 a c x^{12}} \]
Antiderivative was successfully verified.
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Rule 465
Rule 480
Rule 583
Rule 12
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^{13} \left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{c+d x^8}}{12 a c x^{12}}+\frac{\operatorname{Subst}\left (\int \frac{-3 b c-2 a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )}{12 a c}\\ &=-\frac{\sqrt{c+d x^8}}{12 a c x^{12}}+\frac{(3 b c+2 a d) \sqrt{c+d x^8}}{12 a^2 c^2 x^4}-\frac{\operatorname{Subst}\left (\int -\frac{3 b^2 c^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )}{12 a^2 c^2}\\ &=-\frac{\sqrt{c+d x^8}}{12 a c x^{12}}+\frac{(3 b c+2 a d) \sqrt{c+d x^8}}{12 a^2 c^2 x^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )}{4 a^2}\\ &=-\frac{\sqrt{c+d x^8}}{12 a c x^{12}}+\frac{(3 b c+2 a d) \sqrt{c+d x^8}}{12 a^2 c^2 x^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x^4}{\sqrt{c+d x^8}}\right )}{4 a^2}\\ &=-\frac{\sqrt{c+d x^8}}{12 a c x^{12}}+\frac{(3 b c+2 a d) \sqrt{c+d x^8}}{12 a^2 c^2 x^4}+\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^4}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 a^{5/2} \sqrt{b c-a d}}\\ \end{align*}
Mathematica [C] time = 1.75096, size = 253, normalized size = 2.2 \[ -\frac{\left (\frac{d x^8}{c}+1\right ) \left (-\frac{8 x^8 \left (c+d x^8\right )^2 (b c-a d) \text{HypergeometricPFQ}\left (\{2,2,2\},\left \{1,\frac{5}{2}\right \},\frac{x^8 (b c-a d)}{c \left (a+b x^8\right )}\right )}{a+b x^8}+\frac{3 c \left (c^2-4 c d x^8-8 d^2 x^{16}\right ) \sin ^{-1}\left (\sqrt{\frac{x^8 (b c-a d)}{c \left (a+b x^8\right )}}\right )}{\sqrt{\frac{a x^8 \left (c+d x^8\right ) (b c-a d)}{c^2 \left (a+b x^8\right )^2}}}+\frac{24 d x^{16} \left (c+d x^8\right ) (a d-b c) \, _2F_1\left (2,2;\frac{5}{2};\frac{(b c-a d) x^8}{c \left (b x^8+a\right )}\right )}{a+b x^8}\right )}{36 c^3 x^{12} \left (a+b x^8\right ) \sqrt{c+d x^8}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{13} \left ( b{x}^{8}+a \right ) }{\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 )} \sqrt{d x^{8} + c} x^{13}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54158, size = 867, normalized size = 7.54 \begin{align*} \left [-\frac{3 \, \sqrt{-a b c + a^{2} d} b^{2} c^{2} x^{12} \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 ) - 4 \,{\left ({\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{8} - a^{2} b c^{2} + a^{3} c d\right )} \sqrt{d x^{8} + c}}{48 \,{\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{12}}, \frac{3 \, \sqrt{a b c - a^{2} d} b^{2} c^{2} x^{12} \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 ) + 2 \,{\left ({\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{8} - a^{2} b c^{2} + a^{3} c d\right )} \sqrt{d x^{8} + c}}{24 \,{\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{12}}\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 = 1.1727, size = 144, normalized size = 1.25 \begin{align*} -\frac{b^{2} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{4 \, \sqrt{a b c - a^{2} d} a^{2}} + \frac{3 \, a b c^{5} \sqrt{d + \frac{c}{x^{8}}} - a^{2} c^{4}{\left (d + \frac{c}{x^{8}}\right )}^{\frac{3}{2}} + 3 \, a^{2} c^{4} \sqrt{d + \frac{c}{x^{8}}} d}{12 \, a^{3} c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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