Optimal. Leaf size=71 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}-\frac{c \sqrt{a+c x^4}}{16 a x^4}-\frac{\sqrt{a+c x^4}}{8 x^8} \]
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Rubi [A] time = 0.0392335, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}-\frac{c \sqrt{a+c x^4}}{16 a x^4}-\frac{\sqrt{a+c x^4}}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+c x^4}}{x^9} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^3} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{a+c x^4}}{8 x^8}+\frac{1}{16} c \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+c x}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{a+c x^4}}{8 x^8}-\frac{c \sqrt{a+c x^4}}{16 a x^4}-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^4\right )}{32 a}\\ &=-\frac{\sqrt{a+c x^4}}{8 x^8}-\frac{c \sqrt{a+c x^4}}{16 a x^4}-\frac{c \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^4}\right )}{16 a}\\ &=-\frac{\sqrt{a+c x^4}}{8 x^8}-\frac{c \sqrt{a+c x^4}}{16 a x^4}+\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0084714, size = 39, normalized size = 0.55 \[ -\frac{c^2 \left (a+c x^4\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{c x^4}{a}+1\right )}{6 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 85, normalized size = 1.2 \begin{align*} -{\frac{1}{8\,a{x}^{8}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{c}{16\,{a}^{2}{x}^{4}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{c}^{2}}{16\,{a}^{2}}\sqrt{c{x}^{4}+a}} \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 = 1.52014, size = 316, normalized size = 4.45 \begin{align*} \left [\frac{\sqrt{a} c^{2} x^{8} \log \left (\frac{c x^{4} + 2 \, \sqrt{c x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) - 2 \,{\left (a c x^{4} + 2 \, a^{2}\right )} \sqrt{c x^{4} + a}}{32 \, a^{2} x^{8}}, -\frac{\sqrt{-a} c^{2} x^{8} \arctan \left (\frac{\sqrt{c x^{4} + a} \sqrt{-a}}{a}\right ) +{\left (a c x^{4} + 2 \, a^{2}\right )} \sqrt{c x^{4} + a}}{16 \, a^{2} x^{8}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.3359, size = 95, normalized size = 1.34 \begin{align*} - \frac{a}{8 \sqrt{c} x^{10} \sqrt{\frac{a}{c x^{4}} + 1}} - \frac{3 \sqrt{c}}{16 x^{6} \sqrt{\frac{a}{c x^{4}} + 1}} - \frac{c^{\frac{3}{2}}}{16 a x^{2} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{16 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39045, size = 84, normalized size = 1.18 \begin{align*} -\frac{1}{16} \, c^{2}{\left (\frac{\arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}} + \sqrt{c x^{4} + a} a}{a c^{2} x^{8}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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