Optimal. Leaf size=63 \[ -\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{3}{4} c \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0377259, antiderivative size = 63, 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, 50, 63, 208} \[ -\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{3}{4} c \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+c x^4\right )^{3/2}}{x^5} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+c x)^{3/2}}{x^2} \, dx,x,x^4\right )\\ &=-\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{1}{8} (3 c) \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x} \, dx,x,x^4\right )\\ &=\frac{3}{4} c \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{1}{8} (3 a c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^4\right )\\ &=\frac{3}{4} c \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{1}{4} (3 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^4}\right )\\ &=\frac{3}{4} c \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0084739, size = 37, normalized size = 0.59 \[ \frac{c \left (a+c x^4\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{c x^4}{a}+1\right )}{10 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 58, normalized size = 0.9 \begin{align*}{\frac{c}{2}\sqrt{c{x}^{4}+a}}-{\frac{3\,c}{4}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ) }-{\frac{a}{4\,{x}^{4}}\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 = 1.55509, size = 285, normalized size = 4.52 \begin{align*} \left [\frac{3 \, \sqrt{a} c x^{4} \log \left (\frac{c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) + 2 \,{\left (2 \, c x^{4} - a\right )} \sqrt{c x^{4} + a}}{8 \, x^{4}}, \frac{3 \, \sqrt{-a} c x^{4} \arctan \left (\frac{\sqrt{c x^{4} + a} \sqrt{-a}}{a}\right ) +{\left (2 \, c x^{4} - a\right )} \sqrt{c x^{4} + a}}{4 \, x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.75046, size = 95, normalized size = 1.51 \begin{align*} - \frac{3 \sqrt{a} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{4} - \frac{a^{2}}{4 \sqrt{c} x^{6} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{a \sqrt{c}}{4 x^{2} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{c^{\frac{3}{2}} x^{2}}{2 \sqrt{\frac{a}{c x^{4}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10319, size = 77, normalized size = 1.22 \begin{align*} \frac{1}{4} \,{\left (\frac{3 \, a \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{c x^{4} + a} - \frac{\sqrt{c x^{4} + a} a}{c x^{4}}\right )} c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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