Optimal. Leaf size=71 \[ -\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac{b \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right )}{3 \sqrt{a} x^3} \]
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Rubi [A] time = 0.0641282, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {368, 266, 47, 63, 208} \[ -\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac{b \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right )}{3 \sqrt{a} x^3} \]
Antiderivative was successfully verified.
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Rule 368
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx &=\frac{\left (c x^2\right )^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^3}}{x^4} \, dx,x,\sqrt{c x^2}\right )}{x^3}\\ &=\frac{\left (c x^2\right )^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 x^3}\\ &=-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{3 x^3}+\frac{\left (b \left (c x^2\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\left (c x^2\right )^{3/2}\right )}{6 x^3}\\ &=-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{3 x^3}+\frac{\left (c x^2\right )^{3/2} \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \left (c x^2\right )^{3/2}}\right )}{3 x^3}\\ &=-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac{b \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right )}{3 \sqrt{a} x^3}\\ \end{align*}
Mathematica [A] time = 0.0826177, size = 93, normalized size = 1.31 \[ \frac{-b \left (c x^2\right )^{3/2} \sqrt{\frac{b \left (c x^2\right )^{3/2}}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b \left (c x^2\right )^{3/2}}{a}+1}\right )-a-b \left (c x^2\right )^{3/2}}{3 x^3 \sqrt{a+b \left (c x^2\right )^{3/2}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}\, 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{\sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1933, size = 451, normalized size = 6.35 \begin{align*} \left [\frac{b c x^{3} \sqrt{\frac{c}{a}} \log \left (\frac{b c^{2} x^{4} - 2 \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a} a x \sqrt{\frac{c}{a}} + 2 \, \sqrt{c x^{2}} a}{x^{4}}\right ) - 2 \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{6 \, x^{3}}, -\frac{b c x^{3} \sqrt{-\frac{c}{a}} \arctan \left (-\frac{{\left (a b c^{2} x^{4} \sqrt{-\frac{c}{a}} - \sqrt{c x^{2}} a^{2} \sqrt{-\frac{c}{a}}\right )} \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{b^{2} c^{4} x^{7} - a^{2} c x}\right ) + \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{3 \, x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19047, size = 74, normalized size = 1.04 \begin{align*} \frac{1}{3} \, b c^{\frac{3}{2}}{\left (\frac{\arctan \left (\frac{\sqrt{b c^{\frac{3}{2}} x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b c^{\frac{3}{2}} x^{3} + a}}{b c^{\frac{3}{2}} x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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