Optimal. Leaf size=71 \[ \frac{c \sqrt{\frac{c}{a+b x^2}}}{a}-\frac{c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left (\sqrt{\frac{b x^2}{a}+1}\right )}{a} \]
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Rubi [A] time = 0.135793, antiderivative size = 73, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6720, 266, 51, 63, 208} \[ \frac{c \sqrt{\frac{c}{a+b x^2}}}{a}-\frac{c \sqrt{a+b x^2} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (\frac{c}{a+b x^2}\right )^{3/2}}{x} \, dx &=\left (c \sqrt{\frac{c}{a+b x^2}} \sqrt{a+b x^2}\right ) \int \frac{1}{x \left (a+b x^2\right )^{3/2}} \, dx\\ &=\frac{1}{2} \left (c \sqrt{\frac{c}{a+b x^2}} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{c \sqrt{\frac{c}{a+b x^2}}}{a}+\frac{\left (c \sqrt{\frac{c}{a+b x^2}} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{c \sqrt{\frac{c}{a+b x^2}}}{a}+\frac{\left (c \sqrt{\frac{c}{a+b x^2}} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{a b}\\ &=\frac{c \sqrt{\frac{c}{a+b x^2}}}{a}-\frac{c \sqrt{\frac{c}{a+b x^2}} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0089605, size = 38, normalized size = 0.54 \[ \frac{c \sqrt{\frac{c}{a+b x^2}} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^2}{a}+1\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 64, normalized size = 0.9 \begin{align*} -{(b{x}^{2}+a) \left ({\frac{c}{b{x}^{2}+a}} \right ) ^{{\frac{3}{2}}} \left ( \ln \left ( 2\,{\frac{\sqrt{a}\sqrt{b{x}^{2}+a}+a}{x}} \right ) a\sqrt{b{x}^{2}+a}-{a}^{{\frac{3}{2}}} \right ){a}^{-{\frac{5}{2}}}} \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 = 1.53063, size = 289, normalized size = 4.07 \begin{align*} \left [\frac{c \sqrt{\frac{c}{a}} \log \left (-\frac{b c x^{2} + 2 \, a c - 2 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{\frac{c}{b x^{2} + a}} \sqrt{\frac{c}{a}}}{x^{2}}\right ) + 2 \, c \sqrt{\frac{c}{b x^{2} + a}}}{2 \, a}, \frac{c \sqrt{-\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{b x^{2} + a}} \sqrt{-\frac{c}{a}}}{c}\right ) + c \sqrt{\frac{c}{b x^{2} + a}}}{a}\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{\left (\frac{c}{a + b x^{2}}\right )^{\frac{3}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17665, size = 88, normalized size = 1.24 \begin{align*} c^{3}{\left (\frac{\arctan \left (\frac{\sqrt{b c x^{2} + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c} a c} + \frac{1}{\sqrt{b c x^{2} + a c} a c}\right )} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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