Optimal. Leaf size=296 \[ -\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt{2} c^{3/2}}-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}} \]
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Rubi [A] time = 0.274709, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {277, 329, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt{2} c^{3/2}}-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 329
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a-b x^2}}{(c x)^{3/2}} \, dx &=-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}}-\frac{b \int \frac{\sqrt{c x}}{\left (a-b x^2\right )^{3/4}} \, dx}{c^2}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{\left (a-\frac{b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt{c x}\right )}{c^3}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{1+\frac{b x^4}{c^2}} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{c^3}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}}+\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{c-\sqrt{b} x^2}{1+\frac{b x^4}{c^2}} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{c^3}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{c+\sqrt{b} x^2}{1+\frac{b x^4}{c^2}} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{c^3}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{c}}{\sqrt [4]{b}}+2 x}{-\frac{c}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{c} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{c}}{\sqrt [4]{b}}-2 x}{-\frac{c}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{c} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt{2} c^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{c}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{c} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{2 c}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{c}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{c} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{2 c}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}}-\frac{\sqrt [4]{b} \log \left (\sqrt{c}+\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \log \left (\sqrt{c}+\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \log \left (\sqrt{c}+\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \log \left (\sqrt{c}+\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt{2} c^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0128392, size = 55, normalized size = 0.19 \[ -\frac{2 x \sqrt [4]{a-b x^2} \, _2F_1\left (-\frac{1}{4},-\frac{1}{4};\frac{3}{4};\frac{b x^2}{a}\right )}{(c x)^{3/2} \sqrt [4]{1-\frac{b x^2}{a}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [4]{-b{x}^{2}+a} \left ( cx \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{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [C] time = 4.25267, size = 51, normalized size = 0.17 \begin{align*} \frac{\sqrt [4]{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{2 c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.79996, size = 429, normalized size = 1.45 \begin{align*} \frac{2 \, \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} + \frac{2 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right ) + 2 \, \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} - \frac{2 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right ) + \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} \log \left (\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{b}{\left | c \right |} + \frac{\sqrt{-b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right ) - \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} \log \left (-\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{b}{\left | c \right |} + \frac{\sqrt{-b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right ) - \frac{8 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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