Optimal. Leaf size=308 \[ -\frac{a c^{3/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{5/4}}+\frac{a c^{3/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{5/4}}-\frac{a c^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{5/4}}+\frac{a c^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{4 \sqrt{2} b^{5/4}}-\frac{c \sqrt{c x} \left (a-b x^2\right )^{3/4}}{2 b} \]
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Rubi [A] time = 0.265121, antiderivative size = 308, 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 = {321, 329, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{a c^{3/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{5/4}}+\frac{a c^{3/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{5/4}}-\frac{a c^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{5/4}}+\frac{a c^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{4 \sqrt{2} b^{5/4}}-\frac{c \sqrt{c x} \left (a-b x^2\right )^{3/4}}{2 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 329
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(c x)^{3/2}}{\sqrt [4]{a-b x^2}} \, dx &=-\frac{c \sqrt{c x} \left (a-b x^2\right )^{3/4}}{2 b}+\frac{\left (a c^2\right ) \int \frac{1}{\sqrt{c x} \sqrt [4]{a-b x^2}} \, dx}{4 b}\\ &=-\frac{c \sqrt{c x} \left (a-b x^2\right )^{3/4}}{2 b}+\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{a-\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{2 b}\\ &=-\frac{c \sqrt{c x} \left (a-b x^2\right )^{3/4}}{2 b}+\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{1+\frac{b x^4}{c^2}} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{2 b}\\ &=-\frac{c \sqrt{c x} \left (a-b x^2\right )^{3/4}}{2 b}+\frac{a \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 )}{4 b}+\frac{a \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 )}{4 b}\\ &=-\frac{c \sqrt{c x} \left (a-b x^2\right )^{3/4}}{2 b}-\frac{\left (a c^{3/2}\right ) \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 )}{8 \sqrt{2} b^{5/4}}-\frac{\left (a c^{3/2}\right ) \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 )}{8 \sqrt{2} b^{5/4}}+\frac{\left (a c^2\right ) \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 )}{8 b^{3/2}}+\frac{\left (a c^2\right ) \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 )}{8 b^{3/2}}\\ &=-\frac{c \sqrt{c x} \left (a-b x^2\right )^{3/4}}{2 b}-\frac{a c^{3/2} \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 )}{8 \sqrt{2} b^{5/4}}+\frac{a c^{3/2} \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 )}{8 \sqrt{2} b^{5/4}}+\frac{\left (a c^{3/2}\right ) \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 )}{4 \sqrt{2} b^{5/4}}-\frac{\left (a c^{3/2}\right ) \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 )}{4 \sqrt{2} b^{5/4}}\\ &=-\frac{c \sqrt{c x} \left (a-b x^2\right )^{3/4}}{2 b}-\frac{a c^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{5/4}}+\frac{a c^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{5/4}}-\frac{a c^{3/2} \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 )}{8 \sqrt{2} b^{5/4}}+\frac{a c^{3/2} \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 )}{8 \sqrt{2} b^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.144358, size = 241, normalized size = 0.78 \[ -\frac{(c x)^{3/2} \left (8 \sqrt [4]{b} \sqrt{x} \left (a-b x^2\right )^{3/4}+\sqrt{2} a \log \left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a-b x^2}}+1\right )-\sqrt{2} a \log \left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a-b x^2}}+1\right )+2 \sqrt{2} a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a-b x^2}}\right )-2 \sqrt{2} a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a-b x^2}}+1\right )\right )}{16 b^{5/4} x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [4]{-b{x}^{2}+a}}}}\, 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 (c x\right )^{\frac{3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8139, size = 709, normalized size = 2.3 \begin{align*} -\frac{4 \,{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} c + 4 \, \left (-\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}} b \arctan \left (-\frac{\left (-\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{3}{4}}{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} a b^{4} c -{\left (b^{5} x^{2} - a b^{4}\right )} \left (-\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{3}{4}} \sqrt{-\frac{\sqrt{-b x^{2} + a} a^{2} c^{3} x - \sqrt{-\frac{a^{4} c^{6}}{b^{5}}}{\left (b^{3} x^{2} - a b^{2}\right )}}{b x^{2} - a}}}{a^{4} b c^{6} x^{2} - a^{5} c^{6}}\right ) + \left (-\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}} b \log \left (\frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} a c + \left (-\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}}{\left (b^{2} x^{2} - a b\right )}}{b x^{2} - a}\right ) - \left (-\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}} b \log \left (\frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} a c - \left (-\frac{a^{4} c^{6}}{b^{5}}\right )^{\frac{1}{4}}{\left (b^{2} x^{2} - a b\right )}}{b x^{2} - a}\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.8288, size = 46, normalized size = 0.15 \begin{align*} \frac{c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \Gamma \left (\frac{9}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{\frac{3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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