Optimal. Leaf size=308 \[ \frac{3 a c^{5/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{7/4}}-\frac{3 a c^{5/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{7/4}}-\frac{3 a c^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{7/4}}+\frac{3 a c^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{4 \sqrt{2} b^{7/4}}-\frac{c (c x)^{3/2} \sqrt [4]{a-b x^2}}{2 b} \]
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Rubi [A] time = 0.261505, 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, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{3 a c^{5/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{7/4}}-\frac{3 a c^{5/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{7/4}}-\frac{3 a c^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{7/4}}+\frac{3 a c^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{4 \sqrt{2} b^{7/4}}-\frac{c (c x)^{3/2} \sqrt [4]{a-b x^2}}{2 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 329
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(c x)^{5/2}}{\left (a-b x^2\right )^{3/4}} \, dx &=-\frac{c (c x)^{3/2} \sqrt [4]{a-b x^2}}{2 b}+\frac{\left (3 a c^2\right ) \int \frac{\sqrt{c x}}{\left (a-b x^2\right )^{3/4}} \, dx}{4 b}\\ &=-\frac{c (c x)^{3/2} \sqrt [4]{a-b x^2}}{2 b}+\frac{(3 a c) \operatorname{Subst}\left (\int \frac{x^2}{\left (a-\frac{b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt{c x}\right )}{2 b}\\ &=-\frac{c (c x)^{3/2} \sqrt [4]{a-b x^2}}{2 b}+\frac{(3 a c) \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 )}{2 b}\\ &=-\frac{c (c x)^{3/2} \sqrt [4]{a-b x^2}}{2 b}-\frac{(3 a c) \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^{3/2}}+\frac{(3 a c) \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^{3/2}}\\ &=-\frac{c (c x)^{3/2} \sqrt [4]{a-b x^2}}{2 b}+\frac{\left (3 a c^{5/2}\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^{7/4}}+\frac{\left (3 a c^{5/2}\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^{7/4}}+\frac{\left (3 a c^3\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^2}+\frac{\left (3 a c^3\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^2}\\ &=-\frac{c (c x)^{3/2} \sqrt [4]{a-b x^2}}{2 b}+\frac{3 a c^{5/2} \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^{7/4}}-\frac{3 a c^{5/2} \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^{7/4}}+\frac{\left (3 a c^{5/2}\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^{7/4}}-\frac{\left (3 a c^{5/2}\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^{7/4}}\\ &=-\frac{c (c x)^{3/2} \sqrt [4]{a-b x^2}}{2 b}-\frac{3 a c^{5/2} \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^{7/4}}+\frac{3 a c^{5/2} \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^{7/4}}+\frac{3 a c^{5/2} \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^{7/4}}-\frac{3 a c^{5/2} \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^{7/4}}\\ \end{align*}
Mathematica [A] time = 0.0562795, size = 112, normalized size = 0.36 \[ -\frac{(c x)^{5/2} \left (2 b x^{3/2} \sqrt [4]{a-b x^2}-3 a \sqrt [4]{-b} \tan ^{-1}\left (\frac{\sqrt [4]{-b} \sqrt{x}}{\sqrt [4]{a-b x^2}}\right )+3 a \sqrt [4]{-b} \tanh ^{-1}\left (\frac{\sqrt [4]{-b} \sqrt{x}}{\sqrt [4]{a-b x^2}}\right )\right )}{4 b^2 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{{\frac{5}{2}}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, 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{5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}\,{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 = 36.5276, size = 46, normalized size = 0.15 \begin{align*} \frac{c^{\frac{5}{2}} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} \Gamma \left (\frac{11}{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{5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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