Optimal. Leaf size=130 \[ -\frac{8 b^{5/2} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 a^{5/2} c^6 \sqrt [4]{a-b x^2}}-\frac{4 b \left (a-b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{9 a c (c x)^{9/2}} \]
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Rubi [A] time = 0.051073, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {325, 317, 335, 228} \[ -\frac{8 b^{5/2} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 a^{5/2} c^6 \sqrt [4]{a-b x^2}}-\frac{4 b \left (a-b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{9 a c (c x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 317
Rule 335
Rule 228
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{11/2} \sqrt [4]{a-b x^2}} \, dx &=-\frac{2 \left (a-b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}+\frac{(2 b) \int \frac{1}{(c x)^{7/2} \sqrt [4]{a-b x^2}} \, dx}{3 a c^2}\\ &=-\frac{2 \left (a-b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}-\frac{4 b \left (a-b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}+\frac{\left (4 b^2\right ) \int \frac{1}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \, dx}{15 a^2 c^4}\\ &=-\frac{2 \left (a-b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}-\frac{4 b \left (a-b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}+\frac{\left (4 b^2 \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x}\right ) \int \frac{1}{\sqrt [4]{1-\frac{a}{b x^2}} x^2} \, dx}{15 a^2 c^6 \sqrt [4]{a-b x^2}}\\ &=-\frac{2 \left (a-b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}-\frac{4 b \left (a-b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}-\frac{\left (4 b^2 \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{a x^2}{b}}} \, dx,x,\frac{1}{x}\right )}{15 a^2 c^6 \sqrt [4]{a-b x^2}}\\ &=-\frac{2 \left (a-b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}-\frac{4 b \left (a-b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}-\frac{8 b^{5/2} \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 a^{5/2} c^6 \sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.013212, size = 57, normalized size = 0.44 \[ -\frac{2 x \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (-\frac{9}{4},\frac{1}{4};-\frac{5}{4};\frac{b x^2}{a}\right )}{9 (c x)^{11/2} \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{11}{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{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{b c^{6} x^{8} - a c^{6} x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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