Optimal. Leaf size=55 \[ \frac{2}{a c (c x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{3/2}} \]
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Rubi [A] time = 0.0148584, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {273, 264} \[ \frac{2}{a c (c x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx &=\frac{2}{a c (c x)^{3/2} \sqrt [4]{a+b x^2}}+\frac{4 \int \frac{1}{(c x)^{5/2} \sqrt [4]{a+b x^2}} \, dx}{a}\\ &=\frac{2}{a c (c x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0096554, size = 34, normalized size = 0.62 \[ -\frac{2 x \left (a+4 b x^2\right )}{3 a^2 (c x)^{5/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 29, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( 4\,b{x}^{2}+a \right ) }{3\,{a}^{2}} \left ( cx \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}} \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{5}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53265, size = 105, normalized size = 1.91 \begin{align*} -\frac{2 \,{\left (4 \, b x^{2} + a\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{3 \,{\left (a^{2} b c^{3} x^{4} + a^{3} c^{3} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 128.074, size = 78, normalized size = 1.42 \begin{align*} \frac{\Gamma \left (- \frac{3}{4}\right )}{8 a \sqrt [4]{b} c^{\frac{5}{2}} x^{2} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (\frac{5}{4}\right )} + \frac{b^{\frac{3}{4}} \Gamma \left (- \frac{3}{4}\right )}{2 a^{2} c^{\frac{5}{2}} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (\frac{5}{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{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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