Optimal. Leaf size=57 \[ \frac{8 \left (a+b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{9/2}} \]
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Rubi [A] time = 0.0145266, antiderivative size = 57, 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} \[ \frac{8 \left (a+b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^2}}{(c x)^{11/2}} \, dx &=-\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{9/2}}-\frac{4 \int \frac{\left (a+b x^2\right )^{5/4}}{(c x)^{11/2}} \, dx}{5 a}\\ &=-\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{9/2}}+\frac{8 \left (a+b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0148063, size = 41, normalized size = 0.72 \[ \frac{2 \sqrt{c x} \left (a+b x^2\right )^{5/4} \left (4 b x^2-5 a\right )}{45 a^2 c^6 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 31, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( -4\,b{x}^{2}+5\,a \right ) }{45\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{4}}} \left ( cx \right ) ^{-{\frac{11}{2}}}} \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{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69684, size = 105, normalized size = 1.84 \begin{align*} \frac{2 \,{\left (4 \, b^{2} x^{4} - a b x^{2} - 5 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{45 \, a^{2} c^{6} x^{5}} \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 [B] time = 2.25524, size = 144, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (\frac{9 \,{\left (b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}{\left (b c^{2} + \frac{a c^{2}}{x^{2}}\right )} b c^{2}}{\sqrt{c x}} - \frac{5 \,{\left (b^{2} c^{8} x^{4} + 2 \, a b c^{8} x^{2} + a^{2} c^{8}\right )}{\left (b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}}{\sqrt{c x} c^{4} x^{4}}\right )}}{45 \, a^{2} c^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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