Optimal. Leaf size=83 \[ \frac{64 \left (a+b x^2\right )^{7/4}}{21 a^3 c (c x)^{7/2}}-\frac{16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}+\frac{2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0235764, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {273, 264} \[ \frac{64 \left (a+b x^2\right )^{7/4}}{21 a^3 c (c x)^{7/2}}-\frac{16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}+\frac{2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx &=\frac{2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}}+\frac{8 \int \frac{1}{(c x)^{9/2} \sqrt [4]{a+b x^2}} \, dx}{a}\\ &=\frac{2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}-\frac{32 \int \frac{\left (a+b x^2\right )^{3/4}}{(c x)^{9/2}} \, dx}{3 a^2}\\ &=\frac{2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}+\frac{64 \left (a+b x^2\right )^{7/4}}{21 a^3 c (c x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0109534, size = 47, normalized size = 0.57 \[ -\frac{2 x \left (3 a^2-8 a b x^2-32 b^2 x^4\right )}{21 a^3 (c x)^{9/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 42, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( -32\,{b}^{2}{x}^{4}-8\,ab{x}^{2}+3\,{a}^{2} \right ) }{21\,{a}^{3}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( cx \right ) ^{-{\frac{9}{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{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59913, size = 131, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (32 \, b^{2} x^{4} + 8 \, a b x^{2} - 3 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{21 \,{\left (a^{3} b c^{5} x^{6} + a^{4} c^{5} x^{4}\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{5}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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