Optimal. Leaf size=83 \[ -\frac{64 \left (a+b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac{16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}} \]
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Rubi [A] time = 0.024621, 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} \[ -\frac{64 \left (a+b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac{16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}-\frac{8 \int \frac{\sqrt [4]{a+b x^2}}{(c x)^{11/2}} \, dx}{a}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}+\frac{16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}+\frac{32 \int \frac{\left (a+b x^2\right )^{5/4}}{(c x)^{11/2}} \, dx}{5 a^2}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}+\frac{16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac{64 \left (a+b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0233931, size = 52, normalized size = 0.63 \[ -\frac{2 \sqrt{c x} \sqrt [4]{a+b x^2} \left (5 a^2-8 a b x^2+32 b^2 x^4\right )}{45 a^3 c^6 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 42, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( 32\,{b}^{2}{x}^{4}-8\,ab{x}^{2}+5\,{a}^{2} \right ) }{45\,{a}^{3}}\sqrt [4]{b{x}^{2}+a} \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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{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 = 1.29528, size = 111, normalized size = 1.34 \begin{align*} -\frac{2 \,{\left (32 \, b^{2} x^{4} - 8 \, a b x^{2} + 5 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{45 \, a^{3} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{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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