Optimal. Leaf size=55 \[ \frac{8 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}} \]
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Rubi [A] time = 0.0143003, 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} \[ \frac{8 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}}-\frac{4 \int \frac{\sqrt [4]{a+b x^2}}{(c x)^{7/2}} \, dx}{a}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}}+\frac{8 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0169781, size = 34, normalized size = 0.62 \[ -\frac{2 x \left (a-4 b x^2\right ) \sqrt [4]{a+b x^2}}{5 a^2 (c x)^{7/2}} \]
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 ) }{5\,{a}^{2}}\sqrt [4]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{7}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38041, size = 82, normalized size = 1.49 \begin{align*} \frac{2 \,{\left (4 \, b x^{2} - a\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{5 \, a^{2} c^{4} x^{3}} \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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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