Optimal. Leaf size=56 \[ \frac{4 \sqrt [4]{c x} \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{8},\frac{1}{4};\frac{9}{8};-\frac{b x^2}{a}\right )}{c \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0168982, antiderivative size = 56, 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 = {365, 364} \[ \frac{4 \sqrt [4]{c x} \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{8},\frac{1}{4};\frac{9}{8};-\frac{b x^2}{a}\right )}{c \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{3/4} \sqrt [4]{a+b x^2}} \, dx &=\frac{\sqrt [4]{1+\frac{b x^2}{a}} \int \frac{1}{(c x)^{3/4} \sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac{4 \sqrt [4]{c x} \sqrt [4]{1+\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{8},\frac{1}{4};\frac{9}{8};-\frac{b x^2}{a}\right )}{c \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0103394, size = 54, normalized size = 0.96 \[ \frac{4 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{8},\frac{1}{4};\frac{9}{8};-\frac{b x^2}{a}\right )}{(c x)^{3/4} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \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{1}{4}} \left (c x\right )^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{1}{4}}}{b c x^{3} + a c x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.50702, size = 44, normalized size = 0.79 \begin{align*} \frac{\sqrt [4]{x} \Gamma \left (\frac{1}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{8}, \frac{1}{4} \\ \frac{9}{8} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} c^{\frac{3}{4}} \Gamma \left (\frac{9}{8}\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{1}{4}} \left (c x\right )^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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