Optimal. Leaf size=59 \[ -\frac{4 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac{1}{8},\frac{7}{4};\frac{7}{8};-\frac{b x^2}{a}\right )}{a c \sqrt [4]{c x} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.0190109, antiderivative size = 59, 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} \[ -\frac{4 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac{1}{8},\frac{7}{4};\frac{7}{8};-\frac{b x^2}{a}\right )}{a c \sqrt [4]{c x} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{5/4} \left (a+b x^2\right )^{7/4}} \, dx &=\frac{\left (1+\frac{b x^2}{a}\right )^{3/4} \int \frac{1}{(c x)^{5/4} \left (1+\frac{b x^2}{a}\right )^{7/4}} \, dx}{a \left (a+b x^2\right )^{3/4}}\\ &=-\frac{4 \left (1+\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (-\frac{1}{8},\frac{7}{4};\frac{7}{8};-\frac{b x^2}{a}\right )}{a c \sqrt [4]{c x} \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0108342, size = 57, normalized size = 0.97 \[ -\frac{4 x \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac{1}{8},\frac{7}{4};\frac{7}{8};-\frac{b x^2}{a}\right )}{a (c x)^{5/4} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{5}{4}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, 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{7}{4}} \left (c x\right )^{\frac{5}{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{1}{4}} \left (c x\right )^{\frac{3}{4}}}{b^{2} c^{2} x^{6} + 2 \, a b c^{2} x^{4} + a^{2} c^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 133.22, size = 48, normalized size = 0.81 \begin{align*} \frac{\Gamma \left (- \frac{1}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{8}, \frac{7}{4} \\ \frac{7}{8} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{7}{4}} c^{\frac{5}{4}} \sqrt [4]{x} \Gamma \left (\frac{7}{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{7}{4}} \left (c x\right )^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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