Optimal. Leaf size=74 \[ \frac{4 \sqrt [8]{2} \sqrt [8]{1-a^2 x^2} \text{Hypergeometric2F1}\left (-\frac{3}{8},\frac{7}{8},\frac{5}{8},\frac{1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]
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Rubi [A] time = 0.0903437, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {6143, 6140, 69} \[ \frac{4 \sqrt [8]{2} \sqrt [8]{1-a^2 x^2} \, _2F_1\left (-\frac{3}{8},\frac{7}{8};\frac{5}{8};\frac{1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/8}} \, dx &=\frac{\sqrt [8]{1-a^2 x^2} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{9/8}} \, dx}{c \sqrt [8]{c-a^2 c x^2}}\\ &=\frac{\sqrt [8]{1-a^2 x^2} \int \frac{1}{(1-a x)^{11/8} (1+a x)^{7/8}} \, dx}{c \sqrt [8]{c-a^2 c x^2}}\\ &=\frac{4 \sqrt [8]{2} \sqrt [8]{1-a^2 x^2} \, _2F_1\left (-\frac{3}{8},\frac{7}{8};\frac{5}{8};\frac{1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0256603, size = 69, normalized size = 0.93 \[ \frac{4 \sqrt [8]{2-2 a^2 x^2} \text{Hypergeometric2F1}\left (-\frac{3}{8},\frac{7}{8},\frac{5}{8},\frac{1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{9}{8}}}}\, 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{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{8}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{8}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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