Optimal. Leaf size=74 \[ \frac {4 \sqrt [8]{2} \sqrt [8]{1-a^2 x^2} \, _2F_1\left (-\frac {3}{8},\frac {7}{8};\frac {5}{8};\frac {1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 74, normalized size of antiderivative = 1.00, 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 69
Rule 6140
Rule 6143
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*}
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Mathematica [A] time = 0.03, size = 69, normalized size = 0.93 \[ \frac {4 \sqrt [8]{2-2 a^2 x^2} \, _2F_1\left (-\frac {3}{8},\frac {7}{8};\frac {5}{8};\frac {1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{8}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{8}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{{\left (c-a^2\,c\,x^2\right )}^{9/8}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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