Optimal. Leaf size=200 \[ -\frac {4 x^2 \sqrt [8]{a x+1} \sqrt [8]{1-a^2 x^2}}{7 a^2 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}}+\frac {64 \sqrt [8]{2} (1-a x)^{5/8} \sqrt [8]{1-a^2 x^2} \, _2F_1\left (\frac {5}{8},\frac {7}{8};\frac {13}{8};\frac {1}{2} (1-a x)\right )}{105 a^4 c \sqrt [8]{c-a^2 c x^2}}+\frac {8 (6-a x) \sqrt [8]{a x+1} \sqrt [8]{1-a^2 x^2}}{21 a^4 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]
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Rubi [A] time = 0.25, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {6153, 6150, 100, 146, 69} \[ \frac {64 \sqrt [8]{2} (1-a x)^{5/8} \sqrt [8]{1-a^2 x^2} \, _2F_1\left (\frac {5}{8},\frac {7}{8};\frac {13}{8};\frac {1}{2} (1-a x)\right )}{105 a^4 c \sqrt [8]{c-a^2 c x^2}}-\frac {4 x^2 \sqrt [8]{a x+1} \sqrt [8]{1-a^2 x^2}}{7 a^2 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}}+\frac {8 (6-a x) \sqrt [8]{a x+1} \sqrt [8]{1-a^2 x^2}}{21 a^4 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 146
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)} x^3}{\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)} x^3}{\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 {x^3}{(1-a x)^{11/8} (1+a x)^{7/8}} \, dx}{c \sqrt [8]{c-a^2 c x^2}}\\ &=-\frac {4 x^2 \sqrt [8]{1+a x} \sqrt [8]{1-a^2 x^2}}{7 a^2 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}}-\frac {\left (4 \sqrt [8]{1-a^2 x^2}\right ) \int \frac {x \left (-2-\frac {a x}{2}\right )}{(1-a x)^{11/8} (1+a x)^{7/8}} \, dx}{7 a^2 c \sqrt [8]{c-a^2 c x^2}}\\ &=-\frac {4 x^2 \sqrt [8]{1+a x} \sqrt [8]{1-a^2 x^2}}{7 a^2 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}}+\frac {8 (6-a x) \sqrt [8]{1+a x} \sqrt [8]{1-a^2 x^2}}{21 a^4 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}}-\frac {\left (16 \sqrt [8]{1-a^2 x^2}\right ) \int \frac {1}{(1-a x)^{3/8} (1+a x)^{7/8}} \, dx}{21 a^3 c \sqrt [8]{c-a^2 c x^2}}\\ &=-\frac {4 x^2 \sqrt [8]{1+a x} \sqrt [8]{1-a^2 x^2}}{7 a^2 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}}+\frac {8 (6-a x) \sqrt [8]{1+a x} \sqrt [8]{1-a^2 x^2}}{21 a^4 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}}+\frac {64 \sqrt [8]{2} (1-a x)^{5/8} \sqrt [8]{1-a^2 x^2} \, _2F_1\left (\frac {5}{8},\frac {7}{8};\frac {13}{8};\frac {1}{2} (1-a x)\right )}{105 a^4 c \sqrt [8]{c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 107, normalized size = 0.54 \[ -\frac {4 \sqrt [8]{1-a^2 x^2} \left (5 \sqrt [8]{a x+1} \left (3 a^2 x^2+2 a x-12\right )+16 \sqrt [8]{2} (a x-1) \, _2F_1\left (\frac {5}{8},\frac {7}{8};\frac {13}{8};\frac {1}{2} (1-a x)\right )\right )}{105 a^4 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, x^{3}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{8}}}\, dx \]
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 \[ \int \frac {x^{3} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{8}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{{\left (c-a^2\,c\,x^2\right )}^{9/8}} \,d x \]
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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