Optimal. Leaf size=172 \[ \frac {x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 a (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {5 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}-\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 a \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6192, 6193, 88} \[ \frac {x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 a (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {5 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}-\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 a \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {x^3}{(-1+a x)^2 (1+a x)} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \left (\frac {1}{a^3}+\frac {1}{2 a^3 (-1+a x)^2}+\frac {5}{4 a^3 (-1+a x)}-\frac {1}{4 a^3 (1+a x)}\right ) \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 a (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 71, normalized size = 0.41 \[ \frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (4 a x+\frac {2}{1-a x}+5 \log (1-a x)-\log (a x+1)\right )}{4 a \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 69, normalized size = 0.40 \[ \frac {{\left (4 \, a^{2} x^{2} - 4 \, a x - {\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2\right )} \sqrt {-a^{2} c}}{4 \, {\left (a^{6} c^{2} x - a^{5} c^{2}\right )}} \]
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 [A] time = 0.06, size = 97, normalized size = 0.56 \[ \frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (4 a^{2} x^{2}+5 \ln \left (a x -1\right ) x a -a x \ln \left (a x +1\right )-4 a x -5 \ln \left (a x -1\right )+\ln \left (a x +1\right )-2\right )}{4 \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-1\right ) c^{2} a^{4}} \]
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 \[ \int \frac {x^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
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 \[ \int \frac {x^3}{{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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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