Integrand size = 27, antiderivative size = 228 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {4 \sqrt {c-a^2 c x^2}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x^2 \sqrt {c-a^2 c x^2}}{3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 x^3 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^4 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^5 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.19 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6327, 6328, 90} \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {4 x^2 \sqrt {c-a^2 c x^2}}{3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^4 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 x^3 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^5 x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 90
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^4 \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \frac {x^3 (1+a x)^2}{-1+a x} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \left (\frac {4}{a^3}+\frac {4 x}{a^2}+\frac {4 x^2}{a}+3 x^3+a x^4+\frac {4}{a^3 (-1+a x)}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {4 \sqrt {c-a^2 c x^2}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x^2 \sqrt {c-a^2 c x^2}}{3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 x^3 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^4 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^5 \sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.38 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (\frac {4 x}{a^4}+\frac {2 x^2}{a^3}+\frac {4 x^3}{3 a^2}+\frac {3 x^4}{4 a}+\frac {x^5}{5}+\frac {4 \log (1-a x)}{a^5}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.57 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.40
method | result | size |
default | \(\frac {\left (12 a^{5} x^{5}+45 a^{4} x^{4}+80 a^{3} x^{3}+120 a^{2} x^{2}+240 a x +240 \ln \left (a x -1\right )\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x -1\right )}{60 a^{4} \left (a x +1\right )^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(92\) |
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Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.25 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {{\left (12 \, a^{5} x^{5} + 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} + 120 \, a^{2} x^{2} + 240 \, a x + 240 \, \log \left (a x - 1\right )\right )} \sqrt {-a^{2} c}}{60 \, a^{5}} \]
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\[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^{3} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} x^{3}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} x^{3}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^3\,\sqrt {c-a^2\,c\,x^2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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