Optimal. Leaf size=137 \[ \frac{1}{5} x^4 \sqrt{c-a^2 c x^2}-\frac{x^3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 x^2 \sqrt{c-a^2 c x^2}}{5 a^2}+\frac{3 (8-5 a x) \sqrt{c-a^2 c x^2}}{20 a^4}+\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{4 a^4} \]
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Rubi [A] time = 0.429526, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6167, 6152, 1809, 833, 780, 217, 203} \[ \frac{1}{5} x^4 \sqrt{c-a^2 c x^2}-\frac{x^3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 x^2 \sqrt{c-a^2 c x^2}}{5 a^2}+\frac{3 (8-5 a x) \sqrt{c-a^2 c x^2}}{20 a^4}+\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{4 a^4} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6152
Rule 1809
Rule 833
Rule 780
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt{c-a^2 c x^2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x^3 \sqrt{c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac{x^3 (1-a x)^2}{\sqrt{c-a^2 c x^2}} \, dx\right )\\ &=\frac{1}{5} x^4 \sqrt{c-a^2 c x^2}+\frac{\int \frac{x^3 \left (-9 a^2 c+10 a^3 c x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{5 a^2}\\ &=-\frac{x^3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{1}{5} x^4 \sqrt{c-a^2 c x^2}-\frac{\int \frac{x^2 \left (-30 a^3 c^2+36 a^4 c^2 x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{20 a^4 c}\\ &=\frac{3 x^2 \sqrt{c-a^2 c x^2}}{5 a^2}-\frac{x^3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{1}{5} x^4 \sqrt{c-a^2 c x^2}+\frac{\int \frac{x \left (-72 a^4 c^3+90 a^5 c^3 x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{60 a^6 c^2}\\ &=\frac{3 x^2 \sqrt{c-a^2 c x^2}}{5 a^2}-\frac{x^3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{1}{5} x^4 \sqrt{c-a^2 c x^2}+\frac{3 (8-5 a x) \sqrt{c-a^2 c x^2}}{20 a^4}+\frac{(3 c) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{4 a^3}\\ &=\frac{3 x^2 \sqrt{c-a^2 c x^2}}{5 a^2}-\frac{x^3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{1}{5} x^4 \sqrt{c-a^2 c x^2}+\frac{3 (8-5 a x) \sqrt{c-a^2 c x^2}}{20 a^4}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )}{4 a^3}\\ &=\frac{3 x^2 \sqrt{c-a^2 c x^2}}{5 a^2}-\frac{x^3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{1}{5} x^4 \sqrt{c-a^2 c x^2}+\frac{3 (8-5 a x) \sqrt{c-a^2 c x^2}}{20 a^4}+\frac{3 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{4 a^4}\\ \end{align*}
Mathematica [A] time = 0.127231, size = 96, normalized size = 0.7 \[ \frac{\left (4 a^4 x^4-10 a^3 x^3+12 a^2 x^2-15 a x+24\right ) \sqrt{c-a^2 c x^2}-15 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )}{20 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 202, normalized size = 1.5 \begin{align*} -{\frac{{x}^{2}}{5\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{4}{5\,c{a}^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{x}{2\,{a}^{3}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,x}{4\,{a}^{3}}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{5\,c}{4\,{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+2\,{\frac{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}{{a}^{4}}}+2\,{\frac{c}{{a}^{3}\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77678, size = 431, normalized size = 3.15 \begin{align*} \left [\frac{2 \,{\left (4 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 15 \, a x + 24\right )} \sqrt{-a^{2} c x^{2} + c} + 15 \, \sqrt{-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right )}{40 \, a^{4}}, \frac{{\left (4 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 15 \, a x + 24\right )} \sqrt{-a^{2} c x^{2} + c} - 15 \, \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right )}{20 \, a^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16753, size = 124, normalized size = 0.91 \begin{align*} \frac{1}{20} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left ({\left (2 \, x - \frac{5}{a}\right )} x + \frac{6}{a^{2}}\right )} x - \frac{15}{a^{3}}\right )} x + \frac{24}{a^{4}}\right )} - \frac{3 \, c \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{4 \, a^{3} \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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