Optimal. Leaf size=131 \[ \frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}+\frac{7 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{16 a}+\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a} \]
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Rubi [A] time = 0.0968, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6142, 671, 641, 195, 217, 203} \[ \frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}+\frac{7 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{16 a}+\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a} \]
Antiderivative was successfully verified.
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Rule 6142
Rule 671
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=c \int (1-a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=\frac{(1-a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{1}{6} (7 c) \int (1-a x) \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{1}{6} (7 c) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{1}{8} \left (7 c^2\right ) \int \sqrt{c-a^2 c x^2} \, dx\\ &=\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}+\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{1}{16} \left (7 c^3\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}+\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{1}{16} \left (7 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}+\frac{7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac{7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac{7 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{16 a}\\ \end{align*}
Mathematica [A] time = 0.139402, size = 135, normalized size = 1.03 \[ \frac{c^2 \sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (40 a^6 x^6-136 a^5 x^5+86 a^4 x^4+202 a^3 x^3-327 a^2 x^2+39 a x+96\right )-210 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{240 a \sqrt{1-a x} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.036, size = 226, normalized size = 1.7 \begin{align*} -{\frac{x}{6} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,cx}{24} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,x{c}^{2}}{16}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{5\,{c}^{3}}{16}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+{\frac{2}{5\,a} \left ( -c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{cx}{2} \left ( -c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,x{c}^{2}}{4}\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}+{\frac{3\,{c}^{3}}{4}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}} \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 = 2.67876, size = 545, normalized size = 4.16 \begin{align*} \left [\frac{105 \, \sqrt{-c} c^{2} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) - 2 \,{\left (40 \, a^{5} c^{2} x^{5} - 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x - 96 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{480 \, a}, -\frac{105 \, c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) +{\left (40 \, a^{5} c^{2} x^{5} - 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x - 96 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{240 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 13.758, size = 478, normalized size = 3.65 \begin{align*} - a^{4} c^{2} \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{16 a^{5}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{16 a^{5}} & \text{otherwise} \end{cases}\right ) + 2 a^{3} c^{2} \left (\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right ) - 2 a c^{2} \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} c x^{2} + c\right )^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right ) + c^{2} \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{2 a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{2 a} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38837, size = 432, normalized size = 3.3 \begin{align*} -\frac{{\left (6720 \, a^{7} c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{-c + \frac{2 \, c}{a x + 1}}}{\sqrt{c}}\right ) \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - \frac{{\left (105 \, a^{7}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{5} c^{3} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - 595 \, a^{7}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{4} c^{4} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - 1686 \, a^{7}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{3} c^{5} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 1386 \, a^{7}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{2} c^{6} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 105 \, a^{7} c^{8} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 595 \, a^{7} c^{7}{\left (-c + \frac{2 \, c}{a x + 1}\right )}^{\frac{3}{2}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )\right )}{\left (a x + 1\right )}^{6}}{c^{6}}\right )}{\left | a \right |}}{7680 \, a^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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