Optimal. Leaf size=107 \[ \frac{5 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}+\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{(a x+1) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a} \]
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Rubi [A] time = 0.0855183, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6141, 671, 641, 195, 217, 203} \[ \frac{5 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}+\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{(a x+1) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a} \]
Antiderivative was successfully verified.
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Rule 6141
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 )^{3/2} \, dx &=c \int (1+a x)^2 \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{(1+a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac{1}{4} (5 c) \int (1+a x) \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1+a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac{1}{4} (5 c) \int \sqrt{c-a^2 c x^2} \, dx\\ &=\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1+a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac{1}{8} \left (5 c^2\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1+a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac{1}{8} \left (5 c^2\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{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac{(1+a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac{5 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.0899084, size = 117, normalized size = 1.09 \[ -\frac{c \sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (6 a^4 x^4+10 a^3 x^3-7 a^2 x^2-25 a x+16\right )+30 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{24 a \sqrt{1-a x} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.036, size = 186, normalized size = 1.7 \begin{align*} -{\frac{x}{4} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,cx}{8}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{3\,{c}^{2}}{8}\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}{3\,a} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+c\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }x+{{c}^{2}\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 = 2.68867, size = 413, normalized size = 3.86 \begin{align*} \left [\frac{15 \, \sqrt{-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) + 2 \,{\left (6 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} + 9 \, a c x - 16 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{48 \, a}, -\frac{15 \, c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) -{\left (6 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} + 9 \, a c x - 16 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{24 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 9.04011, size = 340, normalized size = 3.18 \begin{align*} a^{2} c \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{8 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{8 a^{3}} & \text{otherwise} \end{cases}\right ) + 2 a c \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 \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 [A] time = 1.13948, size = 115, normalized size = 1.07 \begin{align*} \frac{1}{24} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left (3 \, a^{2} c x + 8 \, a c\right )} x + 9 \, c\right )} x - \frac{16 \, c}{a}\right )} - \frac{5 \, c^{2} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{8 \, \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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