Optimal. Leaf size=111 \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{4 a^2}-\frac{1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}+\frac{c x \sqrt{c-a^2 c x^2}}{4 a}-\frac{(15 a x+14) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2} \]
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Rubi [A] time = 0.194993, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {6151, 1809, 780, 195, 217, 203} \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{4 a^2}-\frac{1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}+\frac{c x \sqrt{c-a^2 c x^2}}{4 a}-\frac{(15 a x+14) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1809
Rule 780
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right )^{3/2} \, dx &=c \int x (1+a x)^2 \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac{\int x \left (-7 a^2 c-10 a^3 c x\right ) \sqrt{c-a^2 c x^2} \, dx}{5 a^2}\\ &=-\frac{1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac{(14+15 a x) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2}+\frac{c \int \sqrt{c-a^2 c x^2} \, dx}{2 a}\\ &=\frac{c x \sqrt{c-a^2 c x^2}}{4 a}-\frac{1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac{(14+15 a x) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2}+\frac{c^2 \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{4 a}\\ &=\frac{c x \sqrt{c-a^2 c x^2}}{4 a}-\frac{1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac{(14+15 a x) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2}+\frac{c^2 \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}\\ &=\frac{c x \sqrt{c-a^2 c x^2}}{4 a}-\frac{1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac{(14+15 a x) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2}+\frac{c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.0991286, size = 97, normalized size = 0.87 \[ \frac{c \left (12 a^4 x^4+30 a^3 x^3+16 a^2 x^2-15 a x-28\right ) \sqrt{c-a^2 c x^2}-15 c^{3/2} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )}{60 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 222, normalized size = 2. \begin{align*}{\frac{1}{5\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{x}{2\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,cx}{4\,a}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{3\,{c}^{2}}{4\,a}\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}^{2}} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{cx}{a}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}+{\frac{{c}^{2}}{a}\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 = 2.853, size = 466, normalized size = 4.2 \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 (12 \, a^{4} c x^{4} + 30 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} - 15 \, a c x - 28 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{120 \, a^{2}}, -\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 (12 \, a^{4} c x^{4} + 30 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} - 15 \, a c x - 28 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{60 \, a^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.7533, size = 306, normalized size = 2.76 \begin{align*} a^{2} c \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 \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 ) + 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16434, size = 132, normalized size = 1.19 \begin{align*} \frac{1}{60} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left (3 \,{\left (2 \, a^{2} c x + 5 \, a c\right )} x + 8 \, c\right )} x - \frac{15 \, c}{a}\right )} x - \frac{28 \, c}{a^{2}}\right )} - \frac{c^{2} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{4 \, a \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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