Optimal. Leaf size=35 \[ \frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a (c-a c x)^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6262, 663}
\begin {gather*} \frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a (c-a c x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 6262
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a (c-a c x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 37, normalized size = 1.06 \begin {gather*} \frac {2 (1+a x)^{5/2} (c-a c x)^{3/2}}{5 a (1-a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.26, size = 43, normalized size = 1.23
method | result | size |
gosper | \(\frac {2 \left (a x +1\right )^{4} \left (-c x a +c \right )^{\frac {3}{2}}}{5 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\) | \(34\) |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (a x +1\right )^{2} c}{5 \left (a x -1\right ) a}\) | \(43\) |
risch | \(-\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c^{2} \left (a^{2} x^{2}+2 a x +1\right ) \left (a x +1\right )}{5 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a \sqrt {\left (a x +1\right ) c}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (29) = 58\).
time = 0.30, size = 121, normalized size = 3.46 \begin {gather*} -\frac {2 \, c^{\frac {3}{2}}}{\sqrt {a x + 1} a} + \frac {2 \, {\left (a^{3} c^{\frac {3}{2}} x^{3} - 2 \, a^{2} c^{\frac {3}{2}} x^{2} + 8 \, a c^{\frac {3}{2}} x + 16 \, c^{\frac {3}{2}}\right )}}{5 \, \sqrt {a x + 1} a} + \frac {2 \, {\left (a^{2} c^{\frac {3}{2}} x^{2} - 4 \, a c^{\frac {3}{2}} x - 8 \, c^{\frac {3}{2}}\right )}}{\sqrt {a x + 1} a} + \frac {6 \, {\left (a c^{\frac {3}{2}} x + 2 \, c^{\frac {3}{2}}\right )}}{\sqrt {a x + 1} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 49, normalized size = 1.40 \begin {gather*} -\frac {2 \, {\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{5 \, {\left (a^{2} x - a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.92, size = 49, normalized size = 1.40 \begin {gather*} \frac {\sqrt {c-a\,c\,x}\,\left (\frac {6\,c\,x}{5}+\frac {2\,c}{5\,a}+\frac {2\,a^2\,c\,x^3}{5}+\frac {6\,a\,c\,x^2}{5}\right )}{\sqrt {1-a^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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