Optimal. Leaf size=71 \[ \frac {8 c^5 \left (1-a^2 x^2\right )^{5/2}}{35 a (c-a c x)^{5/2}}+\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{7 a (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6127, 657, 649} \[ \frac {8 c^5 \left (1-a^2 x^2\right )^{5/2}}{35 a (c-a c x)^{5/2}}+\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{7 a (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 649
Rule 657
Rule 6127
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{\sqrt {c-a c x}} \, dx\\ &=\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{7 a (c-a c x)^{3/2}}+\frac {1}{7} \left (4 c^4\right ) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac {8 c^5 \left (1-a^2 x^2\right )^{5/2}}{35 a (c-a c x)^{5/2}}+\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{7 a (c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.65 \[ -\frac {2 c^2 (a x+1)^{5/2} (5 a x-9) \sqrt {c-a c x}}{35 a \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 68, normalized size = 0.96 \[ \frac {2 \, {\left (5 \, a^{3} c^{2} x^{3} + a^{2} c^{2} x^{2} - 13 \, a c^{2} x - 9 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{35 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 47, normalized size = 0.66 \[ -\frac {2 \, {\left (16 \, \sqrt {2} c^{\frac {3}{2}} + \frac {5 \, {\left (a c x + c\right )}^{\frac {7}{2}} - 14 \, {\left (a c x + c\right )}^{\frac {5}{2}} c}{c^{2}}\right )} c^{2}}{35 \, a {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 47, normalized size = 0.66 \[ \frac {2 \left (a x +1\right )^{4} \left (5 a x -9\right ) \left (-a c x +c \right )^{\frac {5}{2}}}{35 a \left (a x -1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 164, normalized size = 2.31 \[ -\frac {2 \, {\left (a^{4} c^{\frac {5}{2}} x^{4} - 3 \, a^{3} c^{\frac {5}{2}} x^{3} + 6 \, a^{2} c^{\frac {5}{2}} x^{2} - 24 \, a c^{\frac {5}{2}} x - 48 \, c^{\frac {5}{2}}\right )}}{7 \, \sqrt {a x + 1} a} - \frac {2 \, {\left (3 \, a^{3} c^{\frac {5}{2}} x^{3} - 11 \, a^{2} c^{\frac {5}{2}} x^{2} + 44 \, a c^{\frac {5}{2}} x + 88 \, c^{\frac {5}{2}}\right )}}{5 \, \sqrt {a x + 1} a} - \frac {2 \, {\left (a^{2} c^{\frac {5}{2}} x^{2} - 7 \, a c^{\frac {5}{2}} x - 14 \, c^{\frac {5}{2}}\right )}}{\sqrt {a x + 1} a} - \frac {2 \, {\left (a c^{\frac {5}{2}} x + 3 \, c^{\frac {5}{2}}\right )}}{\sqrt {a x + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 68, normalized size = 0.96 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {44\,c^2\,x}{35}+\frac {18\,c^2}{35\,a}+\frac {24\,a\,c^2\,x^2}{35}-\frac {12\,a^2\,c^2\,x^3}{35}-\frac {2\,a^3\,c^2\,x^4}{7}\right )}{\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {5}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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