Optimal. Leaf size=137 \[ \frac {32 \sqrt {2} c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {32 c^3 \sqrt {c-a c x}}{a}-\frac {16 c^2 (c-a c x)^{3/2}}{3 a}-\frac {2 (c-a c x)^{9/2}}{9 a c}-\frac {4 (c-a c x)^{7/2}}{7 a}-\frac {8 c (c-a c x)^{5/2}}{5 a} \]
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Rubi [A] time = 0.16, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6167, 6130, 21, 50, 63, 206} \[ -\frac {16 c^2 (c-a c x)^{3/2}}{3 a}-\frac {32 c^3 \sqrt {c-a c x}}{a}+\frac {32 \sqrt {2} c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {2 (c-a c x)^{9/2}}{9 a c}-\frac {4 (c-a c x)^{7/2}}{7 a}-\frac {8 c (c-a c x)^{5/2}}{5 a} \]
Antiderivative was successfully verified.
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Rule 21
Rule 50
Rule 63
Rule 206
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^{7/2} \, dx\\ &=-\int \frac {(1-a x) (c-a c x)^{7/2}}{1+a x} \, dx\\ &=-\frac {\int \frac {(c-a c x)^{9/2}}{1+a x} \, dx}{c}\\ &=-\frac {2 (c-a c x)^{9/2}}{9 a c}-2 \int \frac {(c-a c x)^{7/2}}{1+a x} \, dx\\ &=-\frac {4 (c-a c x)^{7/2}}{7 a}-\frac {2 (c-a c x)^{9/2}}{9 a c}-(4 c) \int \frac {(c-a c x)^{5/2}}{1+a x} \, dx\\ &=-\frac {8 c (c-a c x)^{5/2}}{5 a}-\frac {4 (c-a c x)^{7/2}}{7 a}-\frac {2 (c-a c x)^{9/2}}{9 a c}-\left (8 c^2\right ) \int \frac {(c-a c x)^{3/2}}{1+a x} \, dx\\ &=-\frac {16 c^2 (c-a c x)^{3/2}}{3 a}-\frac {8 c (c-a c x)^{5/2}}{5 a}-\frac {4 (c-a c x)^{7/2}}{7 a}-\frac {2 (c-a c x)^{9/2}}{9 a c}-\left (16 c^3\right ) \int \frac {\sqrt {c-a c x}}{1+a x} \, dx\\ &=-\frac {32 c^3 \sqrt {c-a c x}}{a}-\frac {16 c^2 (c-a c x)^{3/2}}{3 a}-\frac {8 c (c-a c x)^{5/2}}{5 a}-\frac {4 (c-a c x)^{7/2}}{7 a}-\frac {2 (c-a c x)^{9/2}}{9 a c}-\left (32 c^4\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=-\frac {32 c^3 \sqrt {c-a c x}}{a}-\frac {16 c^2 (c-a c x)^{3/2}}{3 a}-\frac {8 c (c-a c x)^{5/2}}{5 a}-\frac {4 (c-a c x)^{7/2}}{7 a}-\frac {2 (c-a c x)^{9/2}}{9 a c}+\frac {\left (64 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a}\\ &=-\frac {32 c^3 \sqrt {c-a c x}}{a}-\frac {16 c^2 (c-a c x)^{3/2}}{3 a}-\frac {8 c (c-a c x)^{5/2}}{5 a}-\frac {4 (c-a c x)^{7/2}}{7 a}-\frac {2 (c-a c x)^{9/2}}{9 a c}+\frac {32 \sqrt {2} c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 88, normalized size = 0.64 \[ \frac {2 c^3 \left (\left (-35 a^4 x^4+230 a^3 x^3-732 a^2 x^2+1754 a x-6257\right ) \sqrt {c-a c x}+5040 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{315 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 204, normalized size = 1.49 \[ \left [\frac {2 \, {\left (2520 \, \sqrt {2} c^{\frac {7}{2}} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) - {\left (35 \, a^{4} c^{3} x^{4} - 230 \, a^{3} c^{3} x^{3} + 732 \, a^{2} c^{3} x^{2} - 1754 \, a c^{3} x + 6257 \, c^{3}\right )} \sqrt {-a c x + c}\right )}}{315 \, a}, -\frac {2 \, {\left (5040 \, \sqrt {2} \sqrt {-c} c^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) + {\left (35 \, a^{4} c^{3} x^{4} - 230 \, a^{3} c^{3} x^{3} + 732 \, a^{2} c^{3} x^{2} - 1754 \, a c^{3} x + 6257 \, c^{3}\right )} \sqrt {-a c x + c}\right )}}{315 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 161, normalized size = 1.18 \[ -\frac {32 \, \sqrt {2} c^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {2 \, {\left (35 \, {\left (a c x - c\right )}^{4} \sqrt {-a c x + c} a^{8} c^{8} - 90 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{8} c^{9} + 252 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{8} c^{10} + 840 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{8} c^{11} + 5040 \, \sqrt {-a c x + c} a^{8} c^{12}\right )}}{315 \, a^{9} c^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 101, normalized size = 0.74 \[ -\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-a c x +c \right )^{\frac {7}{2}} c}{7}+\frac {4 \left (-a c x +c \right )^{\frac {5}{2}} c^{2}}{5}+\frac {8 c^{3} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+16 \sqrt {-a c x +c}\, c^{4}-16 c^{\frac {9}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 123, normalized size = 0.90 \[ -\frac {2 \, {\left (2520 \, \sqrt {2} c^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 35 \, {\left (-a c x + c\right )}^{\frac {9}{2}} + 90 \, {\left (-a c x + c\right )}^{\frac {7}{2}} c + 252 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c^{2} + 840 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{3} + 5040 \, \sqrt {-a c x + c} c^{4}\right )}}{315 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 112, normalized size = 0.82 \[ -\frac {4\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a}-\frac {8\,c\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a}-\frac {32\,c^3\,\sqrt {c-a\,c\,x}}{a}-\frac {16\,c^2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a}-\frac {2\,{\left (c-a\,c\,x\right )}^{9/2}}{9\,a\,c}-\frac {\sqrt {2}\,c^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,32{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 76.39, size = 129, normalized size = 0.94 \[ - \frac {32 \sqrt {2} c^{4} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {32 c^{3} \sqrt {- a c x + c}}{a} - \frac {16 c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3 a} - \frac {8 c \left (- a c x + c\right )^{\frac {5}{2}}}{5 a} - \frac {4 \left (- a c x + c\right )^{\frac {7}{2}}}{7 a} - \frac {2 \left (- a c x + c\right )^{\frac {9}{2}}}{9 a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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