Optimal. Leaf size=89 \[ \frac {2 x \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{5/2}}{7 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {18 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6176, 6181, 78, 37} \[ \frac {2 x \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{5/2}}{7 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {18 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=\frac {(c-a c x)^{5/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{5/2} x^{5/2} \, dx}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x^{9/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{5/2}}\\ &=\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{5/2}}{7 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\left (9 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 a \left (1-\frac {1}{a x}\right )^{5/2}}\\ &=-\frac {18 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{5/2}}{7 \left (1-\frac {1}{a x}\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 59, normalized size = 0.66 \[ \frac {2 \sqrt {\frac {1}{a x}+1} (5 a x-9) \sqrt {c-a c x} (a c x+c)^2}{35 a \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 83, normalized size = 0.93 \[ \frac {2 \, {\left (5 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 12 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x - 9 \, c^{2}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 84, normalized size = 0.94 \[ -\frac {2 \, {\left (\frac {16 \, \sqrt {2} \sqrt {-c} c^{2}}{\mathrm {sgn}\relax (c)} - \frac {5 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} - 14 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{35 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 48, normalized size = 0.54 \[ \frac {2 \left (a x +1\right ) \left (5 a x -9\right ) \left (-a c x +c \right )^{\frac {5}{2}}}{35 a \left (a x -1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 74, normalized size = 0.83 \[ \frac {2 \, {\left (5 \, a^{3} \sqrt {-c} c^{2} x^{3} - 9 \, a^{2} \sqrt {-c} c^{2} x^{2} - 5 \, a \sqrt {-c} c^{2} x + 9 \, \sqrt {-c} c^{2}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}}{35 \, {\left (a x - 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 93, normalized size = 1.04 \[ -\frac {2\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (-5\,a^3\,x^3-11\,a^2\,x^2+a\,x+23\right )}{35\,a}-\frac {64\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{35\,a\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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