Optimal. Leaf size=197 \[ -\frac {856 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 x^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 x \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {16 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 x \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {8 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac {856 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 x^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 x \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {16 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 x \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {8 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 89
Rule 94
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=\frac {(c-a c x)^{9/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{9/2} x^{9/2} \, dx}{\left (1-\frac {1}{a x}\right )^{9/2} x^{9/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3 \left (1+\frac {x}{a}\right )^{3/2}}{x^{13/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (12 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2 \left (1+\frac {x}{a}\right )^{3/2}}{x^{11/2}} \, dx,x,\frac {1}{x}\right )}{11 a \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {8 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (8 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {11}{a}+\frac {9 x}{2 a^2}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x^{9/2}} \, dx,x,\frac {1}{x}\right )}{33 a \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {8 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {16 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (428 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{231 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {8 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {856 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {16 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.39 \[ \frac {2 c^4 \sqrt {\frac {1}{a x}+1} (a x+1)^2 \left (105 a^3 x^3-455 a^2 x^2+755 a x-533\right ) \sqrt {c-a c x}}{1155 a \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 105, normalized size = 0.53 \[ \frac {2 \, {\left (105 \, a^{6} c^{4} x^{6} - 140 \, a^{5} c^{4} x^{5} - 295 \, a^{4} c^{4} x^{4} + 472 \, a^{3} c^{4} x^{3} + 211 \, a^{2} c^{4} x^{2} - 844 \, a c^{4} x - 533 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{1155 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 132, normalized size = 0.67 \[ -\frac {2 \, {\left (\frac {512 \, \sqrt {2} \sqrt {-c} c^{4}}{\mathrm {sgn}\relax (c)} - \frac {105 \, {\left (a c x + c\right )}^{5} \sqrt {-a c x - c} - 770 \, {\left (a c x + c\right )}^{4} \sqrt {-a c x - c} c + 1980 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} c^{2} - 1848 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c^{3}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{1155 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 64, normalized size = 0.32 \[ \frac {2 \left (a x +1\right ) \left (105 x^{3} a^{3}-455 a^{2} x^{2}+755 a x -533\right ) \left (-a c x +c \right )^{\frac {9}{2}}}{1155 a \left (a x -1\right )^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 106, normalized size = 0.54 \[ \frac {2 \, {\left (105 \, a^{5} \sqrt {-c} c^{4} x^{5} - 455 \, a^{4} \sqrt {-c} c^{4} x^{4} + 650 \, a^{3} \sqrt {-c} c^{4} x^{3} - 78 \, a^{2} \sqrt {-c} c^{4} x^{2} - 755 \, a \sqrt {-c} c^{4} x + 533 \, \sqrt {-c} c^{4}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}}{1155 \, {\left (a x - 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 110, normalized size = 0.56 \[ \frac {2\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (105\,a^5\,x^5-35\,a^4\,x^4-330\,a^3\,x^3+142\,a^2\,x^2+353\,a\,x-491\right )}{1155\,a}-\frac {2048\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{1155\,a\,\left (a\,x-1\right )} \]
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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