Optimal. Leaf size=141 \[ \frac {256 c^7 \left (1-a^2 x^2\right )^{5/2}}{1155 a (c-a c x)^{5/2}}+\frac {64 c^6 \left (1-a^2 x^2\right )^{5/2}}{231 a (c-a c x)^{3/2}}+\frac {8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt {c-a c x}}+\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a c x}}{11 a} \]
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Rubi [A] time = 0.11, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6127, 657, 649} \[ \frac {256 c^7 \left (1-a^2 x^2\right )^{5/2}}{1155 a (c-a c x)^{5/2}}+\frac {64 c^6 \left (1-a^2 x^2\right )^{5/2}}{231 a (c-a c x)^{3/2}}+\frac {8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt {c-a c x}}+\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a c x}}{11 a} \]
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)^{9/2} \, dx &=c^3 \int (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {2 c^4 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{5/2}}{11 a}+\frac {1}{11} \left (12 c^4\right ) \int \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt {c-a c x}}+\frac {2 c^4 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{5/2}}{11 a}+\frac {1}{33} \left (32 c^5\right ) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{\sqrt {c-a c x}} \, dx\\ &=\frac {64 c^6 \left (1-a^2 x^2\right )^{5/2}}{231 a (c-a c x)^{3/2}}+\frac {8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt {c-a c x}}+\frac {2 c^4 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{5/2}}{11 a}+\frac {1}{231} \left (128 c^6\right ) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac {256 c^7 \left (1-a^2 x^2\right )^{5/2}}{1155 a (c-a c x)^{5/2}}+\frac {64 c^6 \left (1-a^2 x^2\right )^{5/2}}{231 a (c-a c x)^{3/2}}+\frac {8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt {c-a c x}}+\frac {2 c^4 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{5/2}}{11 a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 62, normalized size = 0.44 \[ -\frac {2 c^4 (a x+1)^{5/2} \left (105 a^3 x^3-455 a^2 x^2+755 a x-533\right ) \sqrt {c-a c x}}{1155 a \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 91, normalized size = 0.65 \[ \frac {2 \, {\left (105 \, a^{5} c^{4} x^{5} - 245 \, a^{4} c^{4} x^{4} - 50 \, a^{3} c^{4} x^{3} + 522 \, a^{2} c^{4} x^{2} - 311 \, a c^{4} x - 533 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{1155 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 73, normalized size = 0.52 \[ -\frac {2 \, {\left (512 \, \sqrt {2} c^{\frac {7}{2}} + \frac {105 \, {\left (a c x + c\right )}^{\frac {11}{2}} - 770 \, {\left (a c x + c\right )}^{\frac {9}{2}} c + 1980 \, {\left (a c x + c\right )}^{\frac {7}{2}} c^{2} - 1848 \, {\left (a c x + c\right )}^{\frac {5}{2}} c^{3}}{c^{2}}\right )} c^{2}}{1155 \, a {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 63, normalized size = 0.45 \[ \frac {2 \left (a x +1\right )^{4} \left (105 x^{3} a^{3}-455 a^{2} x^{2}+755 a x -533\right ) \left (-a c x +c \right )^{\frac {9}{2}}}{1155 a \left (a x -1\right )^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 254, normalized size = 1.80 \[ -\frac {2 \, {\left (35 \, a^{6} c^{\frac {9}{2}} x^{6} - 175 \, a^{5} c^{\frac {9}{2}} x^{5} + 415 \, a^{4} c^{\frac {9}{2}} x^{4} - 741 \, a^{3} c^{\frac {9}{2}} x^{3} + 1482 \, a^{2} c^{\frac {9}{2}} x^{2} - 5928 \, a c^{\frac {9}{2}} x - 11856 \, c^{\frac {9}{2}}\right )}}{385 \, \sqrt {a x + 1} a} - \frac {2 \, {\left (7 \, a^{5} c^{\frac {9}{2}} x^{5} - 37 \, a^{4} c^{\frac {9}{2}} x^{4} + 97 \, a^{3} c^{\frac {9}{2}} x^{3} - 215 \, a^{2} c^{\frac {9}{2}} x^{2} + 860 \, a c^{\frac {9}{2}} x + 1720 \, c^{\frac {9}{2}}\right )}}{21 \, \sqrt {a x + 1} a} - \frac {6 \, {\left (5 \, a^{4} c^{\frac {9}{2}} x^{4} - 29 \, a^{3} c^{\frac {9}{2}} x^{3} + 93 \, a^{2} c^{\frac {9}{2}} x^{2} - 407 \, a c^{\frac {9}{2}} x - 814 \, c^{\frac {9}{2}}\right )}}{35 \, \sqrt {a x + 1} a} - \frac {2 \, {\left (a^{3} c^{\frac {9}{2}} x^{3} - 7 \, a^{2} c^{\frac {9}{2}} x^{2} + 43 \, a c^{\frac {9}{2}} x + 91 \, c^{\frac {9}{2}}\right )}}{5 \, \sqrt {a x + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 90, normalized size = 0.64 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {1688\,c^4\,x}{1155}+\frac {1066\,c^4}{1155\,a}-\frac {422\,a\,c^4\,x^2}{1155}-\frac {944\,a^2\,c^4\,x^3}{1155}+\frac {118\,a^3\,c^4\,x^4}{231}+\frac {8\,a^4\,c^4\,x^5}{33}-\frac {2\,a^5\,c^4\,x^6}{11}\right )}{\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {9}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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