Optimal. Leaf size=206 \[ \frac {16384 c^5 \sqrt {1-a^2 x^2}}{693 a \sqrt {c-a c x}}+\frac {4096 c^4 \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{693 a}+\frac {512 c^3 \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{231 a}+\frac {640 c^2 \sqrt {1-a^2 x^2} (c-a c x)^{5/2}}{693 a}+\frac {40 c \sqrt {1-a^2 x^2} (c-a c x)^{7/2}}{99 a}+\frac {2 \sqrt {1-a^2 x^2} (c-a c x)^{9/2}}{11 a} \]
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Rubi [A] time = 0.16, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6127, 657, 649} \[ \frac {16384 c^5 \sqrt {1-a^2 x^2}}{693 a \sqrt {c-a c x}}+\frac {4096 c^4 \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{693 a}+\frac {512 c^3 \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{231 a}+\frac {640 c^2 \sqrt {1-a^2 x^2} (c-a c x)^{5/2}}{693 a}+\frac {40 c \sqrt {1-a^2 x^2} (c-a c x)^{7/2}}{99 a}+\frac {2 \sqrt {1-a^2 x^2} (c-a c x)^{9/2}}{11 a} \]
Antiderivative was successfully verified.
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Rule 649
Rule 657
Rule 6127
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=\frac {\int \frac {(c-a c x)^{11/2}}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {2 (c-a c x)^{9/2} \sqrt {1-a^2 x^2}}{11 a}+\frac {20}{11} \int \frac {(c-a c x)^{9/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {40 c (c-a c x)^{7/2} \sqrt {1-a^2 x^2}}{99 a}+\frac {2 (c-a c x)^{9/2} \sqrt {1-a^2 x^2}}{11 a}+\frac {1}{99} (320 c) \int \frac {(c-a c x)^{7/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {640 c^2 (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{693 a}+\frac {40 c (c-a c x)^{7/2} \sqrt {1-a^2 x^2}}{99 a}+\frac {2 (c-a c x)^{9/2} \sqrt {1-a^2 x^2}}{11 a}+\frac {1}{231} \left (1280 c^2\right ) \int \frac {(c-a c x)^{5/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {512 c^3 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{231 a}+\frac {640 c^2 (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{693 a}+\frac {40 c (c-a c x)^{7/2} \sqrt {1-a^2 x^2}}{99 a}+\frac {2 (c-a c x)^{9/2} \sqrt {1-a^2 x^2}}{11 a}+\frac {1}{231} \left (2048 c^3\right ) \int \frac {(c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {4096 c^4 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{693 a}+\frac {512 c^3 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{231 a}+\frac {640 c^2 (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{693 a}+\frac {40 c (c-a c x)^{7/2} \sqrt {1-a^2 x^2}}{99 a}+\frac {2 (c-a c x)^{9/2} \sqrt {1-a^2 x^2}}{11 a}+\frac {1}{693} \left (8192 c^4\right ) \int \frac {\sqrt {c-a c x}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {16384 c^5 \sqrt {1-a^2 x^2}}{693 a \sqrt {c-a c x}}+\frac {4096 c^4 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{693 a}+\frac {512 c^3 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{231 a}+\frac {640 c^2 (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{693 a}+\frac {40 c (c-a c x)^{7/2} \sqrt {1-a^2 x^2}}{99 a}+\frac {2 (c-a c x)^{9/2} \sqrt {1-a^2 x^2}}{11 a}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 73, normalized size = 0.35 \[ -\frac {2 c^5 \sqrt {1-a^2 x^2} \left (63 a^5 x^5-455 a^4 x^4+1510 a^3 x^3-3198 a^2 x^2+5419 a x-11531\right )}{693 a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 91, normalized size = 0.44 \[ \frac {2 \, {\left (63 \, a^{5} c^{4} x^{5} - 455 \, a^{4} c^{4} x^{4} + 1510 \, a^{3} c^{4} x^{3} - 3198 \, a^{2} c^{4} x^{2} + 5419 \, a c^{4} x - 11531 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{693 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 72, normalized size = 0.35 \[ \frac {2 \sqrt {-a^{2} x^{2}+1}\, \left (-a c x +c \right )^{\frac {9}{2}} \left (63 x^{5} a^{5}-455 x^{4} a^{4}+1510 x^{3} a^{3}-3198 a^{2} x^{2}+5419 a x -11531\right )}{693 \left (a x -1\right )^{5} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 82, normalized size = 0.40 \[ -\frac {2 \, {\left (63 \, a^{5} c^{\frac {9}{2}} x^{5} - 455 \, a^{4} c^{\frac {9}{2}} x^{4} + 1510 \, a^{3} c^{\frac {9}{2}} x^{3} - 3198 \, a^{2} c^{\frac {9}{2}} x^{2} + 5419 \, a c^{\frac {9}{2}} x - 11531 \, c^{\frac {9}{2}}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{693 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 96, normalized size = 0.47 \[ \frac {2\,c^4\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}\,\left (63\,a^4\,x^4-392\,a^3\,x^3+1118\,a^2\,x^2-2080\,a\,x+3339\right )}{693\,a}-\frac {16384\,c^4\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{693\,a\,\left (a\,x-1\right )} \]
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 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {9}{2}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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