Optimal. Leaf size=194 \[ \frac {16384 c^5 x \sqrt {1-\frac {1}{a^2 x^2}}}{693 \sqrt {c-a c x}}+\frac {4096}{693} c^4 x \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-a c x}+\frac {512}{231} c^3 x \sqrt {1-\frac {1}{a^2 x^2}} (c-a c x)^{3/2}+\frac {640}{693} c^2 x \sqrt {1-\frac {1}{a^2 x^2}} (c-a c x)^{5/2}+\frac {40}{99} c x \sqrt {1-\frac {1}{a^2 x^2}} (c-a c x)^{7/2}+\frac {2}{11} x \sqrt {1-\frac {1}{a^2 x^2}} (c-a c x)^{9/2} \]
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Rubi [A] time = 0.23, antiderivative size = 311, normalized size of antiderivative = 1.60, number of steps used = 8, 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} \[ -\frac {512 \sqrt {\frac {1}{a x}+1} (c-a c x)^{9/2}}{231 a^3 x^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {1024 \sqrt {\frac {1}{a x}+1} (c-a c x)^{9/2}}{99 a^4 x^3 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {22016 \sqrt {\frac {1}{a x}+1} (c-a c x)^{9/2}}{693 a^5 x^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 x \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{11 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {40 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^4 (c-a c x)^{9/2}}{99 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {640 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3 (c-a c x)^{9/2}}{693 a^5 x \left (1-\frac {1}{a x}\right )^{9/2}} \]
Warning: Unable to verify antiderivative.
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Rule 37
Rule 78
Rule 89
Rule 94
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=\frac {(c-a c x)^{9/2} \int e^{-\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 )^5}{x^{13/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {2 \left (a-\frac {1}{x}\right )^5 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (20 \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 )^4}{x^{11/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{11 a \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {40 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 \left (a-\frac {1}{x}\right )^5 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {\left (320 \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}{x^{9/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{99 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {40 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {640 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (1280 \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}{x^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{231 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {40 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {512 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{231 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {640 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (512 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {-\frac {7}{a}+\frac {5 x}{2 a^2}}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{231 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {40 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {1024 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2} x^3}-\frac {512 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{231 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {640 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (11008 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{693 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {40 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {22016 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac {1}{a x}\right )^{9/2} x^4}+\frac {1024 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2} x^3}-\frac {512 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{231 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {640 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 86, normalized size = 0.44 \[ \frac {2 c^4 \sqrt {\frac {1}{a x}+1} \left (63 a^5 x^5-455 a^4 x^4+1510 a^3 x^3-3198 a^2 x^2+5419 a x-11531\right ) \sqrt {c-a c x}}{693 a \sqrt {1-\frac {1}{a x}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.53, size = 105, normalized size = 0.54 \[ \frac {2 \, {\left (63 \, a^{6} c^{4} x^{6} - 392 \, a^{5} c^{4} x^{5} + 1055 \, a^{4} c^{4} x^{4} - 1688 \, a^{3} c^{4} x^{3} + 2221 \, a^{2} c^{4} x^{2} - 6112 \, a c^{4} x - 11531 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{693 \, {\left (a^{2} x - a\right )}} \]
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 = 0.04, size = 80, normalized size = 0.41 \[ \frac {2 \left (a x +1\right ) \left (63 x^{5} a^{5}-455 x^{4} a^{4}+1510 x^{3} a^{3}-3198 a^{2} x^{2}+5419 a x -11531\right ) \left (-a c x +c \right )^{\frac {9}{2}} \sqrt {\frac {a x -1}{a x +1}}}{693 a \left (a x -1\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 128, normalized size = 0.66 \[ \frac {2 \, {\left (63 \, a^{6} \sqrt {-c} c^{4} x^{6} - 392 \, a^{5} \sqrt {-c} c^{4} x^{5} + 1055 \, a^{4} \sqrt {-c} c^{4} x^{4} - 1688 \, a^{3} \sqrt {-c} c^{4} x^{3} + 2221 \, a^{2} \sqrt {-c} c^{4} x^{2} - 6112 \, a \sqrt {-c} c^{4} x - 11531 \, \sqrt {-c} c^{4}\right )} {\left (a x - 1\right )}}{693 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 110, normalized size = 0.57 \[ \frac {2\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (63\,a^5\,x^5-329\,a^4\,x^4+726\,a^3\,x^3-962\,a^2\,x^2+1259\,a\,x-4853\right )}{693\,a}-\frac {32768\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{693\,a\,\left (a\,x-1\right )} \]
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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