Optimal. Leaf size=225 \[ \frac {11 a^{7/2} x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}+\frac {a^3 x^4 \sqrt {a x+1} (521 a x+2718) \left (c-\frac {c}{a x}\right )^{9/2}}{105 (1-a x)^{9/2}}-\frac {94 a^2 x^3 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{21 (1-a x)^{5/2}}+\frac {6 a x^2 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{5 (1-a x)^{3/2}}-\frac {2 x \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{7 \sqrt {1-a x}} \]
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Rubi [A] time = 0.19, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6134, 6129, 98, 150, 143, 54, 215} \[ \frac {a^3 x^4 \sqrt {a x+1} (521 a x+2718) \left (c-\frac {c}{a x}\right )^{9/2}}{105 (1-a x)^{9/2}}-\frac {94 a^2 x^3 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{21 (1-a x)^{5/2}}+\frac {11 a^{7/2} x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}+\frac {6 a x^2 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{5 (1-a x)^{3/2}}-\frac {2 x \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{7 \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 143
Rule 150
Rule 215
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {e^{-\tanh ^{-1}(a x)} (1-a x)^{9/2}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^5}{x^{9/2} \sqrt {1+a x}} \, dx}{(1-a x)^{9/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}-\frac {\left (2 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^3 \left (\frac {21 a}{2}-\frac {5 a^2 x}{2}\right )}{x^{7/2} \sqrt {1+a x}} \, dx}{7 (1-a x)^{9/2}}\\ &=\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}-\frac {\left (4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^2 \left (-\frac {235 a^2}{4}-\frac {17 a^3 x}{4}\right )}{x^{5/2} \sqrt {1+a x}} \, dx}{35 (1-a x)^{9/2}}\\ &=-\frac {94 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}-\frac {\left (8 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x) \left (\frac {1359 a^3}{8}+\frac {521 a^4 x}{8}\right )}{x^{3/2} \sqrt {1+a x}} \, dx}{105 (1-a x)^{9/2}}\\ &=-\frac {94 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}+\frac {a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 \sqrt {1+a x} (2718+521 a x)}{105 (1-a x)^{9/2}}+\frac {\left (11 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{9/2}}\\ &=-\frac {94 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}+\frac {a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 \sqrt {1+a x} (2718+521 a x)}{105 (1-a x)^{9/2}}+\frac {\left (11 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{(1-a x)^{9/2}}\\ &=-\frac {94 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}+\frac {a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 \sqrt {1+a x} (2718+521 a x)}{105 (1-a x)^{9/2}}+\frac {11 a^{7/2} \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}\\ \end {align*}
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Mathematica [A] time = 2.83, size = 108, normalized size = 0.48 \[ \frac {c^4 \sqrt {c-\frac {c}{a x}} \left (\sqrt {-a x (a x+1)} \left (105 a^4 x^4-4156 a^3 x^3+1028 a^2 x^2-246 a x+30\right )-1155 a^4 x^4 \sin ^{-1}\left (\sqrt {-a x}\right )\right )}{105 a^3 x^2 (-a x)^{3/2} \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 386, normalized size = 1.72 \[ \left [\frac {1155 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (105 \, a^{4} c^{4} x^{4} - 4156 \, a^{3} c^{4} x^{3} + 1028 \, a^{2} c^{4} x^{2} - 246 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{420 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, -\frac {1155 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (105 \, a^{4} c^{4} x^{4} - 4156 \, a^{3} c^{4} x^{3} + 1028 \, a^{2} c^{4} x^{2} - 246 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{210 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}\right ] \]
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 = 0.06, size = 172, normalized size = 0.76 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \sqrt {-a^{2} x^{2}+1}\, \left (210 a^{\frac {9}{2}} \sqrt {-\left (a x +1\right ) x}\, x^{4}+1155 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) x^{4} a^{4}-8312 a^{\frac {7}{2}} x^{3} \sqrt {-\left (a x +1\right ) x}+2056 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}-492 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+60 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right )}{210 x^{3} a^{\frac {9}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]
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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