Optimal. Leaf size=185 \[ -\frac {c^4 (a x+1)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}+\frac {2 c^4 (a x+1)^9 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (a x+1)^8 \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (a x+1)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6143, 6140, 43} \[ -\frac {c^4 (a x+1)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}+\frac {2 c^4 (a x+1)^9 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (a x+1)^8 \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (a x+1)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{9/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^3 (1+a x)^6 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int \left (8 (1+a x)^6-12 (1+a x)^7+6 (1+a x)^8-(1+a x)^9\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {8 c^4 (1+a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (1+a x)^8 \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-a^2 x^2}}+\frac {2 c^4 (1+a x)^9 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {c^4 (1+a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 68, normalized size = 0.37 \[ -\frac {c^4 (a x+1)^7 \left (21 a^3 x^3-77 a^2 x^2+98 a x-44\right ) \sqrt {c-a^2 c x^2}}{210 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 120, normalized size = 0.65 \[ \frac {{\left (21 \, a^{9} c^{4} x^{10} + 70 \, a^{8} c^{4} x^{9} - 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} + 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} - 210 \, c^{4} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{210 \, {\left (a^{2} x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 97, normalized size = 0.52 \[ \frac {x \left (21 a^{9} x^{9}+70 x^{8} a^{8}-240 x^{6} a^{6}-210 x^{5} a^{5}+252 x^{4} a^{4}+420 x^{3} a^{3}-315 a x -210\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}{210 \left (a x -1\right )^{3} \left (a x +1\right )^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 409, normalized size = 2.21 \[ -\frac {1}{7} \, a^{6} c^{\frac {9}{2}} x^{7} + \frac {3}{5} \, a^{4} c^{\frac {9}{2}} x^{5} - a^{2} c^{\frac {9}{2}} x^{3} + c^{\frac {9}{2}} x + \frac {1}{40} \, {\left (\frac {4 \, a^{8} c^{5} x^{12}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {19 \, a^{6} c^{5} x^{10}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {35 \, a^{4} c^{5} x^{8}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {30 \, a^{2} c^{5} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {10 \, c^{5} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}\right )} a^{3} - \frac {1}{105} \, {\left (35 \, a^{6} c^{\frac {9}{2}} x^{9} - 135 \, a^{4} c^{\frac {9}{2}} x^{7} + 189 \, a^{2} c^{\frac {9}{2}} x^{5} - 105 \, c^{\frac {9}{2}} x^{3}\right )} a^{2} + \frac {3}{8} \, {\left (\frac {a^{8} c^{5} x^{10}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {5 \, a^{6} c^{5} x^{8}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {10 \, a^{4} c^{5} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {10 \, a^{2} c^{5} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {4 \, c^{5}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 106, normalized size = 0.57 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (-\frac {a^9\,c^4\,x^{10}}{10}-\frac {a^8\,c^4\,x^9}{3}+\frac {8\,a^6\,c^4\,x^7}{7}+a^5\,c^4\,x^6-\frac {6\,a^4\,c^4\,x^5}{5}-2\,a^3\,c^4\,x^4+\frac {3\,a\,c^4\,x^2}{2}+c^4\,x\right )}{\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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