Optimal. Leaf size=60 \[ -\frac {(1-5 a x) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{15} (a x+1)^{10} \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.23, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6153, 6150, 81} \[ -\frac {(1-5 a x) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{15} (a x+1)^{10} \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 81
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{5 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{5 \tanh ^{-1}(a x)} x^2}{\left (1-a^2 x^2\right )^{27/2}} \, dx}{c^{13} \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx}{c^{13} \sqrt {c-a^2 c x^2}}\\ &=-\frac {(1-5 a x) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{15} (1+a x)^{10} \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 59, normalized size = 0.98 \[ \frac {(1-5 a x) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (a x-1)^{15} (a x+1)^{10} \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 496, normalized size = 8.27 \[ -\frac {{\left (a^{22} x^{25} - 5 \, a^{21} x^{24} + 40 \, a^{19} x^{22} - 50 \, a^{18} x^{21} - 126 \, a^{17} x^{20} + 280 \, a^{16} x^{19} + 160 \, a^{15} x^{18} - 765 \, a^{14} x^{17} + 105 \, a^{13} x^{16} + 1248 \, a^{12} x^{15} - 720 \, a^{11} x^{14} - 1260 \, a^{10} x^{13} + 1260 \, a^{9} x^{12} + 720 \, a^{8} x^{11} - 1248 \, a^{7} x^{10} - 105 \, a^{6} x^{9} + 765 \, a^{5} x^{8} - 160 \, a^{4} x^{7} - 280 \, a^{3} x^{6} + 126 \, a^{2} x^{5} + 50 \, a x^{4} - 40 \, x^{3}\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{120 \, {\left (a^{27} c^{14} x^{27} - 5 \, a^{26} c^{14} x^{26} - a^{25} c^{14} x^{25} + 45 \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} - 166 \, a^{22} c^{14} x^{22} + 330 \, a^{21} c^{14} x^{21} + 286 \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} - 55 \, a^{18} c^{14} x^{18} + 2013 \, a^{17} c^{14} x^{17} - 825 \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} + 1980 \, a^{14} c^{14} x^{14} + 1980 \, a^{13} c^{14} x^{13} - 2508 \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} + 2013 \, a^{10} c^{14} x^{10} - 55 \, a^{9} c^{14} x^{9} - 1045 \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} + 330 \, a^{6} c^{14} x^{6} - 166 \, a^{5} c^{14} x^{5} - 50 \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} - a^{2} c^{14} x^{2} - 5 \, a c^{14} x + c^{14}\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{5} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {27}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 49, normalized size = 0.82 \[ -\frac {\left (a x -1\right ) \left (a x +1\right )^{6} \left (5 a x -1\right )}{120 a^{3} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (-a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{5} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {27}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.40, size = 583, normalized size = 9.72 \[ -\frac {\sqrt {c-a^2\,c\,x^2}-5\,a\,x\,\sqrt {c-a^2\,c\,x^2}}{120\,a^3\,c^{14}\,\sqrt {1-a^2\,x^2}-600\,a^4\,c^{14}\,x\,\sqrt {1-a^2\,x^2}+4800\,a^6\,c^{14}\,x^3\,\sqrt {1-a^2\,x^2}-6000\,a^7\,c^{14}\,x^4\,\sqrt {1-a^2\,x^2}-15120\,a^8\,c^{14}\,x^5\,\sqrt {1-a^2\,x^2}+33600\,a^9\,c^{14}\,x^6\,\sqrt {1-a^2\,x^2}+19200\,a^{10}\,c^{14}\,x^7\,\sqrt {1-a^2\,x^2}-91800\,a^{11}\,c^{14}\,x^8\,\sqrt {1-a^2\,x^2}+12600\,a^{12}\,c^{14}\,x^9\,\sqrt {1-a^2\,x^2}+149760\,a^{13}\,c^{14}\,x^{10}\,\sqrt {1-a^2\,x^2}-86400\,a^{14}\,c^{14}\,x^{11}\,\sqrt {1-a^2\,x^2}-151200\,a^{15}\,c^{14}\,x^{12}\,\sqrt {1-a^2\,x^2}+151200\,a^{16}\,c^{14}\,x^{13}\,\sqrt {1-a^2\,x^2}+86400\,a^{17}\,c^{14}\,x^{14}\,\sqrt {1-a^2\,x^2}-149760\,a^{18}\,c^{14}\,x^{15}\,\sqrt {1-a^2\,x^2}-12600\,a^{19}\,c^{14}\,x^{16}\,\sqrt {1-a^2\,x^2}+91800\,a^{20}\,c^{14}\,x^{17}\,\sqrt {1-a^2\,x^2}-19200\,a^{21}\,c^{14}\,x^{18}\,\sqrt {1-a^2\,x^2}-33600\,a^{22}\,c^{14}\,x^{19}\,\sqrt {1-a^2\,x^2}+15120\,a^{23}\,c^{14}\,x^{20}\,\sqrt {1-a^2\,x^2}+6000\,a^{24}\,c^{14}\,x^{21}\,\sqrt {1-a^2\,x^2}-4800\,a^{25}\,c^{14}\,x^{22}\,\sqrt {1-a^2\,x^2}+600\,a^{27}\,c^{14}\,x^{24}\,\sqrt {1-a^2\,x^2}-120\,a^{28}\,c^{14}\,x^{25}\,\sqrt {1-a^2\,x^2}} \]
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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