Optimal. Leaf size=60 \[ \frac {(5 a x+1) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{10} (a x+1)^{15} \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.24, 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} \[ \frac {(5 a x+1) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{10} (a x+1)^{15} \sqrt {c-a^2 c x^2}} \]
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)^{11} (1+a x)^{16}} \, 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)^{10} (1+a x)^{15} \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 59, normalized size = 0.98 \[ \frac {(5 a x+1) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (a x-1)^{10} (a x+1)^{15} \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 497, normalized size = 8.28 \[ \frac {{\left (a^{22} x^{25} + 5 \, a^{21} x^{24} - 40 \, a^{19} x^{22} - 50 \, a^{18} x^{21} + 126 \, a^{17} x^{20} + 280 \, a^{16} x^{19} - 160 \, a^{15} x^{18} - 765 \, a^{14} x^{17} - 105 \, a^{13} x^{16} + 1248 \, a^{12} x^{15} + 720 \, a^{11} x^{14} - 1260 \, a^{10} x^{13} - 1260 \, a^{9} x^{12} + 720 \, a^{8} x^{11} + 1248 \, a^{7} x^{10} - 105 \, a^{6} x^{9} - 765 \, a^{5} x^{8} - 160 \, a^{4} x^{7} + 280 \, a^{3} x^{6} + 126 \, a^{2} x^{5} - 50 \, a x^{4} - 40 \, x^{3}\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{120 \, {\left (a^{27} c^{14} x^{27} + 5 \, a^{26} c^{14} x^{26} - a^{25} c^{14} x^{25} - 45 \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} + 166 \, a^{22} c^{14} x^{22} + 330 \, a^{21} c^{14} x^{21} - 286 \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} + 55 \, a^{18} c^{14} x^{18} + 2013 \, a^{17} c^{14} x^{17} + 825 \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} - 1980 \, a^{14} c^{14} x^{14} + 1980 \, a^{13} c^{14} x^{13} + 2508 \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} - 2013 \, a^{10} c^{14} x^{10} - 55 \, a^{9} c^{14} x^{9} + 1045 \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} - 330 \, a^{6} c^{14} x^{6} - 166 \, a^{5} c^{14} x^{5} + 50 \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} + a^{2} c^{14} x^{2} - 5 \, a c^{14} x - c^{14}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.60, size = 489, normalized size = 8.15 \[ \frac {2451570 \, \sqrt {c} {\left (\frac {2}{a x + 1} - 1\right )}^{9} + 1514205 \, \sqrt {c} {\left (\frac {2}{a x + 1} - 1\right )}^{8} + 769120 \, \sqrt {c} {\left (\frac {2}{a x + 1} - 1\right )}^{7} + 318780 \, \sqrt {c} {\left (\frac {2}{a x + 1} - 1\right )}^{6} + 106260 \, \sqrt {c} {\left (\frac {2}{a x + 1} - 1\right )}^{5} + 27830 \, \sqrt {c} {\left (\frac {2}{a x + 1} - 1\right )}^{4} + 5520 \, \sqrt {c} {\left (\frac {2}{a x + 1} - 1\right )}^{3} + 780 \, \sqrt {c} {\left (\frac {2}{a x + 1} - 1\right )}^{2} + 70 \, \sqrt {c} {\left (\frac {2}{a x + 1} - 1\right )} + 3 \, \sqrt {c}}{2013265920 \, a^{3} c^{14} {\left (\frac {2}{a x + 1} - 1\right )}^{10}} - \frac {2 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{15} + 45 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{14} + 480 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{13} + 3220 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{12} + 15180 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{11} + 53130 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{10} + 141680 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{9} + 288420 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{8} + 432630 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{7} + 408595 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{6} - 891480 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{4} - 2080120 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{3} - 3120180 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{2} - 3565920 \, a^{42} c^{\frac {393}{2}} {\left (\frac {2}{a x + 1} - 1\right )}}{2013265920 \, a^{45} c^{210}} \]
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 \[ -\frac {\left (a x -1\right ) \left (5 a x +1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{120 \left (a x +1\right )^{4} a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 273, normalized size = 4.55 \[ \frac {5 \, a \sqrt {c} x + \sqrt {c}}{120 \, {\left (a^{28} c^{14} x^{25} + 5 \, a^{27} c^{14} x^{24} - 40 \, a^{25} c^{14} x^{22} - 50 \, a^{24} c^{14} x^{21} + 126 \, a^{23} c^{14} x^{20} + 280 \, a^{22} c^{14} x^{19} - 160 \, a^{21} c^{14} x^{18} - 765 \, a^{20} c^{14} x^{17} - 105 \, a^{19} c^{14} x^{16} + 1248 \, a^{18} c^{14} x^{15} + 720 \, a^{17} c^{14} x^{14} - 1260 \, a^{16} c^{14} x^{13} - 1260 \, a^{15} c^{14} x^{12} + 720 \, a^{14} c^{14} x^{11} + 1248 \, a^{13} c^{14} x^{10} - 105 \, a^{12} c^{14} x^{9} - 765 \, a^{11} c^{14} x^{8} - 160 \, a^{10} c^{14} x^{7} + 280 \, a^{9} c^{14} x^{6} + 126 \, a^{8} c^{14} x^{5} - 50 \, a^{7} c^{14} x^{4} - 40 \, a^{6} c^{14} x^{3} + 5 \, a^{4} c^{14} x + a^{3} c^{14}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 363, normalized size = 6.05 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}+5\,a\,x\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}}{-120\,a^{30}\,c^{14}\,x^{27}-600\,a^{29}\,c^{14}\,x^{26}+120\,a^{28}\,c^{14}\,x^{25}+5400\,a^{27}\,c^{14}\,x^{24}+6000\,a^{26}\,c^{14}\,x^{23}-19920\,a^{25}\,c^{14}\,x^{22}-39600\,a^{24}\,c^{14}\,x^{21}+34320\,a^{23}\,c^{14}\,x^{20}+125400\,a^{22}\,c^{14}\,x^{19}-6600\,a^{21}\,c^{14}\,x^{18}-241560\,a^{20}\,c^{14}\,x^{17}-99000\,a^{19}\,c^{14}\,x^{16}+300960\,a^{18}\,c^{14}\,x^{15}+237600\,a^{17}\,c^{14}\,x^{14}-237600\,a^{16}\,c^{14}\,x^{13}-300960\,a^{15}\,c^{14}\,x^{12}+99000\,a^{14}\,c^{14}\,x^{11}+241560\,a^{13}\,c^{14}\,x^{10}+6600\,a^{12}\,c^{14}\,x^9-125400\,a^{11}\,c^{14}\,x^8-34320\,a^{10}\,c^{14}\,x^7+39600\,a^9\,c^{14}\,x^6+19920\,a^8\,c^{14}\,x^5-6000\,a^7\,c^{14}\,x^4-5400\,a^6\,c^{14}\,x^3-120\,a^5\,c^{14}\,x^2+600\,a^4\,c^{14}\,x+120\,a^3\,c^{14}} \]
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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