Optimal. Leaf size=324 \[ -\frac {b \sqrt {1-d^2 x^2} \left (45 a^2 d^4+60 a c d^2+10 b^2 d^2+24 c^2\right )}{15 d^6}-\frac {x \sqrt {1-d^2 x^2} \left (24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^6}+\frac {\sin ^{-1}(d x) \left (16 a^3 d^6+24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^7}-\frac {b x^2 \sqrt {1-d^2 x^2} \left (30 a c d^2+5 b^2 d^2+12 c^2\right )}{15 d^4}-\frac {c x^3 \sqrt {1-d^2 x^2} \left (18 a c d^2+18 b^2 d^2+5 c^2\right )}{24 d^4}-\frac {3 b c^2 x^4 \sqrt {1-d^2 x^2}}{5 d^2}-\frac {c^3 x^5 \sqrt {1-d^2 x^2}}{6 d^2} \]
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Rubi [A] time = 0.93, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {899, 1815, 641, 216} \begin {gather*} -\frac {x \sqrt {1-d^2 x^2} \left (24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^6}-\frac {b \sqrt {1-d^2 x^2} \left (45 a^2 d^4+60 a c d^2+10 b^2 d^2+24 c^2\right )}{15 d^6}+\frac {\sin ^{-1}(d x) \left (24 a^2 c d^4+16 a^3 d^6+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^7}-\frac {c x^3 \sqrt {1-d^2 x^2} \left (18 a c d^2+18 b^2 d^2+5 c^2\right )}{24 d^4}-\frac {b x^2 \sqrt {1-d^2 x^2} \left (30 a c d^2+5 b^2 d^2+12 c^2\right )}{15 d^4}-\frac {3 b c^2 x^4 \sqrt {1-d^2 x^2}}{5 d^2}-\frac {c^3 x^5 \sqrt {1-d^2 x^2}}{6 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 899
Rule 1815
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {1-d x} \sqrt {1+d x}} \, dx &=\int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {c^3 x^5 \sqrt {1-d^2 x^2}}{6 d^2}-\frac {\int \frac {-6 a^3 d^2-18 a^2 b d^2 x-18 a \left (b^2+a c\right ) d^2 x^2-6 b \left (b^2+6 a c\right ) d^2 x^3-c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^4-18 b c^2 d^2 x^5}{\sqrt {1-d^2 x^2}} \, dx}{6 d^2}\\ &=-\frac {3 b c^2 x^4 \sqrt {1-d^2 x^2}}{5 d^2}-\frac {c^3 x^5 \sqrt {1-d^2 x^2}}{6 d^2}+\frac {\int \frac {30 a^3 d^4+90 a^2 b d^4 x+90 a \left (b^2+a c\right ) d^4 x^2+6 b d^2 \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^3+5 c d^2 \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^4}{\sqrt {1-d^2 x^2}} \, dx}{30 d^4}\\ &=-\frac {c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt {1-d^2 x^2}}{24 d^4}-\frac {3 b c^2 x^4 \sqrt {1-d^2 x^2}}{5 d^2}-\frac {c^3 x^5 \sqrt {1-d^2 x^2}}{6 d^2}-\frac {\int \frac {-120 a^3 d^6-360 a^2 b d^6 x-15 d^2 \left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x^2-24 b d^4 \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^3}{\sqrt {1-d^2 x^2}} \, dx}{120 d^6}\\ &=-\frac {b \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^2 \sqrt {1-d^2 x^2}}{15 d^4}-\frac {c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt {1-d^2 x^2}}{24 d^4}-\frac {3 b c^2 x^4 \sqrt {1-d^2 x^2}}{5 d^2}-\frac {c^3 x^5 \sqrt {1-d^2 x^2}}{6 d^2}+\frac {\int \frac {360 a^3 d^8+24 b d^4 \left (24 c^2+10 b^2 d^2+60 a c d^2+45 a^2 d^4\right ) x+45 d^4 \left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x^2}{\sqrt {1-d^2 x^2}} \, dx}{360 d^8}\\ &=-\frac {\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x \sqrt {1-d^2 x^2}}{16 d^6}-\frac {b \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^2 \sqrt {1-d^2 x^2}}{15 d^4}-\frac {c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt {1-d^2 x^2}}{24 d^4}-\frac {3 b c^2 x^4 \sqrt {1-d^2 x^2}}{5 d^2}-\frac {c^3 x^5 \sqrt {1-d^2 x^2}}{6 d^2}-\frac {\int \frac {-45 d^4 \left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4+16 a^3 d^6\right )-48 b d^6 \left (24 c^2+10 b^2 d^2+60 a c d^2+45 a^2 d^4\right ) x}{\sqrt {1-d^2 x^2}} \, dx}{720 d^{10}}\\ &=-\frac {b \left (24 c^2+10 b^2 d^2+60 a c d^2+45 a^2 d^4\right ) \sqrt {1-d^2 x^2}}{15 d^6}-\frac {\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x \sqrt {1-d^2 x^2}}{16 d^6}-\frac {b \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^2 \sqrt {1-d^2 x^2}}{15 d^4}-\frac {c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt {1-d^2 x^2}}{24 d^4}-\frac {3 b c^2 x^4 \sqrt {1-d^2 x^2}}{5 d^2}-\frac {c^3 x^5 \sqrt {1-d^2 x^2}}{6 d^2}+\frac {\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4+16 a^3 d^6\right ) \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{16 d^6}\\ &=-\frac {b \left (24 c^2+10 b^2 d^2+60 a c d^2+45 a^2 d^4\right ) \sqrt {1-d^2 x^2}}{15 d^6}-\frac {\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x \sqrt {1-d^2 x^2}}{16 d^6}-\frac {b \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^2 \sqrt {1-d^2 x^2}}{15 d^4}-\frac {c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt {1-d^2 x^2}}{24 d^4}-\frac {3 b c^2 x^4 \sqrt {1-d^2 x^2}}{5 d^2}-\frac {c^3 x^5 \sqrt {1-d^2 x^2}}{6 d^2}+\frac {\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4+16 a^3 d^6\right ) \sin ^{-1}(d x)}{16 d^7}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 229, normalized size = 0.71 \begin {gather*} \frac {15 \sin ^{-1}(d x) \left (16 a^3 d^6+24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )-d \sqrt {1-d^2 x^2} \left (48 b \left (15 a^2 d^4+10 a c d^2 \left (d^2 x^2+2\right )+c^2 \left (3 d^4 x^4+4 d^2 x^2+8\right )\right )+5 c x \left (72 a^2 d^4+18 a c d^2 \left (2 d^2 x^2+3\right )+c^2 \left (8 d^4 x^4+10 d^2 x^2+15\right )\right )+90 b^2 d^2 x \left (4 a d^2+c \left (2 d^2 x^2+3\right )\right )+80 b^3 d^2 \left (d^2 x^2+2\right )\right )}{240 d^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.63, size = 1219, normalized size = 3.76 \begin {gather*} \frac {\sqrt {1-d x} \left (-720 a^2 b d^5-\frac {3600 a^2 b (1-d x) d^5}{d x+1}-\frac {7200 a^2 b (1-d x)^2 d^5}{(d x+1)^2}-\frac {7200 a^2 b (1-d x)^3 d^5}{(d x+1)^3}-\frac {3600 a^2 b (1-d x)^4 d^5}{(d x+1)^4}-\frac {720 a^2 b (1-d x)^5 d^5}{(d x+1)^5}-360 a b^2 d^4-360 a^2 c d^4-\frac {1080 a b^2 (1-d x) d^4}{d x+1}-\frac {1080 a^2 c (1-d x) d^4}{d x+1}-\frac {720 a b^2 (1-d x)^2 d^4}{(d x+1)^2}-\frac {720 a^2 c (1-d x)^2 d^4}{(d x+1)^2}+\frac {720 a b^2 (1-d x)^3 d^4}{(d x+1)^3}+\frac {720 a^2 c (1-d x)^3 d^4}{(d x+1)^3}+\frac {1080 a b^2 (1-d x)^4 d^4}{(d x+1)^4}+\frac {1080 a^2 c (1-d x)^4 d^4}{(d x+1)^4}+\frac {360 a b^2 (1-d x)^5 d^4}{(d x+1)^5}+\frac {360 a^2 c (1-d x)^5 d^4}{(d x+1)^5}-240 b^3 d^3-1440 a b c d^3-\frac {880 b^3 (1-d x) d^3}{d x+1}-\frac {5280 a b c (1-d x) d^3}{d x+1}-\frac {1440 b^3 (1-d x)^2 d^3}{(d x+1)^2}-\frac {8640 a b c (1-d x)^2 d^3}{(d x+1)^2}-\frac {1440 b^3 (1-d x)^3 d^3}{(d x+1)^3}-\frac {8640 a b c (1-d x)^3 d^3}{(d x+1)^3}-\frac {880 b^3 (1-d x)^4 d^3}{(d x+1)^4}-\frac {5280 a b c (1-d x)^4 d^3}{(d x+1)^4}-\frac {240 b^3 (1-d x)^5 d^3}{(d x+1)^5}-\frac {1440 a b c (1-d x)^5 d^3}{(d x+1)^5}-450 a c^2 d^2-450 b^2 c d^2-\frac {630 a c^2 (1-d x) d^2}{d x+1}-\frac {630 b^2 c (1-d x) d^2}{d x+1}-\frac {180 a c^2 (1-d x)^2 d^2}{(d x+1)^2}-\frac {180 b^2 c (1-d x)^2 d^2}{(d x+1)^2}+\frac {180 a c^2 (1-d x)^3 d^2}{(d x+1)^3}+\frac {180 b^2 c (1-d x)^3 d^2}{(d x+1)^3}+\frac {630 a c^2 (1-d x)^4 d^2}{(d x+1)^4}+\frac {630 b^2 c (1-d x)^4 d^2}{(d x+1)^4}+\frac {450 a c^2 (1-d x)^5 d^2}{(d x+1)^5}+\frac {450 b^2 c (1-d x)^5 d^2}{(d x+1)^5}-720 b c^2 d-\frac {1680 b c^2 (1-d x) d}{d x+1}-\frac {3744 b c^2 (1-d x)^2 d}{(d x+1)^2}-\frac {3744 b c^2 (1-d x)^3 d}{(d x+1)^3}-\frac {1680 b c^2 (1-d x)^4 d}{(d x+1)^4}-\frac {720 b c^2 (1-d x)^5 d}{(d x+1)^5}-165 c^3+\frac {25 c^3 (1-d x)}{d x+1}-\frac {450 c^3 (1-d x)^2}{(d x+1)^2}+\frac {450 c^3 (1-d x)^3}{(d x+1)^3}-\frac {25 c^3 (1-d x)^4}{(d x+1)^4}+\frac {165 c^3 (1-d x)^5}{(d x+1)^5}\right )}{120 d^7 \sqrt {d x+1} \left (\frac {1-d x}{d x+1}+1\right )^6}+\frac {\left (-16 a^3 d^6-24 a b^2 d^4-24 a^2 c d^4-18 a c^2 d^2-18 b^2 c d^2-5 c^3\right ) \tan ^{-1}\left (\frac {\sqrt {1-d x}}{\sqrt {d x+1}}\right )}{8 d^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 251, normalized size = 0.77 \begin {gather*} -\frac {{\left (40 \, c^{3} d^{5} x^{5} + 144 \, b c^{2} d^{5} x^{4} + 720 \, a^{2} b d^{5} + 384 \, b c^{2} d + 160 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} + 10 \, {\left (5 \, c^{3} d^{3} + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{5}\right )} x^{3} + 16 \, {\left (12 \, b c^{2} d^{3} + 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} x^{2} + 15 \, {\left (24 \, {\left (a b^{2} + a^{2} c\right )} d^{5} + 5 \, c^{3} d + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{3}\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 30 \, {\left (16 \, a^{3} d^{6} + 24 \, {\left (a b^{2} + a^{2} c\right )} d^{4} + 5 \, c^{3} + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{240 \, d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 412, normalized size = 1.27 \begin {gather*} -\frac {{\left ({\left (2 \, {\left ({\left (d x + 1\right )} {\left (4 \, {\left (d x + 1\right )} {\left (\frac {5 \, {\left (d x + 1\right )} c^{3}}{d^{6}} + \frac {18 \, b c^{2} d^{37} - 25 \, c^{3} d^{36}}{d^{42}}\right )} + \frac {9 \, {\left (10 \, b^{2} c d^{38} + 10 \, a c^{2} d^{38} - 32 \, b c^{2} d^{37} + 25 \, c^{3} d^{36}\right )}}{d^{42}}\right )} + \frac {40 \, b^{3} d^{39} + 240 \, a b c d^{39} - 270 \, b^{2} c d^{38} - 270 \, a c^{2} d^{38} + 528 \, b c^{2} d^{37} - 275 \, c^{3} d^{36}}{d^{42}}\right )} {\left (d x + 1\right )} + \frac {5 \, {\left (72 \, a b^{2} d^{40} + 72 \, a^{2} c d^{40} - 32 \, b^{3} d^{39} - 192 \, a b c d^{39} + 162 \, b^{2} c d^{38} + 162 \, a c^{2} d^{38} - 192 \, b c^{2} d^{37} + 85 \, c^{3} d^{36}\right )}}{d^{42}}\right )} {\left (d x + 1\right )} + \frac {15 \, {\left (48 \, a^{2} b d^{41} - 24 \, a b^{2} d^{40} - 24 \, a^{2} c d^{40} + 16 \, b^{3} d^{39} + 96 \, a b c d^{39} - 30 \, b^{2} c d^{38} - 30 \, a c^{2} d^{38} + 48 \, b c^{2} d^{37} - 11 \, c^{3} d^{36}\right )}}{d^{42}}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - \frac {30 \, {\left (16 \, a^{3} d^{6} + 24 \, a b^{2} d^{4} + 24 \, a^{2} c d^{4} + 18 \, b^{2} c d^{2} + 18 \, a c^{2} d^{2} + 5 \, c^{3}\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{6}}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 602, normalized size = 1.86 \begin {gather*} -\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (40 \sqrt {-d^{2} x^{2}+1}\, c^{3} d^{5} x^{5} \mathrm {csgn}\relax (d )+144 \sqrt {-d^{2} x^{2}+1}\, b \,c^{2} d^{5} x^{4} \mathrm {csgn}\relax (d )+180 \sqrt {-d^{2} x^{2}+1}\, a \,c^{2} d^{5} x^{3} \mathrm {csgn}\relax (d )+180 \sqrt {-d^{2} x^{2}+1}\, b^{2} c \,d^{5} x^{3} \mathrm {csgn}\relax (d )+480 \sqrt {-d^{2} x^{2}+1}\, a b c \,d^{5} x^{2} \mathrm {csgn}\relax (d )+80 \sqrt {-d^{2} x^{2}+1}\, b^{3} d^{5} x^{2} \mathrm {csgn}\relax (d )-240 a^{3} d^{6} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+360 \sqrt {-d^{2} x^{2}+1}\, a^{2} c \,d^{5} x \,\mathrm {csgn}\relax (d )+360 \sqrt {-d^{2} x^{2}+1}\, a \,b^{2} d^{5} x \,\mathrm {csgn}\relax (d )+50 \sqrt {-d^{2} x^{2}+1}\, c^{3} d^{3} x^{3} \mathrm {csgn}\relax (d )+720 \sqrt {-d^{2} x^{2}+1}\, a^{2} b \,d^{5} \mathrm {csgn}\relax (d )+192 \sqrt {-d^{2} x^{2}+1}\, b \,c^{2} d^{3} x^{2} \mathrm {csgn}\relax (d )-360 a^{2} c \,d^{4} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-360 a \,b^{2} d^{4} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+270 \sqrt {-d^{2} x^{2}+1}\, a \,c^{2} d^{3} x \,\mathrm {csgn}\relax (d )+270 \sqrt {-d^{2} x^{2}+1}\, b^{2} c \,d^{3} x \,\mathrm {csgn}\relax (d )+960 \sqrt {-d^{2} x^{2}+1}\, a b c \,d^{3} \mathrm {csgn}\relax (d )+160 \sqrt {-d^{2} x^{2}+1}\, b^{3} d^{3} \mathrm {csgn}\relax (d )-270 a \,c^{2} d^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-270 b^{2} c \,d^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+75 \sqrt {-d^{2} x^{2}+1}\, c^{3} d x \,\mathrm {csgn}\relax (d )+384 \sqrt {-d^{2} x^{2}+1}\, b \,c^{2} d \,\mathrm {csgn}\relax (d )-75 c^{3} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )\right ) \mathrm {csgn}\relax (d )}{240 \sqrt {-d^{2} x^{2}+1}\, d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 365, normalized size = 1.13 \begin {gather*} -\frac {\sqrt {-d^{2} x^{2} + 1} c^{3} x^{5}}{6 \, d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} b c^{2} x^{4}}{5 \, d^{2}} + \frac {a^{3} \arcsin \left (d x\right )}{d} - \frac {5 \, \sqrt {-d^{2} x^{2} + 1} c^{3} x^{3}}{24 \, d^{4}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} {\left (b^{2} c + a c^{2}\right )} x^{3}}{4 \, d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} a^{2} b}{d^{2}} - \frac {4 \, \sqrt {-d^{2} x^{2} + 1} b c^{2} x^{2}}{5 \, d^{4}} - \frac {\sqrt {-d^{2} x^{2} + 1} {\left (b^{3} + 6 \, a b c\right )} x^{2}}{3 \, d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} {\left (a b^{2} + a^{2} c\right )} x}{2 \, d^{2}} + \frac {3 \, {\left (a b^{2} + a^{2} c\right )} \arcsin \left (d x\right )}{2 \, d^{3}} - \frac {5 \, \sqrt {-d^{2} x^{2} + 1} c^{3} x}{16 \, d^{6}} - \frac {9 \, \sqrt {-d^{2} x^{2} + 1} {\left (b^{2} c + a c^{2}\right )} x}{8 \, d^{4}} - \frac {8 \, \sqrt {-d^{2} x^{2} + 1} b c^{2}}{5 \, d^{6}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 1} {\left (b^{3} + 6 \, a b c\right )}}{3 \, d^{4}} + \frac {5 \, c^{3} \arcsin \left (d x\right )}{16 \, d^{7}} + \frac {9 \, {\left (b^{2} c + a c^{2}\right )} \arcsin \left (d x\right )}{8 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 31.33, size = 1768, normalized size = 5.46
result too large to display
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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