Optimal. Leaf size=89 \[ -\frac {(c+d x)^{11}}{13 (b c-a d) (a+b x)^{13}}+\frac {d (c+d x)^{11}}{78 (b c-a d)^2 (a+b x)^{12}}-\frac {d^2 (c+d x)^{11}}{858 (b c-a d)^3 (a+b x)^{11}} \]
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Rubi [A]
time = 0.01, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37}
\begin {gather*} -\frac {d^2 (c+d x)^{11}}{858 (a+b x)^{11} (b c-a d)^3}+\frac {d (c+d x)^{11}}{78 (a+b x)^{12} (b c-a d)^2}-\frac {(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx &=-\frac {(c+d x)^{11}}{13 (b c-a d) (a+b x)^{13}}-\frac {(2 d) \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx}{13 (b c-a d)}\\ &=-\frac {(c+d x)^{11}}{13 (b c-a d) (a+b x)^{13}}+\frac {d (c+d x)^{11}}{78 (b c-a d)^2 (a+b x)^{12}}+\frac {d^2 \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{78 (b c-a d)^2}\\ &=-\frac {(c+d x)^{11}}{13 (b c-a d) (a+b x)^{13}}+\frac {d (c+d x)^{11}}{78 (b c-a d)^2 (a+b x)^{12}}-\frac {d^2 (c+d x)^{11}}{858 (b c-a d)^3 (a+b x)^{11}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(690\) vs. \(2(89)=178\).
time = 0.19, size = 690, normalized size = 7.75 \begin {gather*} -\frac {a^{10} d^{10}+a^9 b d^9 (3 c+13 d x)+3 a^8 b^2 d^8 \left (2 c^2+13 c d x+26 d^2 x^2\right )+2 a^7 b^3 d^7 \left (5 c^3+39 c^2 d x+117 c d^2 x^2+143 d^3 x^3\right )+a^6 b^4 d^6 \left (15 c^4+130 c^3 d x+468 c^2 d^2 x^2+858 c d^3 x^3+715 d^4 x^4\right )+3 a^5 b^5 d^5 \left (7 c^5+65 c^4 d x+260 c^3 d^2 x^2+572 c^2 d^3 x^3+715 c d^4 x^4+429 d^5 x^5\right )+a^4 b^6 d^4 \left (28 c^6+273 c^5 d x+1170 c^4 d^2 x^2+2860 c^3 d^3 x^3+4290 c^2 d^4 x^4+3861 c d^5 x^5+1716 d^6 x^6\right )+2 a^3 b^7 d^3 \left (18 c^7+182 c^6 d x+819 c^5 d^2 x^2+2145 c^4 d^3 x^3+3575 c^3 d^4 x^4+3861 c^2 d^5 x^5+2574 c d^6 x^6+858 d^7 x^7\right )+3 a^2 b^8 d^2 \left (15 c^8+156 c^7 d x+728 c^6 d^2 x^2+2002 c^5 d^3 x^3+3575 c^4 d^4 x^4+4290 c^3 d^5 x^5+3432 c^2 d^6 x^6+1716 c d^7 x^7+429 d^8 x^8\right )+a b^9 d \left (55 c^9+585 c^8 d x+2808 c^7 d^2 x^2+8008 c^6 d^3 x^3+15015 c^5 d^4 x^4+19305 c^4 d^5 x^5+17160 c^3 d^6 x^6+10296 c^2 d^7 x^7+3861 c d^8 x^8+715 d^9 x^9\right )+b^{10} \left (66 c^{10}+715 c^9 d x+3510 c^8 d^2 x^2+10296 c^7 d^3 x^3+20020 c^6 d^4 x^4+27027 c^5 d^5 x^5+25740 c^4 d^6 x^6+17160 c^3 d^7 x^7+7722 c^2 d^8 x^8+2145 c d^9 x^9+286 d^{10} x^{10}\right )}{858 b^{11} (a+b x)^{13}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(866\) vs.
\(2(83)=166\).
time = 0.15, size = 867, normalized size = 9.74 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 997 vs.
\(2 (83) = 166\).
time = 0.37, size = 997, normalized size = 11.20 \begin {gather*} -\frac {286 \, b^{10} d^{10} x^{10} + 66 \, b^{10} c^{10} + 55 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 36 \, a^{3} b^{7} c^{7} d^{3} + 28 \, a^{4} b^{6} c^{6} d^{4} + 21 \, a^{5} b^{5} c^{5} d^{5} + 15 \, a^{6} b^{4} c^{4} d^{6} + 10 \, a^{7} b^{3} c^{3} d^{7} + 6 \, a^{8} b^{2} c^{2} d^{8} + 3 \, a^{9} b c d^{9} + a^{10} d^{10} + 715 \, {\left (3 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 1287 \, {\left (6 \, b^{10} c^{2} d^{8} + 3 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 1716 \, {\left (10 \, b^{10} c^{3} d^{7} + 6 \, a b^{9} c^{2} d^{8} + 3 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 1716 \, {\left (15 \, b^{10} c^{4} d^{6} + 10 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} + 3 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 1287 \, {\left (21 \, b^{10} c^{5} d^{5} + 15 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} + 6 \, a^{3} b^{7} c^{2} d^{8} + 3 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 715 \, {\left (28 \, b^{10} c^{6} d^{4} + 21 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} + 10 \, a^{3} b^{7} c^{3} d^{7} + 6 \, a^{4} b^{6} c^{2} d^{8} + 3 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 286 \, {\left (36 \, b^{10} c^{7} d^{3} + 28 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 15 \, a^{3} b^{7} c^{4} d^{6} + 10 \, a^{4} b^{6} c^{3} d^{7} + 6 \, a^{5} b^{5} c^{2} d^{8} + 3 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 78 \, {\left (45 \, b^{10} c^{8} d^{2} + 36 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 21 \, a^{3} b^{7} c^{5} d^{5} + 15 \, a^{4} b^{6} c^{4} d^{6} + 10 \, a^{5} b^{5} c^{3} d^{7} + 6 \, a^{6} b^{4} c^{2} d^{8} + 3 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 13 \, {\left (55 \, b^{10} c^{9} d + 45 \, a b^{9} c^{8} d^{2} + 36 \, a^{2} b^{8} c^{7} d^{3} + 28 \, a^{3} b^{7} c^{6} d^{4} + 21 \, a^{4} b^{6} c^{5} d^{5} + 15 \, a^{5} b^{5} c^{4} d^{6} + 10 \, a^{6} b^{4} c^{3} d^{7} + 6 \, a^{7} b^{3} c^{2} d^{8} + 3 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{858 \, {\left (b^{24} x^{13} + 13 \, a b^{23} x^{12} + 78 \, a^{2} b^{22} x^{11} + 286 \, a^{3} b^{21} x^{10} + 715 \, a^{4} b^{20} x^{9} + 1287 \, a^{5} b^{19} x^{8} + 1716 \, a^{6} b^{18} x^{7} + 1716 \, a^{7} b^{17} x^{6} + 1287 \, a^{8} b^{16} x^{5} + 715 \, a^{9} b^{15} x^{4} + 286 \, a^{10} b^{14} x^{3} + 78 \, a^{11} b^{13} x^{2} + 13 \, a^{12} b^{12} x + a^{13} b^{11}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 997 vs.
\(2 (83) = 166\).
time = 0.90, size = 997, normalized size = 11.20 \begin {gather*} -\frac {286 \, b^{10} d^{10} x^{10} + 66 \, b^{10} c^{10} + 55 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 36 \, a^{3} b^{7} c^{7} d^{3} + 28 \, a^{4} b^{6} c^{6} d^{4} + 21 \, a^{5} b^{5} c^{5} d^{5} + 15 \, a^{6} b^{4} c^{4} d^{6} + 10 \, a^{7} b^{3} c^{3} d^{7} + 6 \, a^{8} b^{2} c^{2} d^{8} + 3 \, a^{9} b c d^{9} + a^{10} d^{10} + 715 \, {\left (3 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 1287 \, {\left (6 \, b^{10} c^{2} d^{8} + 3 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 1716 \, {\left (10 \, b^{10} c^{3} d^{7} + 6 \, a b^{9} c^{2} d^{8} + 3 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 1716 \, {\left (15 \, b^{10} c^{4} d^{6} + 10 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} + 3 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 1287 \, {\left (21 \, b^{10} c^{5} d^{5} + 15 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} + 6 \, a^{3} b^{7} c^{2} d^{8} + 3 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 715 \, {\left (28 \, b^{10} c^{6} d^{4} + 21 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} + 10 \, a^{3} b^{7} c^{3} d^{7} + 6 \, a^{4} b^{6} c^{2} d^{8} + 3 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 286 \, {\left (36 \, b^{10} c^{7} d^{3} + 28 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 15 \, a^{3} b^{7} c^{4} d^{6} + 10 \, a^{4} b^{6} c^{3} d^{7} + 6 \, a^{5} b^{5} c^{2} d^{8} + 3 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 78 \, {\left (45 \, b^{10} c^{8} d^{2} + 36 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 21 \, a^{3} b^{7} c^{5} d^{5} + 15 \, a^{4} b^{6} c^{4} d^{6} + 10 \, a^{5} b^{5} c^{3} d^{7} + 6 \, a^{6} b^{4} c^{2} d^{8} + 3 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 13 \, {\left (55 \, b^{10} c^{9} d + 45 \, a b^{9} c^{8} d^{2} + 36 \, a^{2} b^{8} c^{7} d^{3} + 28 \, a^{3} b^{7} c^{6} d^{4} + 21 \, a^{4} b^{6} c^{5} d^{5} + 15 \, a^{5} b^{5} c^{4} d^{6} + 10 \, a^{6} b^{4} c^{3} d^{7} + 6 \, a^{7} b^{3} c^{2} d^{8} + 3 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{858 \, {\left (b^{24} x^{13} + 13 \, a b^{23} x^{12} + 78 \, a^{2} b^{22} x^{11} + 286 \, a^{3} b^{21} x^{10} + 715 \, a^{4} b^{20} x^{9} + 1287 \, a^{5} b^{19} x^{8} + 1716 \, a^{6} b^{18} x^{7} + 1716 \, a^{7} b^{17} x^{6} + 1287 \, a^{8} b^{16} x^{5} + 715 \, a^{9} b^{15} x^{4} + 286 \, a^{10} b^{14} x^{3} + 78 \, a^{11} b^{13} x^{2} + 13 \, a^{12} b^{12} x + a^{13} b^{11}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 961 vs.
\(2 (83) = 166\).
time = 1.20, size = 961, normalized size = 10.80 \begin {gather*} -\frac {286 \, b^{10} d^{10} x^{10} + 2145 \, b^{10} c d^{9} x^{9} + 715 \, a b^{9} d^{10} x^{9} + 7722 \, b^{10} c^{2} d^{8} x^{8} + 3861 \, a b^{9} c d^{9} x^{8} + 1287 \, a^{2} b^{8} d^{10} x^{8} + 17160 \, b^{10} c^{3} d^{7} x^{7} + 10296 \, a b^{9} c^{2} d^{8} x^{7} + 5148 \, a^{2} b^{8} c d^{9} x^{7} + 1716 \, a^{3} b^{7} d^{10} x^{7} + 25740 \, b^{10} c^{4} d^{6} x^{6} + 17160 \, a b^{9} c^{3} d^{7} x^{6} + 10296 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 5148 \, a^{3} b^{7} c d^{9} x^{6} + 1716 \, a^{4} b^{6} d^{10} x^{6} + 27027 \, b^{10} c^{5} d^{5} x^{5} + 19305 \, a b^{9} c^{4} d^{6} x^{5} + 12870 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 7722 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 3861 \, a^{4} b^{6} c d^{9} x^{5} + 1287 \, a^{5} b^{5} d^{10} x^{5} + 20020 \, b^{10} c^{6} d^{4} x^{4} + 15015 \, a b^{9} c^{5} d^{5} x^{4} + 10725 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 7150 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 4290 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 2145 \, a^{5} b^{5} c d^{9} x^{4} + 715 \, a^{6} b^{4} d^{10} x^{4} + 10296 \, b^{10} c^{7} d^{3} x^{3} + 8008 \, a b^{9} c^{6} d^{4} x^{3} + 6006 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 4290 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 2860 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 1716 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 858 \, a^{6} b^{4} c d^{9} x^{3} + 286 \, a^{7} b^{3} d^{10} x^{3} + 3510 \, b^{10} c^{8} d^{2} x^{2} + 2808 \, a b^{9} c^{7} d^{3} x^{2} + 2184 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 1638 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 1170 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 780 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 468 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 234 \, a^{7} b^{3} c d^{9} x^{2} + 78 \, a^{8} b^{2} d^{10} x^{2} + 715 \, b^{10} c^{9} d x + 585 \, a b^{9} c^{8} d^{2} x + 468 \, a^{2} b^{8} c^{7} d^{3} x + 364 \, a^{3} b^{7} c^{6} d^{4} x + 273 \, a^{4} b^{6} c^{5} d^{5} x + 195 \, a^{5} b^{5} c^{4} d^{6} x + 130 \, a^{6} b^{4} c^{3} d^{7} x + 78 \, a^{7} b^{3} c^{2} d^{8} x + 39 \, a^{8} b^{2} c d^{9} x + 13 \, a^{9} b d^{10} x + 66 \, b^{10} c^{10} + 55 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 36 \, a^{3} b^{7} c^{7} d^{3} + 28 \, a^{4} b^{6} c^{6} d^{4} + 21 \, a^{5} b^{5} c^{5} d^{5} + 15 \, a^{6} b^{4} c^{4} d^{6} + 10 \, a^{7} b^{3} c^{3} d^{7} + 6 \, a^{8} b^{2} c^{2} d^{8} + 3 \, a^{9} b c d^{9} + a^{10} d^{10}}{858 \, {\left (b x + a\right )}^{13} b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 1098, normalized size = 12.34 \begin {gather*} -\frac {a^{10}\,d^{10}+3\,a^9\,b\,c\,d^9+13\,a^9\,b\,d^{10}\,x+6\,a^8\,b^2\,c^2\,d^8+39\,a^8\,b^2\,c\,d^9\,x+78\,a^8\,b^2\,d^{10}\,x^2+10\,a^7\,b^3\,c^3\,d^7+78\,a^7\,b^3\,c^2\,d^8\,x+234\,a^7\,b^3\,c\,d^9\,x^2+286\,a^7\,b^3\,d^{10}\,x^3+15\,a^6\,b^4\,c^4\,d^6+130\,a^6\,b^4\,c^3\,d^7\,x+468\,a^6\,b^4\,c^2\,d^8\,x^2+858\,a^6\,b^4\,c\,d^9\,x^3+715\,a^6\,b^4\,d^{10}\,x^4+21\,a^5\,b^5\,c^5\,d^5+195\,a^5\,b^5\,c^4\,d^6\,x+780\,a^5\,b^5\,c^3\,d^7\,x^2+1716\,a^5\,b^5\,c^2\,d^8\,x^3+2145\,a^5\,b^5\,c\,d^9\,x^4+1287\,a^5\,b^5\,d^{10}\,x^5+28\,a^4\,b^6\,c^6\,d^4+273\,a^4\,b^6\,c^5\,d^5\,x+1170\,a^4\,b^6\,c^4\,d^6\,x^2+2860\,a^4\,b^6\,c^3\,d^7\,x^3+4290\,a^4\,b^6\,c^2\,d^8\,x^4+3861\,a^4\,b^6\,c\,d^9\,x^5+1716\,a^4\,b^6\,d^{10}\,x^6+36\,a^3\,b^7\,c^7\,d^3+364\,a^3\,b^7\,c^6\,d^4\,x+1638\,a^3\,b^7\,c^5\,d^5\,x^2+4290\,a^3\,b^7\,c^4\,d^6\,x^3+7150\,a^3\,b^7\,c^3\,d^7\,x^4+7722\,a^3\,b^7\,c^2\,d^8\,x^5+5148\,a^3\,b^7\,c\,d^9\,x^6+1716\,a^3\,b^7\,d^{10}\,x^7+45\,a^2\,b^8\,c^8\,d^2+468\,a^2\,b^8\,c^7\,d^3\,x+2184\,a^2\,b^8\,c^6\,d^4\,x^2+6006\,a^2\,b^8\,c^5\,d^5\,x^3+10725\,a^2\,b^8\,c^4\,d^6\,x^4+12870\,a^2\,b^8\,c^3\,d^7\,x^5+10296\,a^2\,b^8\,c^2\,d^8\,x^6+5148\,a^2\,b^8\,c\,d^9\,x^7+1287\,a^2\,b^8\,d^{10}\,x^8+55\,a\,b^9\,c^9\,d+585\,a\,b^9\,c^8\,d^2\,x+2808\,a\,b^9\,c^7\,d^3\,x^2+8008\,a\,b^9\,c^6\,d^4\,x^3+15015\,a\,b^9\,c^5\,d^5\,x^4+19305\,a\,b^9\,c^4\,d^6\,x^5+17160\,a\,b^9\,c^3\,d^7\,x^6+10296\,a\,b^9\,c^2\,d^8\,x^7+3861\,a\,b^9\,c\,d^9\,x^8+715\,a\,b^9\,d^{10}\,x^9+66\,b^{10}\,c^{10}+715\,b^{10}\,c^9\,d\,x+3510\,b^{10}\,c^8\,d^2\,x^2+10296\,b^{10}\,c^7\,d^3\,x^3+20020\,b^{10}\,c^6\,d^4\,x^4+27027\,b^{10}\,c^5\,d^5\,x^5+25740\,b^{10}\,c^4\,d^6\,x^6+17160\,b^{10}\,c^3\,d^7\,x^7+7722\,b^{10}\,c^2\,d^8\,x^8+2145\,b^{10}\,c\,d^9\,x^9+286\,b^{10}\,d^{10}\,x^{10}}{858\,a^{13}\,b^{11}+11154\,a^{12}\,b^{12}\,x+66924\,a^{11}\,b^{13}\,x^2+245388\,a^{10}\,b^{14}\,x^3+613470\,a^9\,b^{15}\,x^4+1104246\,a^8\,b^{16}\,x^5+1472328\,a^7\,b^{17}\,x^6+1472328\,a^6\,b^{18}\,x^7+1104246\,a^5\,b^{19}\,x^8+613470\,a^4\,b^{20}\,x^9+245388\,a^3\,b^{21}\,x^{10}+66924\,a^2\,b^{22}\,x^{11}+11154\,a\,b^{23}\,x^{12}+858\,b^{24}\,x^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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