Optimal. Leaf size=58 \[ -\frac {(c+d x)^{11}}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^{11}}{132 (b c-a d)^2 (a+b x)^{11}} \]
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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 2, 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 (c+d x)^{11}}{132 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (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)^{13}} \, dx &=-\frac {(c+d x)^{11}}{12 (b c-a d) (a+b x)^{12}}-\frac {d \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{12 (b c-a d)}\\ &=-\frac {(c+d x)^{11}}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^{11}}{132 (b c-a d)^2 (a+b x)^{11}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(684\) vs. \(2(58)=116\).
time = 0.18, size = 684, normalized size = 11.79 \begin {gather*} -\frac {a^{10} d^{10}+2 a^9 b d^9 (c+6 d x)+3 a^8 b^2 d^8 \left (c^2+8 c d x+22 d^2 x^2\right )+4 a^7 b^3 d^7 \left (c^3+9 c^2 d x+33 c d^2 x^2+55 d^3 x^3\right )+a^6 b^4 d^6 \left (5 c^4+48 c^3 d x+198 c^2 d^2 x^2+440 c d^3 x^3+495 d^4 x^4\right )+6 a^5 b^5 d^5 \left (c^5+10 c^4 d x+44 c^3 d^2 x^2+110 c^2 d^3 x^3+165 c d^4 x^4+132 d^5 x^5\right )+a^4 b^6 d^4 \left (7 c^6+72 c^5 d x+330 c^4 d^2 x^2+880 c^3 d^3 x^3+1485 c^2 d^4 x^4+1584 c d^5 x^5+924 d^6 x^6\right )+4 a^3 b^7 d^3 \left (2 c^7+21 c^6 d x+99 c^5 d^2 x^2+275 c^4 d^3 x^3+495 c^3 d^4 x^4+594 c^2 d^5 x^5+462 c d^6 x^6+198 d^7 x^7\right )+3 a^2 b^8 d^2 \left (3 c^8+32 c^7 d x+154 c^6 d^2 x^2+440 c^5 d^3 x^3+825 c^4 d^4 x^4+1056 c^3 d^5 x^5+924 c^2 d^6 x^6+528 c d^7 x^7+165 d^8 x^8\right )+2 a b^9 d \left (5 c^9+54 c^8 d x+264 c^7 d^2 x^2+770 c^6 d^3 x^3+1485 c^5 d^4 x^4+1980 c^4 d^5 x^5+1848 c^3 d^6 x^6+1188 c^2 d^7 x^7+495 c d^8 x^8+110 d^9 x^9\right )+b^{10} \left (11 c^{10}+120 c^9 d x+594 c^8 d^2 x^2+1760 c^7 d^3 x^3+3465 c^6 d^4 x^4+4752 c^5 d^5 x^5+4620 c^4 d^6 x^6+3168 c^3 d^7 x^7+1485 c^2 d^8 x^8+440 c d^9 x^9+66 d^{10} x^{10}\right )}{132 b^{11} (a+b x)^{12}} \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.
\(2(54)=108\).
time = 0.14, size = 867, normalized size = 14.95 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. 986 vs.
\(2 (54) = 108\).
time = 0.34, size = 986, normalized size = 17.00 \begin {gather*} -\frac {66 \, b^{10} d^{10} x^{10} + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10} + 220 \, {\left (2 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 495 \, {\left (3 \, b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 792 \, {\left (4 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 2 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 924 \, {\left (5 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 3 \, a^{2} b^{8} c^{2} d^{8} + 2 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 792 \, {\left (6 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 4 \, a^{2} b^{8} c^{3} d^{7} + 3 \, a^{3} b^{7} c^{2} d^{8} + 2 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 495 \, {\left (7 \, b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 4 \, a^{3} b^{7} c^{3} d^{7} + 3 \, a^{4} b^{6} c^{2} d^{8} + 2 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 220 \, {\left (8 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 6 \, a^{2} b^{8} c^{5} d^{5} + 5 \, a^{3} b^{7} c^{4} d^{6} + 4 \, a^{4} b^{6} c^{3} d^{7} + 3 \, a^{5} b^{5} c^{2} d^{8} + 2 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 66 \, {\left (9 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 7 \, a^{2} b^{8} c^{6} d^{4} + 6 \, a^{3} b^{7} c^{5} d^{5} + 5 \, a^{4} b^{6} c^{4} d^{6} + 4 \, a^{5} b^{5} c^{3} d^{7} + 3 \, a^{6} b^{4} c^{2} d^{8} + 2 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 12 \, {\left (10 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 8 \, a^{2} b^{8} c^{7} d^{3} + 7 \, a^{3} b^{7} c^{6} d^{4} + 6 \, a^{4} b^{6} c^{5} d^{5} + 5 \, a^{5} b^{5} c^{4} d^{6} + 4 \, a^{6} b^{4} c^{3} d^{7} + 3 \, a^{7} b^{3} c^{2} d^{8} + 2 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{132 \, {\left (b^{23} x^{12} + 12 \, a b^{22} x^{11} + 66 \, a^{2} b^{21} x^{10} + 220 \, a^{3} b^{20} x^{9} + 495 \, a^{4} b^{19} x^{8} + 792 \, a^{5} b^{18} x^{7} + 924 \, a^{6} b^{17} x^{6} + 792 \, a^{7} b^{16} x^{5} + 495 \, a^{8} b^{15} x^{4} + 220 \, a^{9} b^{14} x^{3} + 66 \, a^{10} b^{13} x^{2} + 12 \, a^{11} b^{12} x + a^{12} 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. 986 vs.
\(2 (54) = 108\).
time = 1.49, size = 986, normalized size = 17.00 \begin {gather*} -\frac {66 \, b^{10} d^{10} x^{10} + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10} + 220 \, {\left (2 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 495 \, {\left (3 \, b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 792 \, {\left (4 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 2 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 924 \, {\left (5 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 3 \, a^{2} b^{8} c^{2} d^{8} + 2 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 792 \, {\left (6 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 4 \, a^{2} b^{8} c^{3} d^{7} + 3 \, a^{3} b^{7} c^{2} d^{8} + 2 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 495 \, {\left (7 \, b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 4 \, a^{3} b^{7} c^{3} d^{7} + 3 \, a^{4} b^{6} c^{2} d^{8} + 2 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 220 \, {\left (8 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 6 \, a^{2} b^{8} c^{5} d^{5} + 5 \, a^{3} b^{7} c^{4} d^{6} + 4 \, a^{4} b^{6} c^{3} d^{7} + 3 \, a^{5} b^{5} c^{2} d^{8} + 2 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 66 \, {\left (9 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 7 \, a^{2} b^{8} c^{6} d^{4} + 6 \, a^{3} b^{7} c^{5} d^{5} + 5 \, a^{4} b^{6} c^{4} d^{6} + 4 \, a^{5} b^{5} c^{3} d^{7} + 3 \, a^{6} b^{4} c^{2} d^{8} + 2 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 12 \, {\left (10 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 8 \, a^{2} b^{8} c^{7} d^{3} + 7 \, a^{3} b^{7} c^{6} d^{4} + 6 \, a^{4} b^{6} c^{5} d^{5} + 5 \, a^{5} b^{5} c^{4} d^{6} + 4 \, a^{6} b^{4} c^{3} d^{7} + 3 \, a^{7} b^{3} c^{2} d^{8} + 2 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{132 \, {\left (b^{23} x^{12} + 12 \, a b^{22} x^{11} + 66 \, a^{2} b^{21} x^{10} + 220 \, a^{3} b^{20} x^{9} + 495 \, a^{4} b^{19} x^{8} + 792 \, a^{5} b^{18} x^{7} + 924 \, a^{6} b^{17} x^{6} + 792 \, a^{7} b^{16} x^{5} + 495 \, a^{8} b^{15} x^{4} + 220 \, a^{9} b^{14} x^{3} + 66 \, a^{10} b^{13} x^{2} + 12 \, a^{11} b^{12} x + a^{12} 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.
\(2 (54) = 108\).
time = 0.58, size = 961, normalized size = 16.57 \begin {gather*} -\frac {66 \, b^{10} d^{10} x^{10} + 440 \, b^{10} c d^{9} x^{9} + 220 \, a b^{9} d^{10} x^{9} + 1485 \, b^{10} c^{2} d^{8} x^{8} + 990 \, a b^{9} c d^{9} x^{8} + 495 \, a^{2} b^{8} d^{10} x^{8} + 3168 \, b^{10} c^{3} d^{7} x^{7} + 2376 \, a b^{9} c^{2} d^{8} x^{7} + 1584 \, a^{2} b^{8} c d^{9} x^{7} + 792 \, a^{3} b^{7} d^{10} x^{7} + 4620 \, b^{10} c^{4} d^{6} x^{6} + 3696 \, a b^{9} c^{3} d^{7} x^{6} + 2772 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 1848 \, a^{3} b^{7} c d^{9} x^{6} + 924 \, a^{4} b^{6} d^{10} x^{6} + 4752 \, b^{10} c^{5} d^{5} x^{5} + 3960 \, a b^{9} c^{4} d^{6} x^{5} + 3168 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 2376 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 1584 \, a^{4} b^{6} c d^{9} x^{5} + 792 \, a^{5} b^{5} d^{10} x^{5} + 3465 \, b^{10} c^{6} d^{4} x^{4} + 2970 \, a b^{9} c^{5} d^{5} x^{4} + 2475 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 1980 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 1485 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 990 \, a^{5} b^{5} c d^{9} x^{4} + 495 \, a^{6} b^{4} d^{10} x^{4} + 1760 \, b^{10} c^{7} d^{3} x^{3} + 1540 \, a b^{9} c^{6} d^{4} x^{3} + 1320 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 1100 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 880 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 660 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 440 \, a^{6} b^{4} c d^{9} x^{3} + 220 \, a^{7} b^{3} d^{10} x^{3} + 594 \, b^{10} c^{8} d^{2} x^{2} + 528 \, a b^{9} c^{7} d^{3} x^{2} + 462 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 396 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 330 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 264 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 198 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 132 \, a^{7} b^{3} c d^{9} x^{2} + 66 \, a^{8} b^{2} d^{10} x^{2} + 120 \, b^{10} c^{9} d x + 108 \, a b^{9} c^{8} d^{2} x + 96 \, a^{2} b^{8} c^{7} d^{3} x + 84 \, a^{3} b^{7} c^{6} d^{4} x + 72 \, a^{4} b^{6} c^{5} d^{5} x + 60 \, a^{5} b^{5} c^{4} d^{6} x + 48 \, a^{6} b^{4} c^{3} d^{7} x + 36 \, a^{7} b^{3} c^{2} d^{8} x + 24 \, a^{8} b^{2} c d^{9} x + 12 \, a^{9} b d^{10} x + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10}}{132 \, {\left (b x + a\right )}^{12} b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 39, normalized size = 0.67 \begin {gather*} \frac {{\left (c+d\,x\right )}^{11}\,\left (12\,a\,d-11\,b\,c+b\,d\,x\right )}{132\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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