Optimal. Leaf size=28 \[ -\frac {1-4 a x}{60 a^3 (1-a x)^{10} (1+a x)^6} \]
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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {82}
\begin {gather*} -\frac {1-4 a x}{60 a^3 (1-a x)^{10} (a x+1)^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 82
Rubi steps
\begin {align*} \int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx &=-\frac {1-4 a x}{60 a^3 (1-a x)^{10} (1+a x)^6}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.96 \begin {gather*} \frac {-1+4 a x}{60 a^3 (-1+a x)^{10} (1+a x)^6} \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. \(181\) vs.
\(2(26)=52\).
time = 0.08, size = 182, normalized size = 6.50
method | result | size |
gosper | \(\frac {4 a x -1}{60 \left (a x -1\right )^{10} a^{3} \left (a x +1\right )^{6}}\) | \(26\) |
risch | \(\frac {-\frac {1}{60 a^{3}}+\frac {x}{15 a^{2}}}{\left (a x -1\right )^{10} \left (a x +1\right )^{6}}\) | \(28\) |
norman | \(\frac {\frac {1}{3} x^{3}-\frac {1}{3} a \,x^{4}-\frac {3}{5} a^{2} x^{5}+\frac {16}{15} a^{3} x^{6}+\frac {1}{3} a^{4} x^{7}-\frac {3}{2} a^{5} x^{8}+\frac {1}{3} a^{6} x^{9}+\frac {16}{15} a^{7} x^{10}-\frac {3}{5} a^{8} x^{11}-\frac {1}{3} a^{9} x^{12}+\frac {1}{3} a^{10} x^{13}-\frac {1}{15} a^{12} x^{15}+\frac {1}{60} a^{13} x^{16}}{\left (a x -1\right )^{10} \left (a x +1\right )^{6}}\) | \(116\) |
default | \(\frac {1}{1280 a^{3} \left (a x -1\right )^{10}}-\frac {1}{768 a^{3} \left (a x -1\right )^{9}}-\frac {7}{6144 a^{3} \left (a x -1\right )^{6}}+\frac {21}{10240 a^{3} \left (a x -1\right )^{5}}-\frac {21}{8192 a^{3} \left (a x -1\right )^{4}}+\frac {11}{4096 a^{3} \left (a x -1\right )^{3}}-\frac {165}{65536 a^{3} \left (a x -1\right )^{2}}+\frac {143}{65536 a^{3} \left (a x -1\right )}+\frac {1}{1024 a^{3} \left (a x -1\right )^{8}}-\frac {1}{12288 a^{3} \left (a x +1\right )^{6}}-\frac {7}{20480 a^{3} \left (a x +1\right )^{5}}-\frac {11}{8192 a^{3} \left (a x +1\right )^{3}}-\frac {121}{65536 a^{3} \left (a x +1\right )^{2}}-\frac {143}{65536 a^{3} \left (a x +1\right )}-\frac {13}{16384 a^{3} \left (a x +1\right )^{4}}\) | \(182\) |
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. 123 vs.
\(2 (25) = 50\).
time = 0.44, size = 123, normalized size = 4.39 \begin {gather*} \frac {4 \, a x - 1}{60 \, {\left (a^{19} x^{16} - 4 \, a^{18} x^{15} + 20 \, a^{16} x^{13} - 20 \, a^{15} x^{12} - 36 \, a^{14} x^{11} + 64 \, a^{13} x^{10} + 20 \, a^{12} x^{9} - 90 \, a^{11} x^{8} + 20 \, a^{10} x^{7} + 64 \, a^{9} x^{6} - 36 \, a^{8} x^{5} - 20 \, a^{7} x^{4} + 20 \, a^{6} x^{3} - 4 \, a^{4} x + a^{3}\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. 123 vs.
\(2 (25) = 50\).
time = 0.44, size = 123, normalized size = 4.39 \begin {gather*} \frac {4 \, a x - 1}{60 \, {\left (a^{19} x^{16} - 4 \, a^{18} x^{15} + 20 \, a^{16} x^{13} - 20 \, a^{15} x^{12} - 36 \, a^{14} x^{11} + 64 \, a^{13} x^{10} + 20 \, a^{12} x^{9} - 90 \, a^{11} x^{8} + 20 \, a^{10} x^{7} + 64 \, a^{9} x^{6} - 36 \, a^{8} x^{5} - 20 \, a^{7} x^{4} + 20 \, a^{6} x^{3} - 4 \, a^{4} x + a^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (24) = 48\).
time = 0.70, size = 129, normalized size = 4.61 \begin {gather*} - \frac {- 4 a x + 1}{60 a^{19} x^{16} - 240 a^{18} x^{15} + 1200 a^{16} x^{13} - 1200 a^{15} x^{12} - 2160 a^{14} x^{11} + 3840 a^{13} x^{10} + 1200 a^{12} x^{9} - 5400 a^{11} x^{8} + 1200 a^{10} x^{7} + 3840 a^{9} x^{6} - 2160 a^{8} x^{5} - 1200 a^{7} x^{4} + 1200 a^{6} x^{3} - 240 a^{4} x + 60 a^{3}} \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. 133 vs.
\(2 (25) = 50\).
time = 3.15, size = 133, normalized size = 4.75 \begin {gather*} -\frac {2145 \, a^{5} x^{5} + 12540 \, a^{4} x^{4} + 30030 \, a^{3} x^{3} + 37080 \, a^{2} x^{2} + 23841 \, a x + 6476}{983040 \, {\left (a x + 1\right )}^{6} a^{3}} + \frac {2145 \, a^{9} x^{9} - 21780 \, a^{8} x^{8} + 99660 \, a^{7} x^{7} - 270480 \, a^{6} x^{6} + 481446 \, a^{5} x^{5} - 584920 \, a^{4} x^{4} + 486220 \, a^{3} x^{3} - 265680 \, a^{2} x^{2} + 84065 \, a x - 9908}{983040 \, {\left (a x - 1\right )}^{10} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.33, size = 124, normalized size = 4.43 \begin {gather*} \frac {\frac {x}{15\,a^2}-\frac {1}{60\,a^3}}{a^{16}\,x^{16}-4\,a^{15}\,x^{15}+20\,a^{13}\,x^{13}-20\,a^{12}\,x^{12}-36\,a^{11}\,x^{11}+64\,a^{10}\,x^{10}+20\,a^9\,x^9-90\,a^8\,x^8+20\,a^7\,x^7+64\,a^6\,x^6-36\,a^5\,x^5-20\,a^4\,x^4+20\,a^3\,x^3-4\,a\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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