Optimal. Leaf size=158 \[ -\frac {1}{a^{10} x}-\frac {b}{9 a^2 (a+b x)^9}-\frac {b}{4 a^3 (a+b x)^8}-\frac {3 b}{7 a^4 (a+b x)^7}-\frac {2 b}{3 a^5 (a+b x)^6}-\frac {b}{a^6 (a+b x)^5}-\frac {3 b}{2 a^7 (a+b x)^4}-\frac {7 b}{3 a^8 (a+b x)^3}-\frac {4 b}{a^9 (a+b x)^2}-\frac {9 b}{a^{10} (a+b x)}-\frac {10 b \log (x)}{a^{11}}+\frac {10 b \log (a+b x)}{a^{11}} \]
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Rubi [A]
time = 0.09, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46}
\begin {gather*} -\frac {10 b \log (x)}{a^{11}}+\frac {10 b \log (a+b x)}{a^{11}}-\frac {9 b}{a^{10} (a+b x)}-\frac {1}{a^{10} x}-\frac {4 b}{a^9 (a+b x)^2}-\frac {7 b}{3 a^8 (a+b x)^3}-\frac {3 b}{2 a^7 (a+b x)^4}-\frac {b}{a^6 (a+b x)^5}-\frac {2 b}{3 a^5 (a+b x)^6}-\frac {3 b}{7 a^4 (a+b x)^7}-\frac {b}{4 a^3 (a+b x)^8}-\frac {b}{9 a^2 (a+b x)^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^{10}} \, dx &=\int \left (\frac {1}{a^{10} x^2}-\frac {10 b}{a^{11} x}+\frac {b^2}{a^2 (a+b x)^{10}}+\frac {2 b^2}{a^3 (a+b x)^9}+\frac {3 b^2}{a^4 (a+b x)^8}+\frac {4 b^2}{a^5 (a+b x)^7}+\frac {5 b^2}{a^6 (a+b x)^6}+\frac {6 b^2}{a^7 (a+b x)^5}+\frac {7 b^2}{a^8 (a+b x)^4}+\frac {8 b^2}{a^9 (a+b x)^3}+\frac {9 b^2}{a^{10} (a+b x)^2}+\frac {10 b^2}{a^{11} (a+b x)}\right ) \, dx\\ &=-\frac {1}{a^{10} x}-\frac {b}{9 a^2 (a+b x)^9}-\frac {b}{4 a^3 (a+b x)^8}-\frac {3 b}{7 a^4 (a+b x)^7}-\frac {2 b}{3 a^5 (a+b x)^6}-\frac {b}{a^6 (a+b x)^5}-\frac {3 b}{2 a^7 (a+b x)^4}-\frac {7 b}{3 a^8 (a+b x)^3}-\frac {4 b}{a^9 (a+b x)^2}-\frac {9 b}{a^{10} (a+b x)}-\frac {10 b \log (x)}{a^{11}}+\frac {10 b \log (a+b x)}{a^{11}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 130, normalized size = 0.82 \begin {gather*} -\frac {\frac {a \left (252 a^9+7129 a^8 b x+41481 a^7 b^2 x^2+120564 a^6 b^3 x^3+210756 a^5 b^4 x^4+236754 a^4 b^5 x^5+173250 a^3 b^6 x^6+80220 a^2 b^7 x^7+21420 a b^8 x^8+2520 b^9 x^9\right )}{x (a+b x)^9}+2520 b \log (x)-2520 b \log (a+b x)}{252 a^{11}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 4.17, size = 314, normalized size = 1.99 \begin {gather*} \frac {a \left (-252 a^9-7129 a^8 b x-41481 a^7 b^2 x^2-120564 a^6 b^3 x^3-210756 a^5 b^4 x^4-236754 a^4 b^5 x^5-173250 a^3 b^6 x^6-80220 a^2 b^7 x^7-21420 a b^8 x^8-2520 b^9 x^9\right )-2520 b x \left (a^9+9 a^8 b x+36 a^7 b^2 x^2+84 a^6 b^3 x^3+126 a^5 b^4 x^4+126 a^4 b^5 x^5+84 a^3 b^6 x^6+36 a^2 b^7 x^7+9 a b^8 x^8+b^9 x^9\right ) \left (\text {Log}\left [x\right ]-\text {Log}\left [\frac {a+b x}{b}\right ]\right )}{252 a^{11} x \left (a^9+9 a^8 b x+36 a^7 b^2 x^2+84 a^6 b^3 x^3+126 a^5 b^4 x^4+126 a^4 b^5 x^5+84 a^3 b^6 x^6+36 a^2 b^7 x^7+9 a b^8 x^8+b^9 x^9\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 147, normalized size = 0.93
method | result | size |
risch | \(\frac {-\frac {10 b^{9} x^{9}}{a^{10}}-\frac {85 b^{8} x^{8}}{a^{9}}-\frac {955 b^{7} x^{7}}{3 a^{8}}-\frac {1375 b^{6} x^{6}}{2 a^{7}}-\frac {1879 b^{5} x^{5}}{2 a^{6}}-\frac {2509 b^{4} x^{4}}{3 a^{5}}-\frac {3349 b^{3} x^{3}}{7 a^{4}}-\frac {4609 b^{2} x^{2}}{28 a^{3}}-\frac {7129 b x}{252 a^{2}}-\frac {1}{a}}{x \left (b x +a \right )^{9}}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {10 b \ln \left (-b x -a \right )}{a^{11}}\) | \(137\) |
norman | \(\frac {-\frac {1}{a}+\frac {90 b^{2} x^{2}}{a^{3}}+\frac {540 b^{3} x^{3}}{a^{4}}+\frac {1540 b^{4} x^{4}}{a^{5}}+\frac {2625 b^{5} x^{5}}{a^{6}}+\frac {2877 b^{6} x^{6}}{a^{7}}+\frac {2058 b^{7} x^{7}}{a^{8}}+\frac {6534 b^{8} x^{8}}{7 a^{9}}+\frac {6849 b^{9} x^{9}}{28 a^{10}}+\frac {7129 b^{10} x^{10}}{252 a^{11}}}{x \left (b x +a \right )^{9}}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {10 b \ln \left (b x +a \right )}{a^{11}}\) | \(138\) |
default | \(-\frac {1}{a^{10} x}-\frac {b}{9 a^{2} \left (b x +a \right )^{9}}-\frac {b}{4 a^{3} \left (b x +a \right )^{8}}-\frac {3 b}{7 a^{4} \left (b x +a \right )^{7}}-\frac {2 b}{3 a^{5} \left (b x +a \right )^{6}}-\frac {b}{a^{6} \left (b x +a \right )^{5}}-\frac {3 b}{2 a^{7} \left (b x +a \right )^{4}}-\frac {7 b}{3 a^{8} \left (b x +a \right )^{3}}-\frac {4 b}{a^{9} \left (b x +a \right )^{2}}-\frac {9 b}{a^{10} \left (b x +a \right )}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {10 b \ln \left (b x +a \right )}{a^{11}}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 223, normalized size = 1.41 \begin {gather*} -\frac {2520 \, b^{9} x^{9} + 21420 \, a b^{8} x^{8} + 80220 \, a^{2} b^{7} x^{7} + 173250 \, a^{3} b^{6} x^{6} + 236754 \, a^{4} b^{5} x^{5} + 210756 \, a^{5} b^{4} x^{4} + 120564 \, a^{6} b^{3} x^{3} + 41481 \, a^{7} b^{2} x^{2} + 7129 \, a^{8} b x + 252 \, a^{9}}{252 \, {\left (a^{10} b^{9} x^{10} + 9 \, a^{11} b^{8} x^{9} + 36 \, a^{12} b^{7} x^{8} + 84 \, a^{13} b^{6} x^{7} + 126 \, a^{14} b^{5} x^{6} + 126 \, a^{15} b^{4} x^{5} + 84 \, a^{16} b^{3} x^{4} + 36 \, a^{17} b^{2} x^{3} + 9 \, a^{18} b x^{2} + a^{19} x\right )}} + \frac {10 \, b \log \left (b x + a\right )}{a^{11}} - \frac {10 \, b \log \left (x\right )}{a^{11}} \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. 417 vs.
\(2 (146) = 292\).
time = 0.32, size = 417, normalized size = 2.64 \begin {gather*} -\frac {2520 \, a b^{9} x^{9} + 21420 \, a^{2} b^{8} x^{8} + 80220 \, a^{3} b^{7} x^{7} + 173250 \, a^{4} b^{6} x^{6} + 236754 \, a^{5} b^{5} x^{5} + 210756 \, a^{6} b^{4} x^{4} + 120564 \, a^{7} b^{3} x^{3} + 41481 \, a^{8} b^{2} x^{2} + 7129 \, a^{9} b x + 252 \, a^{10} - 2520 \, {\left (b^{10} x^{10} + 9 \, a b^{9} x^{9} + 36 \, a^{2} b^{8} x^{8} + 84 \, a^{3} b^{7} x^{7} + 126 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} + 36 \, a^{7} b^{3} x^{3} + 9 \, a^{8} b^{2} x^{2} + a^{9} b x\right )} \log \left (b x + a\right ) + 2520 \, {\left (b^{10} x^{10} + 9 \, a b^{9} x^{9} + 36 \, a^{2} b^{8} x^{8} + 84 \, a^{3} b^{7} x^{7} + 126 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} + 36 \, a^{7} b^{3} x^{3} + 9 \, a^{8} b^{2} x^{2} + a^{9} b x\right )} \log \left (x\right )}{252 \, {\left (a^{11} b^{9} x^{10} + 9 \, a^{12} b^{8} x^{9} + 36 \, a^{13} b^{7} x^{8} + 84 \, a^{14} b^{6} x^{7} + 126 \, a^{15} b^{5} x^{6} + 126 \, a^{16} b^{4} x^{5} + 84 \, a^{17} b^{3} x^{4} + 36 \, a^{18} b^{2} x^{3} + 9 \, a^{19} b x^{2} + a^{20} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.53, size = 233, normalized size = 1.47 \begin {gather*} \frac {- 252 a^{9} - 7129 a^{8} b x - 41481 a^{7} b^{2} x^{2} - 120564 a^{6} b^{3} x^{3} - 210756 a^{5} b^{4} x^{4} - 236754 a^{4} b^{5} x^{5} - 173250 a^{3} b^{6} x^{6} - 80220 a^{2} b^{7} x^{7} - 21420 a b^{8} x^{8} - 2520 b^{9} x^{9}}{252 a^{19} x + 2268 a^{18} b x^{2} + 9072 a^{17} b^{2} x^{3} + 21168 a^{16} b^{3} x^{4} + 31752 a^{15} b^{4} x^{5} + 31752 a^{14} b^{5} x^{6} + 21168 a^{13} b^{6} x^{7} + 9072 a^{12} b^{7} x^{8} + 2268 a^{11} b^{8} x^{9} + 252 a^{10} b^{9} x^{10}} + \frac {10 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 157, normalized size = 0.99 \begin {gather*} -\frac {10 b \ln \left |x\right |}{a^{11}}+\frac {10 b^{2} \ln \left |x b+a\right |}{b a^{11}}+\frac {\frac {1}{252} \left (-2520 b^{9} a x^{9}-21420 b^{8} a^{2} x^{8}-80220 b^{7} a^{3} x^{7}-173250 b^{6} a^{4} x^{6}-236754 b^{5} a^{5} x^{5}-210756 b^{4} a^{6} x^{4}-120564 b^{3} a^{7} x^{3}-41481 b^{2} a^{8} x^{2}-7129 b a^{9} x-252 a^{10}\right )}{a^{11} x \left (b x+a\right )^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 217, normalized size = 1.37 \begin {gather*} \frac {20\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{11}}-\frac {\frac {1}{a}+\frac {4609\,b^2\,x^2}{28\,a^3}+\frac {3349\,b^3\,x^3}{7\,a^4}+\frac {2509\,b^4\,x^4}{3\,a^5}+\frac {1879\,b^5\,x^5}{2\,a^6}+\frac {1375\,b^6\,x^6}{2\,a^7}+\frac {955\,b^7\,x^7}{3\,a^8}+\frac {85\,b^8\,x^8}{a^9}+\frac {10\,b^9\,x^9}{a^{10}}+\frac {7129\,b\,x}{252\,a^2}}{a^9\,x+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^3+84\,a^6\,b^3\,x^4+126\,a^5\,b^4\,x^5+126\,a^4\,b^5\,x^6+84\,a^3\,b^6\,x^7+36\,a^2\,b^7\,x^8+9\,a\,b^8\,x^9+b^9\,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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