Integrand size = 11, antiderivative size = 191 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=-\frac {1}{2 a^{10} x^2}+\frac {10 b}{a^{11} x}+\frac {b^2}{9 a^3 (a+b x)^9}+\frac {3 b^2}{8 a^4 (a+b x)^8}+\frac {6 b^2}{7 a^5 (a+b x)^7}+\frac {5 b^2}{3 a^6 (a+b x)^6}+\frac {3 b^2}{a^7 (a+b x)^5}+\frac {21 b^2}{4 a^8 (a+b x)^4}+\frac {28 b^2}{3 a^9 (a+b x)^3}+\frac {18 b^2}{a^{10} (a+b x)^2}+\frac {45 b^2}{a^{11} (a+b x)}+\frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log (a+b x)}{a^{12}} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log (a+b x)}{a^{12}}+\frac {45 b^2}{a^{11} (a+b x)}+\frac {10 b}{a^{11} x}+\frac {18 b^2}{a^{10} (a+b x)^2}-\frac {1}{2 a^{10} x^2}+\frac {28 b^2}{3 a^9 (a+b x)^3}+\frac {21 b^2}{4 a^8 (a+b x)^4}+\frac {3 b^2}{a^7 (a+b x)^5}+\frac {5 b^2}{3 a^6 (a+b x)^6}+\frac {6 b^2}{7 a^5 (a+b x)^7}+\frac {3 b^2}{8 a^4 (a+b x)^8}+\frac {b^2}{9 a^3 (a+b x)^9} \]
[In]
[Out]
Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^{10} x^3}-\frac {10 b}{a^{11} x^2}+\frac {55 b^2}{a^{12} x}-\frac {b^3}{a^3 (a+b x)^{10}}-\frac {3 b^3}{a^4 (a+b x)^9}-\frac {6 b^3}{a^5 (a+b x)^8}-\frac {10 b^3}{a^6 (a+b x)^7}-\frac {15 b^3}{a^7 (a+b x)^6}-\frac {21 b^3}{a^8 (a+b x)^5}-\frac {28 b^3}{a^9 (a+b x)^4}-\frac {36 b^3}{a^{10} (a+b x)^3}-\frac {45 b^3}{a^{11} (a+b x)^2}-\frac {55 b^3}{a^{12} (a+b x)}\right ) \, dx \\ & = -\frac {1}{2 a^{10} x^2}+\frac {10 b}{a^{11} x}+\frac {b^2}{9 a^3 (a+b x)^9}+\frac {3 b^2}{8 a^4 (a+b x)^8}+\frac {6 b^2}{7 a^5 (a+b x)^7}+\frac {5 b^2}{3 a^6 (a+b x)^6}+\frac {3 b^2}{a^7 (a+b x)^5}+\frac {21 b^2}{4 a^8 (a+b x)^4}+\frac {28 b^2}{3 a^9 (a+b x)^3}+\frac {18 b^2}{a^{10} (a+b x)^2}+\frac {45 b^2}{a^{11} (a+b x)}+\frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log (a+b x)}{a^{12}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {\frac {a \left (-252 a^{10}+2772 a^9 b x+78419 a^8 b^2 x^2+456291 a^7 b^3 x^3+1326204 a^6 b^4 x^4+2318316 a^5 b^5 x^5+2604294 a^4 b^6 x^6+1905750 a^3 b^7 x^7+882420 a^2 b^8 x^8+235620 a b^9 x^9+27720 b^{10} x^{10}\right )}{x^2 (a+b x)^9}+27720 b^2 \log (x)-27720 b^2 \log (a+b x)}{504 a^{12}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.78
method | result | size |
norman | \(\frac {-\frac {1}{2 a}+\frac {11 b x}{2 a^{2}}-\frac {495 b^{3} x^{3}}{a^{4}}-\frac {2970 b^{4} x^{4}}{a^{5}}-\frac {8470 b^{5} x^{5}}{a^{6}}-\frac {28875 b^{6} x^{6}}{2 a^{7}}-\frac {31647 b^{7} x^{7}}{2 a^{8}}-\frac {11319 b^{8} x^{8}}{a^{9}}-\frac {35937 b^{9} x^{9}}{7 a^{10}}-\frac {75339 b^{10} x^{10}}{56 a^{11}}-\frac {78419 b^{11} x^{11}}{504 a^{12}}}{x^{2} \left (b x +a \right )^{9}}+\frac {55 b^{2} \ln \left (x \right )}{a^{12}}-\frac {55 b^{2} \ln \left (b x +a \right )}{a^{12}}\) | \(149\) |
risch | \(\frac {\frac {55 b^{10} x^{10}}{a^{11}}+\frac {935 b^{9} x^{9}}{2 a^{10}}+\frac {10505 b^{8} x^{8}}{6 a^{9}}+\frac {15125 b^{7} x^{7}}{4 a^{8}}+\frac {20669 b^{6} x^{6}}{4 a^{7}}+\frac {27599 b^{5} x^{5}}{6 a^{6}}+\frac {36839 b^{4} x^{4}}{14 a^{5}}+\frac {50699 b^{3} x^{3}}{56 a^{4}}+\frac {78419 b^{2} x^{2}}{504 a^{3}}+\frac {11 b x}{2 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b x +a \right )^{9}}-\frac {55 b^{2} \ln \left (b x +a \right )}{a^{12}}+\frac {55 b^{2} \ln \left (-x \right )}{a^{12}}\) | \(151\) |
default | \(-\frac {1}{2 a^{10} x^{2}}+\frac {10 b}{a^{11} x}+\frac {b^{2}}{9 a^{3} \left (b x +a \right )^{9}}+\frac {3 b^{2}}{8 a^{4} \left (b x +a \right )^{8}}+\frac {6 b^{2}}{7 a^{5} \left (b x +a \right )^{7}}+\frac {5 b^{2}}{3 a^{6} \left (b x +a \right )^{6}}+\frac {3 b^{2}}{a^{7} \left (b x +a \right )^{5}}+\frac {21 b^{2}}{4 a^{8} \left (b x +a \right )^{4}}+\frac {28 b^{2}}{3 a^{9} \left (b x +a \right )^{3}}+\frac {18 b^{2}}{a^{10} \left (b x +a \right )^{2}}+\frac {45 b^{2}}{a^{11} \left (b x +a \right )}+\frac {55 b^{2} \ln \left (x \right )}{a^{12}}-\frac {55 b^{2} \ln \left (b x +a \right )}{a^{12}}\) | \(178\) |
parallelrisch | \(\frac {-252 a^{11}+249480 \ln \left (x \right ) x^{10} a \,b^{10}-249480 \ln \left (b x +a \right ) x^{10} a \,b^{10}+997920 \ln \left (x \right ) x^{9} a^{2} b^{9}+2328480 \ln \left (x \right ) x^{8} a^{3} b^{8}+3492720 \ln \left (x \right ) x^{7} a^{4} b^{7}+3492720 \ln \left (x \right ) x^{6} a^{5} b^{6}+2328480 \ln \left (x \right ) x^{5} a^{6} b^{5}+997920 \ln \left (x \right ) x^{4} a^{7} b^{4}+249480 \ln \left (x \right ) x^{3} a^{8} b^{3}+27720 \ln \left (x \right ) x^{2} a^{9} b^{2}-997920 \ln \left (b x +a \right ) x^{9} a^{2} b^{9}-2328480 \ln \left (b x +a \right ) x^{8} a^{3} b^{8}-3492720 \ln \left (b x +a \right ) x^{7} a^{4} b^{7}-3492720 \ln \left (b x +a \right ) x^{6} a^{5} b^{6}-2328480 \ln \left (b x +a \right ) x^{5} a^{6} b^{5}-997920 \ln \left (b x +a \right ) x^{4} a^{7} b^{4}-249480 \ln \left (b x +a \right ) x^{3} a^{8} b^{3}-27720 \ln \left (b x +a \right ) x^{2} a^{9} b^{2}-78419 b^{11} x^{11}-5704776 a^{3} x^{8} b^{8}-27720 \ln \left (b x +a \right ) x^{11} b^{11}-2587464 x^{9} a^{2} b^{9}+27720 \ln \left (x \right ) x^{11} b^{11}-7975044 a^{4} b^{7} x^{7}-7276500 a^{5} b^{6} x^{6}-4268880 a^{6} b^{5} x^{5}-1496880 a^{7} b^{4} x^{4}-249480 b^{3} a^{8} x^{3}+2772 a^{10} b x -678051 a \,x^{10} b^{10}}{504 a^{12} x^{2} \left (b x +a \right )^{9}}\) | \(413\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (177) = 354\).
Time = 0.23 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.29 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11} - 27720 \, {\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 27720 \, {\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \left (x\right )}{504 \, {\left (a^{12} b^{9} x^{11} + 9 \, a^{13} b^{8} x^{10} + 36 \, a^{14} b^{7} x^{9} + 84 \, a^{15} b^{6} x^{8} + 126 \, a^{16} b^{5} x^{7} + 126 \, a^{17} b^{4} x^{6} + 84 \, a^{18} b^{3} x^{5} + 36 \, a^{19} b^{2} x^{4} + 9 \, a^{20} b x^{3} + a^{21} x^{2}\right )}} \]
[In]
[Out]
Time = 0.60 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {- 252 a^{10} + 2772 a^{9} b x + 78419 a^{8} b^{2} x^{2} + 456291 a^{7} b^{3} x^{3} + 1326204 a^{6} b^{4} x^{4} + 2318316 a^{5} b^{5} x^{5} + 2604294 a^{4} b^{6} x^{6} + 1905750 a^{3} b^{7} x^{7} + 882420 a^{2} b^{8} x^{8} + 235620 a b^{9} x^{9} + 27720 b^{10} x^{10}}{504 a^{20} x^{2} + 4536 a^{19} b x^{3} + 18144 a^{18} b^{2} x^{4} + 42336 a^{17} b^{3} x^{5} + 63504 a^{16} b^{4} x^{6} + 63504 a^{15} b^{5} x^{7} + 42336 a^{14} b^{6} x^{8} + 18144 a^{13} b^{7} x^{9} + 4536 a^{12} b^{8} x^{10} + 504 a^{11} b^{9} x^{11}} + \frac {55 b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{12}} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {27720 \, b^{10} x^{10} + 235620 \, a b^{9} x^{9} + 882420 \, a^{2} b^{8} x^{8} + 1905750 \, a^{3} b^{7} x^{7} + 2604294 \, a^{4} b^{6} x^{6} + 2318316 \, a^{5} b^{5} x^{5} + 1326204 \, a^{6} b^{4} x^{4} + 456291 \, a^{7} b^{3} x^{3} + 78419 \, a^{8} b^{2} x^{2} + 2772 \, a^{9} b x - 252 \, a^{10}}{504 \, {\left (a^{11} b^{9} x^{11} + 9 \, a^{12} b^{8} x^{10} + 36 \, a^{13} b^{7} x^{9} + 84 \, a^{14} b^{6} x^{8} + 126 \, a^{15} b^{5} x^{7} + 126 \, a^{16} b^{4} x^{6} + 84 \, a^{17} b^{3} x^{5} + 36 \, a^{18} b^{2} x^{4} + 9 \, a^{19} b x^{3} + a^{20} x^{2}\right )}} - \frac {55 \, b^{2} \log \left (b x + a\right )}{a^{12}} + \frac {55 \, b^{2} \log \left (x\right )}{a^{12}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=-\frac {55 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{12}} + \frac {55 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{12}} + \frac {27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11}}{504 \, {\left (b x + a\right )}^{9} a^{12} x^{2}} \]
[In]
[Out]
Time = 0.43 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {\frac {78419\,b^2\,x^2}{504\,a^3}-\frac {1}{2\,a}+\frac {50699\,b^3\,x^3}{56\,a^4}+\frac {36839\,b^4\,x^4}{14\,a^5}+\frac {27599\,b^5\,x^5}{6\,a^6}+\frac {20669\,b^6\,x^6}{4\,a^7}+\frac {15125\,b^7\,x^7}{4\,a^8}+\frac {10505\,b^8\,x^8}{6\,a^9}+\frac {935\,b^9\,x^9}{2\,a^{10}}+\frac {55\,b^{10}\,x^{10}}{a^{11}}+\frac {11\,b\,x}{2\,a^2}}{a^9\,x^2+9\,a^8\,b\,x^3+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^5+126\,a^5\,b^4\,x^6+126\,a^4\,b^5\,x^7+84\,a^3\,b^6\,x^8+36\,a^2\,b^7\,x^9+9\,a\,b^8\,x^{10}+b^9\,x^{11}}-\frac {110\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{12}} \]
[In]
[Out]