Integrand size = 11, antiderivative size = 115 \[ \int \frac {(a+b x)^{10}}{x^4} \, dx=-\frac {a^{10}}{3 x^3}-\frac {5 a^9 b}{x^2}-\frac {45 a^8 b^2}{x}+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac {5}{3} a b^9 x^6+\frac {b^{10} x^7}{7}+120 a^7 b^3 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 115, 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 = {45} \[ \int \frac {(a+b x)^{10}}{x^4} \, dx=-\frac {a^{10}}{3 x^3}-\frac {5 a^9 b}{x^2}-\frac {45 a^8 b^2}{x}+120 a^7 b^3 \log (x)+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac {5}{3} a b^9 x^6+\frac {b^{10} x^7}{7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (210 a^6 b^4+\frac {a^{10}}{x^4}+\frac {10 a^9 b}{x^3}+\frac {45 a^8 b^2}{x^2}+\frac {120 a^7 b^3}{x}+252 a^5 b^5 x+210 a^4 b^6 x^2+120 a^3 b^7 x^3+45 a^2 b^8 x^4+10 a b^9 x^5+b^{10} x^6\right ) \, dx \\ & = -\frac {a^{10}}{3 x^3}-\frac {5 a^9 b}{x^2}-\frac {45 a^8 b^2}{x}+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac {5}{3} a b^9 x^6+\frac {b^{10} x^7}{7}+120 a^7 b^3 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^4} \, dx=-\frac {a^{10}}{3 x^3}-\frac {5 a^9 b}{x^2}-\frac {45 a^8 b^2}{x}+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac {5}{3} a b^9 x^6+\frac {b^{10} x^7}{7}+120 a^7 b^3 \log (x) \]
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Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {a^{10}}{3 x^{3}}-\frac {5 a^{9} b}{x^{2}}-\frac {45 a^{8} b^{2}}{x}+210 a^{6} b^{4} x +126 a^{5} b^{5} x^{2}+70 a^{4} b^{6} x^{3}+30 a^{3} b^{7} x^{4}+9 a^{2} b^{8} x^{5}+\frac {5 a \,b^{9} x^{6}}{3}+\frac {b^{10} x^{7}}{7}+120 a^{7} b^{3} \ln \left (x \right )\) | \(110\) |
risch | \(\frac {b^{10} x^{7}}{7}+\frac {5 a \,b^{9} x^{6}}{3}+9 a^{2} b^{8} x^{5}+30 a^{3} b^{7} x^{4}+70 a^{4} b^{6} x^{3}+126 a^{5} b^{5} x^{2}+210 a^{6} b^{4} x +\frac {-45 a^{8} b^{2} x^{2}-5 a^{9} b x -\frac {1}{3} a^{10}}{x^{3}}+120 a^{7} b^{3} \ln \left (x \right )\) | \(110\) |
norman | \(\frac {-\frac {1}{3} a^{10}+\frac {1}{7} b^{10} x^{10}+\frac {5}{3} a \,b^{9} x^{9}+9 a^{2} b^{8} x^{8}+30 a^{3} b^{7} x^{7}+70 a^{4} b^{6} x^{6}+126 a^{5} b^{5} x^{5}+210 a^{6} b^{4} x^{4}-45 a^{8} b^{2} x^{2}-5 a^{9} b x}{x^{3}}+120 a^{7} b^{3} \ln \left (x \right )\) | \(112\) |
parallelrisch | \(\frac {3 b^{10} x^{10}+35 a \,b^{9} x^{9}+189 a^{2} b^{8} x^{8}+630 a^{3} b^{7} x^{7}+1470 a^{4} b^{6} x^{6}+2646 a^{5} b^{5} x^{5}+2520 a^{7} b^{3} \ln \left (x \right ) x^{3}+4410 a^{6} b^{4} x^{4}-945 a^{8} b^{2} x^{2}-105 a^{9} b x -7 a^{10}}{21 x^{3}}\) | \(115\) |
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Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{10}}{x^4} \, dx=\frac {3 \, b^{10} x^{10} + 35 \, a b^{9} x^{9} + 189 \, a^{2} b^{8} x^{8} + 630 \, a^{3} b^{7} x^{7} + 1470 \, a^{4} b^{6} x^{6} + 2646 \, a^{5} b^{5} x^{5} + 4410 \, a^{6} b^{4} x^{4} + 2520 \, a^{7} b^{3} x^{3} \log \left (x\right ) - 945 \, a^{8} b^{2} x^{2} - 105 \, a^{9} b x - 7 \, a^{10}}{21 \, x^{3}} \]
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Time = 0.14 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10}}{x^4} \, dx=120 a^{7} b^{3} \log {\left (x \right )} + 210 a^{6} b^{4} x + 126 a^{5} b^{5} x^{2} + 70 a^{4} b^{6} x^{3} + 30 a^{3} b^{7} x^{4} + 9 a^{2} b^{8} x^{5} + \frac {5 a b^{9} x^{6}}{3} + \frac {b^{10} x^{7}}{7} + \frac {- a^{10} - 15 a^{9} b x - 135 a^{8} b^{2} x^{2}}{3 x^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{10}}{x^4} \, dx=\frac {1}{7} \, b^{10} x^{7} + \frac {5}{3} \, a b^{9} x^{6} + 9 \, a^{2} b^{8} x^{5} + 30 \, a^{3} b^{7} x^{4} + 70 \, a^{4} b^{6} x^{3} + 126 \, a^{5} b^{5} x^{2} + 210 \, a^{6} b^{4} x + 120 \, a^{7} b^{3} \log \left (x\right ) - \frac {135 \, a^{8} b^{2} x^{2} + 15 \, a^{9} b x + a^{10}}{3 \, x^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{10}}{x^4} \, dx=\frac {1}{7} \, b^{10} x^{7} + \frac {5}{3} \, a b^{9} x^{6} + 9 \, a^{2} b^{8} x^{5} + 30 \, a^{3} b^{7} x^{4} + 70 \, a^{4} b^{6} x^{3} + 126 \, a^{5} b^{5} x^{2} + 210 \, a^{6} b^{4} x + 120 \, a^{7} b^{3} \log \left ({\left | x \right |}\right ) - \frac {135 \, a^{8} b^{2} x^{2} + 15 \, a^{9} b x + a^{10}}{3 \, x^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^4} \, dx=\frac {b^{10}\,x^7}{7}-\frac {\frac {a^{10}}{3}+5\,a^9\,b\,x+45\,a^8\,b^2\,x^2}{x^3}+210\,a^6\,b^4\,x+\frac {5\,a\,b^9\,x^6}{3}+126\,a^5\,b^5\,x^2+70\,a^4\,b^6\,x^3+30\,a^3\,b^7\,x^4+9\,a^2\,b^8\,x^5+120\,a^7\,b^3\,\ln \left (x\right ) \]
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