Integrand size = 11, antiderivative size = 96 \[ \int \frac {(a+b x)^7}{x^{13}} \, dx=-\frac {(a+b x)^8}{12 a x^{12}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac {b^3 (a+b x)^8}{495 a^4 x^9}-\frac {b^4 (a+b x)^8}{3960 a^5 x^8} \]
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Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^7}{x^{13}} \, dx=-\frac {b^4 (a+b x)^8}{3960 a^5 x^8}+\frac {b^3 (a+b x)^8}{495 a^4 x^9}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {(a+b x)^8}{12 a x^{12}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^8}{12 a x^{12}}-\frac {b \int \frac {(a+b x)^7}{x^{12}} \, dx}{3 a} \\ & = -\frac {(a+b x)^8}{12 a x^{12}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}+\frac {b^2 \int \frac {(a+b x)^7}{x^{11}} \, dx}{11 a^2} \\ & = -\frac {(a+b x)^8}{12 a x^{12}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}-\frac {b^3 \int \frac {(a+b x)^7}{x^{10}} \, dx}{55 a^3} \\ & = -\frac {(a+b x)^8}{12 a x^{12}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac {b^3 (a+b x)^8}{495 a^4 x^9}+\frac {b^4 \int \frac {(a+b x)^7}{x^9} \, dx}{495 a^4} \\ & = -\frac {(a+b x)^8}{12 a x^{12}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac {b^3 (a+b x)^8}{495 a^4 x^9}-\frac {b^4 (a+b x)^8}{3960 a^5 x^8} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^7}{x^{13}} \, dx=-\frac {a^7}{12 x^{12}}-\frac {7 a^6 b}{11 x^{11}}-\frac {21 a^5 b^2}{10 x^{10}}-\frac {35 a^4 b^3}{9 x^9}-\frac {35 a^3 b^4}{8 x^8}-\frac {3 a^2 b^5}{x^7}-\frac {7 a b^6}{6 x^6}-\frac {b^7}{5 x^5} \]
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Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82
method | result | size |
norman | \(\frac {-\frac {1}{5} b^{7} x^{7}-\frac {7}{6} a \,b^{6} x^{6}-3 a^{2} b^{5} x^{5}-\frac {35}{8} a^{3} b^{4} x^{4}-\frac {35}{9} a^{4} b^{3} x^{3}-\frac {21}{10} a^{5} b^{2} x^{2}-\frac {7}{11} a^{6} b x -\frac {1}{12} a^{7}}{x^{12}}\) | \(79\) |
risch | \(\frac {-\frac {1}{5} b^{7} x^{7}-\frac {7}{6} a \,b^{6} x^{6}-3 a^{2} b^{5} x^{5}-\frac {35}{8} a^{3} b^{4} x^{4}-\frac {35}{9} a^{4} b^{3} x^{3}-\frac {21}{10} a^{5} b^{2} x^{2}-\frac {7}{11} a^{6} b x -\frac {1}{12} a^{7}}{x^{12}}\) | \(79\) |
gosper | \(-\frac {792 b^{7} x^{7}+4620 a \,b^{6} x^{6}+11880 a^{2} b^{5} x^{5}+17325 a^{3} b^{4} x^{4}+15400 a^{4} b^{3} x^{3}+8316 a^{5} b^{2} x^{2}+2520 a^{6} b x +330 a^{7}}{3960 x^{12}}\) | \(80\) |
default | \(-\frac {21 a^{5} b^{2}}{10 x^{10}}-\frac {7 a \,b^{6}}{6 x^{6}}-\frac {3 a^{2} b^{5}}{x^{7}}-\frac {35 a^{4} b^{3}}{9 x^{9}}-\frac {a^{7}}{12 x^{12}}-\frac {7 a^{6} b}{11 x^{11}}-\frac {b^{7}}{5 x^{5}}-\frac {35 a^{3} b^{4}}{8 x^{8}}\) | \(80\) |
parallelrisch | \(\frac {-792 b^{7} x^{7}-4620 a \,b^{6} x^{6}-11880 a^{2} b^{5} x^{5}-17325 a^{3} b^{4} x^{4}-15400 a^{4} b^{3} x^{3}-8316 a^{5} b^{2} x^{2}-2520 a^{6} b x -330 a^{7}}{3960 x^{12}}\) | \(80\) |
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Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^7}{x^{13}} \, dx=-\frac {792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \]
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Time = 0.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^7}{x^{13}} \, dx=\frac {- 330 a^{7} - 2520 a^{6} b x - 8316 a^{5} b^{2} x^{2} - 15400 a^{4} b^{3} x^{3} - 17325 a^{3} b^{4} x^{4} - 11880 a^{2} b^{5} x^{5} - 4620 a b^{6} x^{6} - 792 b^{7} x^{7}}{3960 x^{12}} \]
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Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^7}{x^{13}} \, dx=-\frac {792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \]
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Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^7}{x^{13}} \, dx=-\frac {792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \]
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Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^7}{x^{13}} \, dx=-\frac {\frac {a^7}{12}+\frac {7\,a^6\,b\,x}{11}+\frac {21\,a^5\,b^2\,x^2}{10}+\frac {35\,a^4\,b^3\,x^3}{9}+\frac {35\,a^3\,b^4\,x^4}{8}+3\,a^2\,b^5\,x^5+\frac {7\,a\,b^6\,x^6}{6}+\frac {b^7\,x^7}{5}}{x^{12}} \]
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