Integrand size = 20, antiderivative size = 119 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {a^7 c^6}{12 x^{12}}+\frac {5 a^6 b c^6}{11 x^{11}}-\frac {9 a^5 b^2 c^6}{10 x^{10}}+\frac {5 a^4 b^3 c^6}{9 x^9}+\frac {5 a^3 b^4 c^6}{8 x^8}-\frac {9 a^2 b^5 c^6}{7 x^7}+\frac {5 a b^6 c^6}{6 x^6}-\frac {b^7 c^6}{5 x^5} \]
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Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {a^7 c^6}{12 x^{12}}+\frac {5 a^6 b c^6}{11 x^{11}}-\frac {9 a^5 b^2 c^6}{10 x^{10}}+\frac {5 a^4 b^3 c^6}{9 x^9}+\frac {5 a^3 b^4 c^6}{8 x^8}-\frac {9 a^2 b^5 c^6}{7 x^7}+\frac {5 a b^6 c^6}{6 x^6}-\frac {b^7 c^6}{5 x^5} \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^7 c^6}{x^{13}}-\frac {5 a^6 b c^6}{x^{12}}+\frac {9 a^5 b^2 c^6}{x^{11}}-\frac {5 a^4 b^3 c^6}{x^{10}}-\frac {5 a^3 b^4 c^6}{x^9}+\frac {9 a^2 b^5 c^6}{x^8}-\frac {5 a b^6 c^6}{x^7}+\frac {b^7 c^6}{x^6}\right ) \, dx \\ & = -\frac {a^7 c^6}{12 x^{12}}+\frac {5 a^6 b c^6}{11 x^{11}}-\frac {9 a^5 b^2 c^6}{10 x^{10}}+\frac {5 a^4 b^3 c^6}{9 x^9}+\frac {5 a^3 b^4 c^6}{8 x^8}-\frac {9 a^2 b^5 c^6}{7 x^7}+\frac {5 a b^6 c^6}{6 x^6}-\frac {b^7 c^6}{5 x^5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {a^7 c^6}{12 x^{12}}+\frac {5 a^6 b c^6}{11 x^{11}}-\frac {9 a^5 b^2 c^6}{10 x^{10}}+\frac {5 a^4 b^3 c^6}{9 x^9}+\frac {5 a^3 b^4 c^6}{8 x^8}-\frac {9 a^2 b^5 c^6}{7 x^7}+\frac {5 a b^6 c^6}{6 x^6}-\frac {b^7 c^6}{5 x^5} \]
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Time = 0.38 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(-\frac {c^{6} \left (5544 b^{7} x^{7}-23100 a \,b^{6} x^{6}+35640 a^{2} b^{5} x^{5}-17325 a^{3} b^{4} x^{4}-15400 a^{4} b^{3} x^{3}+24948 a^{5} b^{2} x^{2}-12600 a^{6} b x +2310 a^{7}\right )}{27720 x^{12}}\) | \(83\) |
default | \(c^{6} \left (-\frac {a^{7}}{12 x^{12}}+\frac {5 a \,b^{6}}{6 x^{6}}-\frac {9 a^{2} b^{5}}{7 x^{7}}+\frac {5 a^{3} b^{4}}{8 x^{8}}-\frac {9 a^{5} b^{2}}{10 x^{10}}-\frac {b^{7}}{5 x^{5}}+\frac {5 a^{4} b^{3}}{9 x^{9}}+\frac {5 a^{6} b}{11 x^{11}}\right )\) | \(84\) |
norman | \(\frac {-\frac {1}{12} a^{7} c^{6}-\frac {1}{5} b^{7} c^{6} x^{7}+\frac {5}{6} a \,b^{6} c^{6} x^{6}-\frac {9}{7} a^{2} b^{5} c^{6} x^{5}+\frac {5}{8} a^{3} b^{4} c^{6} x^{4}+\frac {5}{9} a^{4} b^{3} c^{6} x^{3}-\frac {9}{10} a^{5} b^{2} c^{6} x^{2}+\frac {5}{11} a^{6} b \,c^{6} x}{x^{12}}\) | \(103\) |
risch | \(\frac {-\frac {1}{12} a^{7} c^{6}-\frac {1}{5} b^{7} c^{6} x^{7}+\frac {5}{6} a \,b^{6} c^{6} x^{6}-\frac {9}{7} a^{2} b^{5} c^{6} x^{5}+\frac {5}{8} a^{3} b^{4} c^{6} x^{4}+\frac {5}{9} a^{4} b^{3} c^{6} x^{3}-\frac {9}{10} a^{5} b^{2} c^{6} x^{2}+\frac {5}{11} a^{6} b \,c^{6} x}{x^{12}}\) | \(103\) |
parallelrisch | \(\frac {-5544 b^{7} c^{6} x^{7}+23100 a \,b^{6} c^{6} x^{6}-35640 a^{2} b^{5} c^{6} x^{5}+17325 a^{3} b^{4} c^{6} x^{4}+15400 a^{4} b^{3} c^{6} x^{3}-24948 a^{5} b^{2} c^{6} x^{2}+12600 a^{6} b \,c^{6} x -2310 a^{7} c^{6}}{27720 x^{12}}\) | \(104\) |
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Time = 0.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {5544 \, b^{7} c^{6} x^{7} - 23100 \, a b^{6} c^{6} x^{6} + 35640 \, a^{2} b^{5} c^{6} x^{5} - 17325 \, a^{3} b^{4} c^{6} x^{4} - 15400 \, a^{4} b^{3} c^{6} x^{3} + 24948 \, a^{5} b^{2} c^{6} x^{2} - 12600 \, a^{6} b c^{6} x + 2310 \, a^{7} c^{6}}{27720 \, x^{12}} \]
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Time = 0.37 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=\frac {- 2310 a^{7} c^{6} + 12600 a^{6} b c^{6} x - 24948 a^{5} b^{2} c^{6} x^{2} + 15400 a^{4} b^{3} c^{6} x^{3} + 17325 a^{3} b^{4} c^{6} x^{4} - 35640 a^{2} b^{5} c^{6} x^{5} + 23100 a b^{6} c^{6} x^{6} - 5544 b^{7} c^{6} x^{7}}{27720 x^{12}} \]
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Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {5544 \, b^{7} c^{6} x^{7} - 23100 \, a b^{6} c^{6} x^{6} + 35640 \, a^{2} b^{5} c^{6} x^{5} - 17325 \, a^{3} b^{4} c^{6} x^{4} - 15400 \, a^{4} b^{3} c^{6} x^{3} + 24948 \, a^{5} b^{2} c^{6} x^{2} - 12600 \, a^{6} b c^{6} x + 2310 \, a^{7} c^{6}}{27720 \, x^{12}} \]
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Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {5544 \, b^{7} c^{6} x^{7} - 23100 \, a b^{6} c^{6} x^{6} + 35640 \, a^{2} b^{5} c^{6} x^{5} - 17325 \, a^{3} b^{4} c^{6} x^{4} - 15400 \, a^{4} b^{3} c^{6} x^{3} + 24948 \, a^{5} b^{2} c^{6} x^{2} - 12600 \, a^{6} b c^{6} x + 2310 \, a^{7} c^{6}}{27720 \, x^{12}} \]
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Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {\frac {a^7\,c^6}{12}-\frac {5\,a^6\,b\,c^6\,x}{11}+\frac {9\,a^5\,b^2\,c^6\,x^2}{10}-\frac {5\,a^4\,b^3\,c^6\,x^3}{9}-\frac {5\,a^3\,b^4\,c^6\,x^4}{8}+\frac {9\,a^2\,b^5\,c^6\,x^5}{7}-\frac {5\,a\,b^6\,c^6\,x^6}{6}+\frac {b^7\,c^6\,x^7}{5}}{x^{12}} \]
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