Integrand size = 20, antiderivative size = 114 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx=-\frac {a^7 c^6}{11 x^{11}}+\frac {a^6 b c^6}{2 x^{10}}-\frac {a^5 b^2 c^6}{x^9}+\frac {5 a^4 b^3 c^6}{8 x^8}+\frac {5 a^3 b^4 c^6}{7 x^7}-\frac {3 a^2 b^5 c^6}{2 x^6}+\frac {a b^6 c^6}{x^5}-\frac {b^7 c^6}{4 x^4} \]
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Time = 0.04 (sec) , antiderivative size = 114, 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} \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx=-\frac {a^7 c^6}{11 x^{11}}+\frac {a^6 b c^6}{2 x^{10}}-\frac {a^5 b^2 c^6}{x^9}+\frac {5 a^4 b^3 c^6}{8 x^8}+\frac {5 a^3 b^4 c^6}{7 x^7}-\frac {3 a^2 b^5 c^6}{2 x^6}+\frac {a b^6 c^6}{x^5}-\frac {b^7 c^6}{4 x^4} \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^7 c^6}{x^{12}}-\frac {5 a^6 b c^6}{x^{11}}+\frac {9 a^5 b^2 c^6}{x^{10}}-\frac {5 a^4 b^3 c^6}{x^9}-\frac {5 a^3 b^4 c^6}{x^8}+\frac {9 a^2 b^5 c^6}{x^7}-\frac {5 a b^6 c^6}{x^6}+\frac {b^7 c^6}{x^5}\right ) \, dx \\ & = -\frac {a^7 c^6}{11 x^{11}}+\frac {a^6 b c^6}{2 x^{10}}-\frac {a^5 b^2 c^6}{x^9}+\frac {5 a^4 b^3 c^6}{8 x^8}+\frac {5 a^3 b^4 c^6}{7 x^7}-\frac {3 a^2 b^5 c^6}{2 x^6}+\frac {a b^6 c^6}{x^5}-\frac {b^7 c^6}{4 x^4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx=-\frac {a^7 c^6}{11 x^{11}}+\frac {a^6 b c^6}{2 x^{10}}-\frac {a^5 b^2 c^6}{x^9}+\frac {5 a^4 b^3 c^6}{8 x^8}+\frac {5 a^3 b^4 c^6}{7 x^7}-\frac {3 a^2 b^5 c^6}{2 x^6}+\frac {a b^6 c^6}{x^5}-\frac {b^7 c^6}{4 x^4} \]
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Time = 0.37 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(-\frac {c^{6} \left (154 b^{7} x^{7}-616 a \,b^{6} x^{6}+924 a^{2} b^{5} x^{5}-440 a^{3} b^{4} x^{4}-385 a^{4} b^{3} x^{3}+616 a^{5} b^{2} x^{2}-308 a^{6} b x +56 a^{7}\right )}{616 x^{11}}\) | \(83\) |
default | \(c^{6} \left (-\frac {3 a^{2} b^{5}}{2 x^{6}}+\frac {5 a^{3} b^{4}}{7 x^{7}}+\frac {5 a^{4} b^{3}}{8 x^{8}}+\frac {a^{6} b}{2 x^{10}}-\frac {b^{7}}{4 x^{4}}+\frac {a \,b^{6}}{x^{5}}-\frac {a^{5} b^{2}}{x^{9}}-\frac {a^{7}}{11 x^{11}}\right )\) | \(83\) |
norman | \(\frac {a \,b^{6} c^{6} x^{6}-\frac {1}{11} a^{7} c^{6}-\frac {1}{4} b^{7} c^{6} x^{7}-\frac {3}{2} a^{2} b^{5} c^{6} x^{5}+\frac {5}{7} a^{3} b^{4} c^{6} x^{4}+\frac {5}{8} a^{4} b^{3} c^{6} x^{3}-a^{5} b^{2} c^{6} x^{2}+\frac {1}{2} a^{6} b \,c^{6} x}{x^{11}}\) | \(102\) |
risch | \(\frac {a \,b^{6} c^{6} x^{6}-\frac {1}{11} a^{7} c^{6}-\frac {1}{4} b^{7} c^{6} x^{7}-\frac {3}{2} a^{2} b^{5} c^{6} x^{5}+\frac {5}{7} a^{3} b^{4} c^{6} x^{4}+\frac {5}{8} a^{4} b^{3} c^{6} x^{3}-a^{5} b^{2} c^{6} x^{2}+\frac {1}{2} a^{6} b \,c^{6} x}{x^{11}}\) | \(102\) |
parallelrisch | \(\frac {-154 b^{7} c^{6} x^{7}+616 a \,b^{6} c^{6} x^{6}-924 a^{2} b^{5} c^{6} x^{5}+440 a^{3} b^{4} c^{6} x^{4}+385 a^{4} b^{3} c^{6} x^{3}-616 a^{5} b^{2} c^{6} x^{2}+308 a^{6} b \,c^{6} x -56 a^{7} c^{6}}{616 x^{11}}\) | \(104\) |
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Time = 0.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx=-\frac {154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \]
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Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx=\frac {- 56 a^{7} c^{6} + 308 a^{6} b c^{6} x - 616 a^{5} b^{2} c^{6} x^{2} + 385 a^{4} b^{3} c^{6} x^{3} + 440 a^{3} b^{4} c^{6} x^{4} - 924 a^{2} b^{5} c^{6} x^{5} + 616 a b^{6} c^{6} x^{6} - 154 b^{7} c^{6} x^{7}}{616 x^{11}} \]
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Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx=-\frac {154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \]
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Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx=-\frac {154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \]
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Time = 0.06 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx=-\frac {\frac {a^7\,c^6}{11}-\frac {a^6\,b\,c^6\,x}{2}+a^5\,b^2\,c^6\,x^2-\frac {5\,a^4\,b^3\,c^6\,x^3}{8}-\frac {5\,a^3\,b^4\,c^6\,x^4}{7}+\frac {3\,a^2\,b^5\,c^6\,x^5}{2}-a\,b^6\,c^6\,x^6+\frac {b^7\,c^6\,x^7}{4}}{x^{11}} \]
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