Integrand size = 27, antiderivative size = 101 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx=-\frac {a^6 B}{6 x^6}-\frac {6 a^5 b B}{5 x^5}-\frac {15 a^4 b^2 B}{4 x^4}-\frac {20 a^3 b^3 B}{3 x^3}-\frac {15 a^2 b^4 B}{2 x^2}-\frac {6 a b^5 B}{x}-\frac {A (a+b x)^7}{7 a x^7}+b^6 B \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 79, 45} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx=-\frac {a^6 B}{6 x^6}-\frac {6 a^5 b B}{5 x^5}-\frac {15 a^4 b^2 B}{4 x^4}-\frac {20 a^3 b^3 B}{3 x^3}-\frac {15 a^2 b^4 B}{2 x^2}-\frac {A (a+b x)^7}{7 a x^7}-\frac {6 a b^5 B}{x}+b^6 B \log (x) \]
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Rule 27
Rule 45
Rule 79
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^6 (A+B x)}{x^8} \, dx \\ & = -\frac {A (a+b x)^7}{7 a x^7}+B \int \frac {(a+b x)^6}{x^7} \, dx \\ & = -\frac {A (a+b x)^7}{7 a x^7}+B \int \left (\frac {a^6}{x^7}+\frac {6 a^5 b}{x^6}+\frac {15 a^4 b^2}{x^5}+\frac {20 a^3 b^3}{x^4}+\frac {15 a^2 b^4}{x^3}+\frac {6 a b^5}{x^2}+\frac {b^6}{x}\right ) \, dx \\ & = -\frac {a^6 B}{6 x^6}-\frac {6 a^5 b B}{5 x^5}-\frac {15 a^4 b^2 B}{4 x^4}-\frac {20 a^3 b^3 B}{3 x^3}-\frac {15 a^2 b^4 B}{2 x^2}-\frac {6 a b^5 B}{x}-\frac {A (a+b x)^7}{7 a x^7}+b^6 B \log (x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx=-\frac {A b^6}{x}-\frac {3 a b^5 (A+2 B x)}{x^2}-\frac {5 a^2 b^4 (2 A+3 B x)}{2 x^3}-\frac {5 a^3 b^3 (3 A+4 B x)}{3 x^4}-\frac {3 a^4 b^2 (4 A+5 B x)}{4 x^5}-\frac {a^5 b (5 A+6 B x)}{5 x^6}-\frac {a^6 (6 A+7 B x)}{42 x^7}+b^6 B \log (x) \]
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Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.25
method | result | size |
default | \(-\frac {a^{5} \left (6 A b +B a \right )}{6 x^{6}}-\frac {5 a^{3} b^{2} \left (4 A b +3 B a \right )}{4 x^{4}}-\frac {3 a^{4} b \left (5 A b +2 B a \right )}{5 x^{5}}+b^{6} B \ln \left (x \right )-\frac {3 a \,b^{4} \left (2 A b +5 B a \right )}{2 x^{2}}-\frac {A \,a^{6}}{7 x^{7}}-\frac {b^{5} \left (A b +6 B a \right )}{x}-\frac {5 a^{2} b^{3} \left (3 A b +4 B a \right )}{3 x^{3}}\) | \(126\) |
norman | \(\frac {\left (-3 A a \,b^{5}-\frac {15}{2} B \,b^{4} a^{2}\right ) x^{5}+\left (-5 A \,a^{3} b^{3}-\frac {15}{4} B \,a^{4} b^{2}\right ) x^{3}+\left (-3 A \,a^{4} b^{2}-\frac {6}{5} B \,a^{5} b \right ) x^{2}+\left (-A \,a^{5} b -\frac {1}{6} B \,a^{6}\right ) x +\left (-5 A \,b^{4} a^{2}-\frac {20}{3} B \,a^{3} b^{3}\right ) x^{4}+\left (-A \,b^{6}-6 B a \,b^{5}\right ) x^{6}-\frac {A \,a^{6}}{7}}{x^{7}}+b^{6} B \ln \left (x \right )\) | \(142\) |
risch | \(\frac {\left (-3 A a \,b^{5}-\frac {15}{2} B \,b^{4} a^{2}\right ) x^{5}+\left (-5 A \,a^{3} b^{3}-\frac {15}{4} B \,a^{4} b^{2}\right ) x^{3}+\left (-3 A \,a^{4} b^{2}-\frac {6}{5} B \,a^{5} b \right ) x^{2}+\left (-A \,a^{5} b -\frac {1}{6} B \,a^{6}\right ) x +\left (-5 A \,b^{4} a^{2}-\frac {20}{3} B \,a^{3} b^{3}\right ) x^{4}+\left (-A \,b^{6}-6 B a \,b^{5}\right ) x^{6}-\frac {A \,a^{6}}{7}}{x^{7}}+b^{6} B \ln \left (x \right )\) | \(142\) |
parallelrisch | \(-\frac {-420 B \,b^{6} \ln \left (x \right ) x^{7}+420 A \,b^{6} x^{6}+2520 x^{6} B a \,b^{5}+1260 a A \,b^{5} x^{5}+3150 x^{5} B \,b^{4} a^{2}+2100 a^{2} A \,b^{4} x^{4}+2800 x^{4} B \,a^{3} b^{3}+2100 a^{3} A \,b^{3} x^{3}+1575 x^{3} B \,a^{4} b^{2}+1260 a^{4} A \,b^{2} x^{2}+504 x^{2} B \,a^{5} b +420 a^{5} A b x +70 x B \,a^{6}+60 A \,a^{6}}{420 x^{7}}\) | \(150\) |
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Time = 0.60 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx=\frac {420 \, B b^{6} x^{7} \log \left (x\right ) - 60 \, A a^{6} - 420 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} - 630 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 700 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 70 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{420 \, x^{7}} \]
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Time = 2.57 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx=B b^{6} \log {\left (x \right )} + \frac {- 60 A a^{6} + x^{6} \left (- 420 A b^{6} - 2520 B a b^{5}\right ) + x^{5} \left (- 1260 A a b^{5} - 3150 B a^{2} b^{4}\right ) + x^{4} \left (- 2100 A a^{2} b^{4} - 2800 B a^{3} b^{3}\right ) + x^{3} \left (- 2100 A a^{3} b^{3} - 1575 B a^{4} b^{2}\right ) + x^{2} \left (- 1260 A a^{4} b^{2} - 504 B a^{5} b\right ) + x \left (- 420 A a^{5} b - 70 B a^{6}\right )}{420 x^{7}} \]
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Time = 0.19 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx=B b^{6} \log \left (x\right ) - \frac {60 \, A a^{6} + 420 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 630 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 700 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 70 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{420 \, x^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx=B b^{6} \log \left ({\left | x \right |}\right ) - \frac {60 \, A a^{6} + 420 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 630 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 700 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 70 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{420 \, x^{7}} \]
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Time = 10.00 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.39 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx=B\,b^6\,\ln \left (x\right )-\frac {x\,\left (\frac {B\,a^6}{6}+A\,b\,a^5\right )+\frac {A\,a^6}{7}+x^2\,\left (\frac {6\,B\,a^5\,b}{5}+3\,A\,a^4\,b^2\right )+x^5\,\left (\frac {15\,B\,a^2\,b^4}{2}+3\,A\,a\,b^5\right )+x^6\,\left (A\,b^6+6\,B\,a\,b^5\right )+x^3\,\left (\frac {15\,B\,a^4\,b^2}{4}+5\,A\,a^3\,b^3\right )+x^4\,\left (\frac {20\,B\,a^3\,b^3}{3}+5\,A\,a^2\,b^4\right )}{x^7} \]
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