Integrand size = 20, antiderivative size = 41 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx=-\frac {c^5 (a-b x)^6}{7 x^7}-\frac {4 b c^5 (a-b x)^6}{21 a x^6} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx=-\frac {c^5 (a-b x)^6}{7 x^7}-\frac {4 b c^5 (a-b x)^6}{21 a x^6} \]
[In]
[Out]
Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {c^5 (a-b x)^6}{7 x^7}+\frac {1}{7} (8 b) \int \frac {(a c-b c x)^5}{x^7} \, dx \\ & = -\frac {c^5 (a-b x)^6}{7 x^7}-\frac {4 b c^5 (a-b x)^6}{21 a x^6} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx=c^5 \left (-\frac {a^6}{7 x^7}+\frac {2 a^5 b}{3 x^6}-\frac {a^4 b^2}{x^5}+\frac {5 a^2 b^4}{3 x^3}-\frac {2 a b^5}{x^2}+\frac {b^6}{x}\right ) \]
[In]
[Out]
Time = 0.39 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.49
method | result | size |
gosper | \(-\frac {c^{5} \left (-21 b^{6} x^{6}+42 a \,x^{5} b^{5}-35 a^{2} x^{4} b^{4}+21 a^{4} x^{2} b^{2}-14 a^{5} x b +3 a^{6}\right )}{21 x^{7}}\) | \(61\) |
default | \(c^{5} \left (\frac {2 a^{5} b}{3 x^{6}}-\frac {a^{6}}{7 x^{7}}+\frac {5 a^{2} b^{4}}{3 x^{3}}+\frac {b^{6}}{x}-\frac {2 a \,b^{5}}{x^{2}}-\frac {a^{4} b^{2}}{x^{5}}\right )\) | \(61\) |
norman | \(\frac {b^{6} c^{5} x^{6}-\frac {1}{7} a^{6} c^{5}-2 a \,b^{5} c^{5} x^{5}+\frac {5}{3} a^{2} b^{4} c^{5} x^{4}-a^{4} b^{2} c^{5} x^{2}+\frac {2}{3} a^{5} b \,c^{5} x}{x^{7}}\) | \(74\) |
risch | \(\frac {b^{6} c^{5} x^{6}-\frac {1}{7} a^{6} c^{5}-2 a \,b^{5} c^{5} x^{5}+\frac {5}{3} a^{2} b^{4} c^{5} x^{4}-a^{4} b^{2} c^{5} x^{2}+\frac {2}{3} a^{5} b \,c^{5} x}{x^{7}}\) | \(74\) |
parallelrisch | \(\frac {21 b^{6} c^{5} x^{6}-42 a \,b^{5} c^{5} x^{5}+35 a^{2} b^{4} c^{5} x^{4}-21 a^{4} b^{2} c^{5} x^{2}+14 a^{5} b \,c^{5} x -3 a^{6} c^{5}}{21 x^{7}}\) | \(76\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx=\frac {21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (36) = 72\).
Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx=- \frac {3 a^{6} c^{5} - 14 a^{5} b c^{5} x + 21 a^{4} b^{2} c^{5} x^{2} - 35 a^{2} b^{4} c^{5} x^{4} + 42 a b^{5} c^{5} x^{5} - 21 b^{6} c^{5} x^{6}}{21 x^{7}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx=\frac {21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx=\frac {21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx=-\frac {\frac {a^6\,c^5}{7}-\frac {2\,a^5\,b\,c^5\,x}{3}+a^4\,b^2\,c^5\,x^2-\frac {5\,a^2\,b^4\,c^5\,x^4}{3}+2\,a\,b^5\,c^5\,x^5-b^6\,c^5\,x^6}{x^7} \]
[In]
[Out]