Integrand size = 7, antiderivative size = 14 \[ \int \frac {1}{(c+d x)^8} \, dx=-\frac {1}{7 d (c+d x)^7} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int \frac {1}{(c+d x)^8} \, dx=-\frac {1}{7 d (c+d x)^7} \]
[In]
[Out]
Rule 32
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{7 d (c+d x)^7} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^8} \, dx=-\frac {1}{7 d (c+d x)^7} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(-\frac {1}{7 d \left (d x +c \right )^{7}}\) | \(13\) |
default | \(-\frac {1}{7 d \left (d x +c \right )^{7}}\) | \(13\) |
norman | \(-\frac {1}{7 d \left (d x +c \right )^{7}}\) | \(13\) |
risch | \(-\frac {1}{7 d \left (d x +c \right )^{7}}\) | \(13\) |
parallelrisch | \(-\frac {1}{7 d \left (d x +c \right )^{7}}\) | \(13\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (12) = 24\).
Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 5.64 \[ \int \frac {1}{(c+d x)^8} \, dx=-\frac {1}{7 \, {\left (d^{8} x^{7} + 7 \, c d^{7} x^{6} + 21 \, c^{2} d^{6} x^{5} + 35 \, c^{3} d^{5} x^{4} + 35 \, c^{4} d^{4} x^{3} + 21 \, c^{5} d^{3} x^{2} + 7 \, c^{6} d^{2} x + c^{7} d\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (12) = 24\).
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 6.07 \[ \int \frac {1}{(c+d x)^8} \, dx=- \frac {1}{7 c^{7} d + 49 c^{6} d^{2} x + 147 c^{5} d^{3} x^{2} + 245 c^{4} d^{4} x^{3} + 245 c^{3} d^{5} x^{4} + 147 c^{2} d^{6} x^{5} + 49 c d^{7} x^{6} + 7 d^{8} x^{7}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(c+d x)^8} \, dx=-\frac {1}{7 \, {\left (d x + c\right )}^{7} d} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(c+d x)^8} \, dx=-\frac {1}{7 \, {\left (d x + c\right )}^{7} d} \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 5.79 \[ \int \frac {1}{(c+d x)^8} \, dx=-\frac {1}{7\,c^7\,d+49\,c^6\,d^2\,x+147\,c^5\,d^3\,x^2+245\,c^4\,d^4\,x^3+245\,c^3\,d^5\,x^4+147\,c^2\,d^6\,x^5+49\,c\,d^7\,x^6+7\,d^8\,x^7} \]
[In]
[Out]