Integrand size = 15, antiderivative size = 28 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=\frac {(a+b x)^7}{7 (b c-a d) (c+d x)^7} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=\frac {(a+b x)^7}{7 (c+d x)^7 (b c-a d)} \]
[In]
[Out]
Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^7}{7 (b c-a d) (c+d x)^7} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(271\) vs. \(2(28)=56\).
Time = 0.06 (sec) , antiderivative size = 271, normalized size of antiderivative = 9.68 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {a^6 d^6+a^5 b d^5 (c+7 d x)+a^4 b^2 d^4 \left (c^2+7 c d x+21 d^2 x^2\right )+a^3 b^3 d^3 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+a^2 b^4 d^2 \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+a b^5 d \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )+b^6 \left (c^6+7 c^5 d x+21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+21 c d^5 x^5+7 d^6 x^6\right )}{7 d^7 (c+d x)^7} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(26)=52\).
Time = 0.23 (sec) , antiderivative size = 314, normalized size of antiderivative = 11.21
method | result | size |
risch | \(\frac {-\frac {b^{6} x^{6}}{d}-\frac {3 b^{5} \left (a d +b c \right ) x^{5}}{d^{2}}-\frac {5 b^{4} \left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{4}}{d^{3}}-\frac {5 b^{3} \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{3}}{d^{4}}-\frac {3 b^{2} \left (a^{4} d^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x^{2}}{d^{5}}-\frac {b \left (a^{5} d^{5}+a^{4} b c \,d^{4}+a^{3} b^{2} c^{2} d^{3}+a^{2} b^{3} c^{3} d^{2}+a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x}{d^{6}}-\frac {a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +b^{6} c^{6}}{7 d^{7}}}{\left (d x +c \right )^{7}}\) | \(314\) |
norman | \(\frac {-\frac {b^{6} x^{6}}{d}-\frac {3 \left (a \,b^{5} d +b^{6} c \right ) x^{5}}{d^{2}}-\frac {5 \left (a^{2} b^{4} d^{2}+a \,b^{5} c d +b^{6} c^{2}\right ) x^{4}}{d^{3}}-\frac {5 \left (a^{3} b^{3} d^{3}+a^{2} b^{4} c \,d^{2}+a \,c^{2} b^{5} d +b^{6} c^{3}\right ) x^{3}}{d^{4}}-\frac {3 \left (a^{4} b^{2} d^{4}+a^{3} b^{3} d^{3} c +a^{2} b^{4} d^{2} c^{2}+a \,b^{5} d \,c^{3}+b^{6} c^{4}\right ) x^{2}}{d^{5}}-\frac {\left (a^{5} b \,d^{5}+a^{4} b^{2} c \,d^{4}+a^{3} c^{2} b^{3} d^{3}+a^{2} c^{3} b^{4} d^{2}+a \,c^{4} b^{5} d +b^{6} c^{5}\right ) x}{d^{6}}-\frac {a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +b^{6} c^{6}}{7 d^{7}}}{\left (d x +c \right )^{7}}\) | \(324\) |
default | \(-\frac {3 b^{5} \left (a d -b c \right )}{d^{7} \left (d x +c \right )^{2}}-\frac {b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{d^{7} \left (d x +c \right )^{6}}-\frac {a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}}{7 d^{7} \left (d x +c \right )^{7}}-\frac {5 b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{7} \left (d x +c \right )^{4}}-\frac {3 b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{d^{7} \left (d x +c \right )^{5}}-\frac {5 b^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{7} \left (d x +c \right )^{3}}-\frac {b^{6}}{d^{7} \left (d x +c \right )}\) | \(357\) |
gosper | \(-\frac {7 x^{6} b^{6} d^{6}+21 x^{5} a \,b^{5} d^{6}+21 x^{5} b^{6} c \,d^{5}+35 x^{4} a^{2} b^{4} d^{6}+35 x^{4} a \,b^{5} c \,d^{5}+35 x^{4} b^{6} c^{2} d^{4}+35 x^{3} a^{3} b^{3} d^{6}+35 x^{3} a^{2} b^{4} c \,d^{5}+35 x^{3} a \,b^{5} c^{2} d^{4}+35 x^{3} b^{6} c^{3} d^{3}+21 a^{4} b^{2} d^{6} x^{2}+21 a^{3} b^{3} c \,d^{5} x^{2}+21 a^{2} b^{4} c^{2} d^{4} x^{2}+21 a \,b^{5} c^{3} d^{3} x^{2}+21 b^{6} c^{4} d^{2} x^{2}+7 a^{5} b \,d^{6} x +7 x \,a^{4} b^{2} c \,d^{5}+7 a^{3} b^{3} c^{2} d^{4} x +7 a^{2} b^{4} c^{3} d^{3} x +7 a \,b^{5} c^{4} d^{2} x +7 b^{6} c^{5} d x +a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +b^{6} c^{6}}{7 \left (d x +c \right )^{7} d^{7}}\) | \(370\) |
parallelrisch | \(\frac {-7 x^{6} b^{6} d^{6}-21 x^{5} a \,b^{5} d^{6}-21 x^{5} b^{6} c \,d^{5}-35 x^{4} a^{2} b^{4} d^{6}-35 x^{4} a \,b^{5} c \,d^{5}-35 x^{4} b^{6} c^{2} d^{4}-35 x^{3} a^{3} b^{3} d^{6}-35 x^{3} a^{2} b^{4} c \,d^{5}-35 x^{3} a \,b^{5} c^{2} d^{4}-35 x^{3} b^{6} c^{3} d^{3}-21 a^{4} b^{2} d^{6} x^{2}-21 a^{3} b^{3} c \,d^{5} x^{2}-21 a^{2} b^{4} c^{2} d^{4} x^{2}-21 a \,b^{5} c^{3} d^{3} x^{2}-21 b^{6} c^{4} d^{2} x^{2}-7 a^{5} b \,d^{6} x -7 x \,a^{4} b^{2} c \,d^{5}-7 a^{3} b^{3} c^{2} d^{4} x -7 a^{2} b^{4} c^{3} d^{3} x -7 a \,b^{5} c^{4} d^{2} x -7 b^{6} c^{5} d x -a^{6} d^{6}-a^{5} b c \,d^{5}-a^{4} b^{2} c^{2} d^{4}-a^{3} b^{3} c^{3} d^{3}-a^{2} b^{4} c^{4} d^{2}-a \,b^{5} c^{5} d -b^{6} c^{6}}{7 d^{7} \left (d x +c \right )^{7}}\) | \(377\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 398, normalized size of antiderivative = 14.21 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \, {\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \, {\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \, {\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \, {\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \, {\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 398, normalized size of antiderivative = 14.21 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \, {\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \, {\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \, {\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \, {\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \, {\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (26) = 52\).
Time = 0.31 (sec) , antiderivative size = 369, normalized size of antiderivative = 13.18 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {7 \, b^{6} d^{6} x^{6} + 21 \, b^{6} c d^{5} x^{5} + 21 \, a b^{5} d^{6} x^{5} + 35 \, b^{6} c^{2} d^{4} x^{4} + 35 \, a b^{5} c d^{5} x^{4} + 35 \, a^{2} b^{4} d^{6} x^{4} + 35 \, b^{6} c^{3} d^{3} x^{3} + 35 \, a b^{5} c^{2} d^{4} x^{3} + 35 \, a^{2} b^{4} c d^{5} x^{3} + 35 \, a^{3} b^{3} d^{6} x^{3} + 21 \, b^{6} c^{4} d^{2} x^{2} + 21 \, a b^{5} c^{3} d^{3} x^{2} + 21 \, a^{2} b^{4} c^{2} d^{4} x^{2} + 21 \, a^{3} b^{3} c d^{5} x^{2} + 21 \, a^{4} b^{2} d^{6} x^{2} + 7 \, b^{6} c^{5} d x + 7 \, a b^{5} c^{4} d^{2} x + 7 \, a^{2} b^{4} c^{3} d^{3} x + 7 \, a^{3} b^{3} c^{2} d^{4} x + 7 \, a^{4} b^{2} c d^{5} x + 7 \, a^{5} b d^{6} x + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6}}{7 \, {\left (d x + c\right )}^{7} d^{7}} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 378, normalized size of antiderivative = 13.50 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {\frac {a^6\,d^6+a^5\,b\,c\,d^5+a^4\,b^2\,c^2\,d^4+a^3\,b^3\,c^3\,d^3+a^2\,b^4\,c^4\,d^2+a\,b^5\,c^5\,d+b^6\,c^6}{7\,d^7}+\frac {b^6\,x^6}{d}+\frac {5\,b^3\,x^3\,\left (a^3\,d^3+a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3\right )}{d^4}+\frac {b\,x\,\left (a^5\,d^5+a^4\,b\,c\,d^4+a^3\,b^2\,c^2\,d^3+a^2\,b^3\,c^3\,d^2+a\,b^4\,c^4\,d+b^5\,c^5\right )}{d^6}+\frac {3\,b^5\,x^5\,\left (a\,d+b\,c\right )}{d^2}+\frac {3\,b^2\,x^2\,\left (a^4\,d^4+a^3\,b\,c\,d^3+a^2\,b^2\,c^2\,d^2+a\,b^3\,c^3\,d+b^4\,c^4\right )}{d^5}+\frac {5\,b^4\,x^4\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{d^3}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \]
[In]
[Out]