Integrand size = 15, antiderivative size = 58 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=\frac {(a+b x)^6}{7 (b c-a d) (c+d x)^7}+\frac {b (a+b x)^6}{42 (b c-a d)^2 (c+d x)^6} \]
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Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=\frac {b (a+b x)^6}{42 (c+d x)^6 (b c-a d)^2}+\frac {(a+b x)^6}{7 (c+d x)^7 (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^6}{7 (b c-a d) (c+d x)^7}+\frac {b \int \frac {(a+b x)^5}{(c+d x)^7} \, dx}{7 (b c-a d)} \\ & = \frac {(a+b x)^6}{7 (b c-a d) (c+d x)^7}+\frac {b (a+b x)^6}{42 (b c-a d)^2 (c+d x)^6} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(58)=116\).
Time = 0.04 (sec) , antiderivative size = 205, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=-\frac {6 a^5 d^5+5 a^4 b d^4 (c+7 d x)+4 a^3 b^2 d^3 \left (c^2+7 c d x+21 d^2 x^2\right )+3 a^2 b^3 d^2 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+2 a b^4 d \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+b^5 \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )}{42 d^6 (c+d x)^7} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(54)=108\).
Time = 0.23 (sec) , antiderivative size = 246, normalized size of antiderivative = 4.24
method | result | size |
risch | \(\frac {-\frac {b^{5} x^{5}}{2 d}-\frac {5 b^{4} \left (2 a d +b c \right ) x^{4}}{6 d^{2}}-\frac {5 b^{3} \left (3 a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) x^{3}}{6 d^{3}}-\frac {b^{2} \left (4 a^{3} d^{3}+3 a^{2} b c \,d^{2}+2 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{2}}{2 d^{4}}-\frac {b \left (5 a^{4} d^{4}+4 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{6 d^{5}}-\frac {6 a^{5} d^{5}+5 a^{4} b c \,d^{4}+4 a^{3} b^{2} c^{2} d^{3}+3 a^{2} b^{3} c^{3} d^{2}+2 a \,b^{4} c^{4} d +b^{5} c^{5}}{42 d^{6}}}{\left (d x +c \right )^{7}}\) | \(246\) |
default | \(-\frac {5 b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d^{6} \left (d x +c \right )^{4}}-\frac {b^{5}}{2 d^{6} \left (d x +c \right )^{2}}-\frac {5 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{6 d^{6} \left (d x +c \right )^{6}}-\frac {a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}{7 d^{6} \left (d x +c \right )^{7}}-\frac {5 b^{4} \left (a d -b c \right )}{3 d^{6} \left (d x +c \right )^{3}}-\frac {2 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{6} \left (d x +c \right )^{5}}\) | \(265\) |
norman | \(\frac {-\frac {b^{5} x^{5}}{2 d}-\frac {5 \left (2 a \,b^{4} d^{2}+b^{5} c d \right ) x^{4}}{6 d^{3}}-\frac {5 \left (3 a^{2} b^{3} d^{3}+2 a \,b^{4} c \,d^{2}+b^{5} c^{2} d \right ) x^{3}}{6 d^{4}}-\frac {\left (4 a^{3} b^{2} d^{4}+3 a^{2} c \,b^{3} d^{3}+2 a \,b^{4} c^{2} d^{2}+b^{5} c^{3} d \right ) x^{2}}{2 d^{5}}-\frac {\left (5 a^{4} b \,d^{5}+4 a^{3} b^{2} c \,d^{4}+3 a^{2} b^{3} c^{2} d^{3}+2 a \,b^{4} c^{3} d^{2}+b^{5} c^{4} d \right ) x}{6 d^{6}}-\frac {6 a^{5} d^{6}+5 a^{4} b c \,d^{5}+4 a^{3} b^{2} c^{2} d^{4}+3 a^{2} b^{3} c^{3} d^{3}+2 a \,b^{4} c^{4} d^{2}+b^{5} c^{5} d}{42 d^{7}}}{\left (d x +c \right )^{7}}\) | \(269\) |
gosper | \(-\frac {21 x^{5} b^{5} d^{5}+70 x^{4} a \,b^{4} d^{5}+35 x^{4} b^{5} c \,d^{4}+105 x^{3} a^{2} b^{3} d^{5}+70 x^{3} a \,b^{4} c \,d^{4}+35 x^{3} b^{5} c^{2} d^{3}+84 x^{2} a^{3} b^{2} d^{5}+63 x^{2} a^{2} b^{3} c \,d^{4}+42 x^{2} a \,b^{4} c^{2} d^{3}+21 x^{2} b^{5} c^{3} d^{2}+35 x \,a^{4} b \,d^{5}+28 x \,a^{3} b^{2} c \,d^{4}+21 x \,a^{2} b^{3} c^{2} d^{3}+14 x a \,b^{4} c^{3} d^{2}+7 x \,b^{5} c^{4} d +6 a^{5} d^{5}+5 a^{4} b c \,d^{4}+4 a^{3} b^{2} c^{2} d^{3}+3 a^{2} b^{3} c^{3} d^{2}+2 a \,b^{4} c^{4} d +b^{5} c^{5}}{42 d^{6} \left (d x +c \right )^{7}}\) | \(272\) |
parallelrisch | \(\frac {-21 b^{5} x^{5} d^{6}-70 a \,b^{4} d^{6} x^{4}-35 b^{5} c \,d^{5} x^{4}-105 a^{2} b^{3} d^{6} x^{3}-70 a \,b^{4} c \,d^{5} x^{3}-35 b^{5} c^{2} d^{4} x^{3}-84 a^{3} b^{2} d^{6} x^{2}-63 a^{2} b^{3} c \,d^{5} x^{2}-42 a \,b^{4} c^{2} d^{4} x^{2}-21 b^{5} c^{3} d^{3} x^{2}-35 a^{4} b \,d^{6} x -28 a^{3} b^{2} c \,d^{5} x -21 a^{2} b^{3} c^{2} d^{4} x -14 a \,b^{4} c^{3} d^{3} x -7 b^{5} c^{4} d^{2} x -6 a^{5} d^{6}-5 a^{4} b c \,d^{5}-4 a^{3} b^{2} c^{2} d^{4}-3 a^{2} b^{3} c^{3} d^{3}-2 a \,b^{4} c^{4} d^{2}-b^{5} c^{5} d}{42 d^{7} \left (d x +c \right )^{7}}\) | \(278\) |
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Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (54) = 108\).
Time = 0.22 (sec) , antiderivative size = 326, normalized size of antiderivative = 5.62 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=-\frac {21 \, b^{5} d^{5} x^{5} + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5} + 35 \, {\left (b^{5} c d^{4} + 2 \, a b^{4} d^{5}\right )} x^{4} + 35 \, {\left (b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 21 \, {\left (b^{5} c^{3} d^{2} + 2 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + 4 \, a^{3} b^{2} d^{5}\right )} x^{2} + 7 \, {\left (b^{5} c^{4} d + 2 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x}{42 \, {\left (d^{13} x^{7} + 7 \, c d^{12} x^{6} + 21 \, c^{2} d^{11} x^{5} + 35 \, c^{3} d^{10} x^{4} + 35 \, c^{4} d^{9} x^{3} + 21 \, c^{5} d^{8} x^{2} + 7 \, c^{6} d^{7} x + c^{7} d^{6}\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (54) = 108\).
Time = 0.23 (sec) , antiderivative size = 326, normalized size of antiderivative = 5.62 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=-\frac {21 \, b^{5} d^{5} x^{5} + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5} + 35 \, {\left (b^{5} c d^{4} + 2 \, a b^{4} d^{5}\right )} x^{4} + 35 \, {\left (b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 21 \, {\left (b^{5} c^{3} d^{2} + 2 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + 4 \, a^{3} b^{2} d^{5}\right )} x^{2} + 7 \, {\left (b^{5} c^{4} d + 2 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x}{42 \, {\left (d^{13} x^{7} + 7 \, c d^{12} x^{6} + 21 \, c^{2} d^{11} x^{5} + 35 \, c^{3} d^{10} x^{4} + 35 \, c^{4} d^{9} x^{3} + 21 \, c^{5} d^{8} x^{2} + 7 \, c^{6} d^{7} x + c^{7} d^{6}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (54) = 108\).
Time = 0.29 (sec) , antiderivative size = 271, normalized size of antiderivative = 4.67 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=-\frac {21 \, b^{5} d^{5} x^{5} + 35 \, b^{5} c d^{4} x^{4} + 70 \, a b^{4} d^{5} x^{4} + 35 \, b^{5} c^{2} d^{3} x^{3} + 70 \, a b^{4} c d^{4} x^{3} + 105 \, a^{2} b^{3} d^{5} x^{3} + 21 \, b^{5} c^{3} d^{2} x^{2} + 42 \, a b^{4} c^{2} d^{3} x^{2} + 63 \, a^{2} b^{3} c d^{4} x^{2} + 84 \, a^{3} b^{2} d^{5} x^{2} + 7 \, b^{5} c^{4} d x + 14 \, a b^{4} c^{3} d^{2} x + 21 \, a^{2} b^{3} c^{2} d^{3} x + 28 \, a^{3} b^{2} c d^{4} x + 35 \, a^{4} b d^{5} x + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5}}{42 \, {\left (d x + c\right )}^{7} d^{6}} \]
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Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=\frac {{\left (a+b\,x\right )}^6\,\left (7\,b\,c-6\,a\,d+b\,d\,x\right )}{42\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^7} \]
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