Integrand size = 15, antiderivative size = 92 \[ \int \frac {(a+b x)^3}{(c+d x)^8} \, dx=\frac {(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac {b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac {3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac {b^3}{4 d^4 (c+d x)^4} \]
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Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(a+b x)^3}{(c+d x)^8} \, dx=\frac {3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac {b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac {(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac {b^3}{4 d^4 (c+d x)^4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^3}{d^3 (c+d x)^8}+\frac {3 b (b c-a d)^2}{d^3 (c+d x)^7}-\frac {3 b^2 (b c-a d)}{d^3 (c+d x)^6}+\frac {b^3}{d^3 (c+d x)^5}\right ) \, dx \\ & = \frac {(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac {b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac {3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac {b^3}{4 d^4 (c+d x)^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^3}{(c+d x)^8} \, dx=-\frac {20 a^3 d^3+10 a^2 b d^2 (c+7 d x)+4 a b^2 d \left (c^2+7 c d x+21 d^2 x^2\right )+b^3 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )}{140 d^4 (c+d x)^7} \]
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Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\frac {-\frac {b^{3} x^{3}}{4 d}-\frac {3 b^{2} \left (4 a d +b c \right ) x^{2}}{20 d^{2}}-\frac {b \left (10 a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x}{20 d^{3}}-\frac {20 a^{3} d^{3}+10 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d +b^{3} c^{3}}{140 d^{4}}}{\left (d x +c \right )^{7}}\) | \(110\) |
gosper | \(-\frac {35 d^{3} x^{3} b^{3}+84 x^{2} a \,b^{2} d^{3}+21 x^{2} b^{3} c \,d^{2}+70 x \,a^{2} b \,d^{3}+28 x a \,b^{2} c \,d^{2}+7 x \,b^{3} c^{2} d +20 a^{3} d^{3}+10 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d +b^{3} c^{3}}{140 d^{4} \left (d x +c \right )^{7}}\) | \(115\) |
default | \(-\frac {b^{3}}{4 d^{4} \left (d x +c \right )^{4}}-\frac {b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d^{4} \left (d x +c \right )^{6}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{7 d^{4} \left (d x +c \right )^{7}}-\frac {3 b^{2} \left (a d -b c \right )}{5 d^{4} \left (d x +c \right )^{5}}\) | \(122\) |
parallelrisch | \(\frac {-35 b^{3} x^{3} d^{6}-84 a \,b^{2} d^{6} x^{2}-21 b^{3} c \,d^{5} x^{2}-70 a^{2} b \,d^{6} x -28 a \,b^{2} c \,d^{5} x -7 b^{3} c^{2} d^{4} x -20 a^{3} d^{6}-10 a^{2} b c \,d^{5}-4 a \,b^{2} c^{2} d^{4}-b^{3} c^{3} d^{3}}{140 d^{7} \left (d x +c \right )^{7}}\) | \(123\) |
norman | \(\frac {-\frac {b^{3} x^{3}}{4 d}-\frac {3 \left (4 a \,b^{2} d^{4}+b^{3} c \,d^{3}\right ) x^{2}}{20 d^{5}}-\frac {\left (10 a^{2} b \,d^{5}+4 a \,b^{2} c \,d^{4}+b^{3} c^{2} d^{3}\right ) x}{20 d^{6}}-\frac {20 a^{3} d^{6}+10 a^{2} b c \,d^{5}+4 a \,b^{2} c^{2} d^{4}+b^{3} c^{3} d^{3}}{140 d^{7}}}{\left (d x +c \right )^{7}}\) | \(129\) |
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Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (84) = 168\).
Time = 0.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^3}{(c+d x)^8} \, dx=-\frac {35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \, {\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \, {\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \, {\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (80) = 160\).
Time = 1.94 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.13 \[ \int \frac {(a+b x)^3}{(c+d x)^8} \, dx=\frac {- 20 a^{3} d^{3} - 10 a^{2} b c d^{2} - 4 a b^{2} c^{2} d - b^{3} c^{3} - 35 b^{3} d^{3} x^{3} + x^{2} \left (- 84 a b^{2} d^{3} - 21 b^{3} c d^{2}\right ) + x \left (- 70 a^{2} b d^{3} - 28 a b^{2} c d^{2} - 7 b^{3} c^{2} d\right )}{140 c^{7} d^{4} + 980 c^{6} d^{5} x + 2940 c^{5} d^{6} x^{2} + 4900 c^{4} d^{7} x^{3} + 4900 c^{3} d^{8} x^{4} + 2940 c^{2} d^{9} x^{5} + 980 c d^{10} x^{6} + 140 d^{11} x^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (84) = 168\).
Time = 0.23 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^3}{(c+d x)^8} \, dx=-\frac {35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \, {\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \, {\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \, {\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x)^3}{(c+d x)^8} \, dx=-\frac {35 \, b^{3} d^{3} x^{3} + 21 \, b^{3} c d^{2} x^{2} + 84 \, a b^{2} d^{3} x^{2} + 7 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 70 \, a^{2} b d^{3} x + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3}}{140 \, {\left (d x + c\right )}^{7} d^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b x)^3}{(c+d x)^8} \, dx=-\frac {\frac {20\,a^3\,d^3+10\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d+b^3\,c^3}{140\,d^4}+\frac {b^3\,x^3}{4\,d}+\frac {b\,x\,\left (10\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{20\,d^3}+\frac {3\,b^2\,x^2\,\left (4\,a\,d+b\,c\right )}{20\,d^2}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \]
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