Integrand size = 15, antiderivative size = 65 \[ \int \frac {(a+b x)^2}{(c+d x)^8} \, dx=-\frac {(b c-a d)^2}{7 d^3 (c+d x)^7}+\frac {b (b c-a d)}{3 d^3 (c+d x)^6}-\frac {b^2}{5 d^3 (c+d x)^5} \]
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Time = 0.03 (sec) , antiderivative size = 65, 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} \[ \int \frac {(a+b x)^2}{(c+d x)^8} \, dx=\frac {b (b c-a d)}{3 d^3 (c+d x)^6}-\frac {(b c-a d)^2}{7 d^3 (c+d x)^7}-\frac {b^2}{5 d^3 (c+d x)^5} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^2}{d^2 (c+d x)^8}-\frac {2 b (b c-a d)}{d^2 (c+d x)^7}+\frac {b^2}{d^2 (c+d x)^6}\right ) \, dx \\ & = -\frac {(b c-a d)^2}{7 d^3 (c+d x)^7}+\frac {b (b c-a d)}{3 d^3 (c+d x)^6}-\frac {b^2}{5 d^3 (c+d x)^5} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^2}{(c+d x)^8} \, dx=-\frac {15 a^2 d^2+5 a b d (c+7 d x)+b^2 \left (c^2+7 c d x+21 d^2 x^2\right )}{105 d^3 (c+d x)^7} \]
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Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(-\frac {21 d^{2} x^{2} b^{2}+35 x a b \,d^{2}+7 x \,b^{2} c d +15 a^{2} d^{2}+5 a b c d +b^{2} c^{2}}{105 d^{3} \left (d x +c \right )^{7}}\) | \(62\) |
risch | \(\frac {-\frac {b^{2} x^{2}}{5 d}-\frac {b \left (5 a d +b c \right ) x}{15 d^{2}}-\frac {15 a^{2} d^{2}+5 a b c d +b^{2} c^{2}}{105 d^{3}}}{\left (d x +c \right )^{7}}\) | \(63\) |
parallelrisch | \(\frac {-21 b^{2} x^{2} d^{6}-35 a b \,d^{6} x -7 b^{2} c \,d^{5} x -15 a^{2} d^{6}-5 a b c \,d^{5}-b^{2} c^{2} d^{4}}{105 d^{7} \left (d x +c \right )^{7}}\) | \(70\) |
default | \(-\frac {b \left (a d -b c \right )}{3 d^{3} \left (d x +c \right )^{6}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{7 d^{3} \left (d x +c \right )^{7}}-\frac {b^{2}}{5 d^{3} \left (d x +c \right )^{5}}\) | \(71\) |
norman | \(\frac {-\frac {b^{2} x^{2}}{5 d}-\frac {\left (5 a b \,d^{5}+b^{2} c \,d^{4}\right ) x}{15 d^{6}}-\frac {15 a^{2} d^{6}+5 a b c \,d^{5}+b^{2} c^{2} d^{4}}{105 d^{7}}}{\left (d x +c \right )^{7}}\) | \(75\) |
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (59) = 118\).
Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b x)^2}{(c+d x)^8} \, dx=-\frac {21 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + 7 \, {\left (b^{2} c d + 5 \, a b d^{2}\right )} x}{105 \, {\left (d^{10} x^{7} + 7 \, c d^{9} x^{6} + 21 \, c^{2} d^{8} x^{5} + 35 \, c^{3} d^{7} x^{4} + 35 \, c^{4} d^{6} x^{3} + 21 \, c^{5} d^{5} x^{2} + 7 \, c^{6} d^{4} x + c^{7} d^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (54) = 108\).
Time = 0.82 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.14 \[ \int \frac {(a+b x)^2}{(c+d x)^8} \, dx=\frac {- 15 a^{2} d^{2} - 5 a b c d - b^{2} c^{2} - 21 b^{2} d^{2} x^{2} + x \left (- 35 a b d^{2} - 7 b^{2} c d\right )}{105 c^{7} d^{3} + 735 c^{6} d^{4} x + 2205 c^{5} d^{5} x^{2} + 3675 c^{4} d^{6} x^{3} + 3675 c^{3} d^{7} x^{4} + 2205 c^{2} d^{8} x^{5} + 735 c d^{9} x^{6} + 105 d^{10} x^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (59) = 118\).
Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b x)^2}{(c+d x)^8} \, dx=-\frac {21 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + 7 \, {\left (b^{2} c d + 5 \, a b d^{2}\right )} x}{105 \, {\left (d^{10} x^{7} + 7 \, c d^{9} x^{6} + 21 \, c^{2} d^{8} x^{5} + 35 \, c^{3} d^{7} x^{4} + 35 \, c^{4} d^{6} x^{3} + 21 \, c^{5} d^{5} x^{2} + 7 \, c^{6} d^{4} x + c^{7} d^{3}\right )}} \]
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none
Time = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^2}{(c+d x)^8} \, dx=-\frac {21 \, b^{2} d^{2} x^{2} + 7 \, b^{2} c d x + 35 \, a b d^{2} x + b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2}}{105 \, {\left (d x + c\right )}^{7} d^{3}} \]
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Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^2}{(c+d x)^8} \, dx=-\frac {\frac {15\,a^2\,d^2+5\,a\,b\,c\,d+b^2\,c^2}{105\,d^3}+\frac {b^2\,x^2}{5\,d}+\frac {b\,x\,\left (5\,a\,d+b\,c\right )}{15\,d^2}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \]
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