Integrand size = 15, antiderivative size = 89 \[ \int \frac {(a+b x)^4}{(c+d x)^8} \, dx=\frac {(a+b x)^5}{7 (b c-a d) (c+d x)^7}+\frac {b (a+b x)^5}{21 (b c-a d)^2 (c+d x)^6}+\frac {b^2 (a+b x)^5}{105 (b c-a d)^3 (c+d x)^5} \]
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Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(a+b x)^4}{(c+d x)^8} \, dx=\frac {b^2 (a+b x)^5}{105 (c+d x)^5 (b c-a d)^3}+\frac {b (a+b x)^5}{21 (c+d x)^6 (b c-a d)^2}+\frac {(a+b x)^5}{7 (c+d x)^7 (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^5}{7 (b c-a d) (c+d x)^7}+\frac {(2 b) \int \frac {(a+b x)^4}{(c+d x)^7} \, dx}{7 (b c-a d)} \\ & = \frac {(a+b x)^5}{7 (b c-a d) (c+d x)^7}+\frac {b (a+b x)^5}{21 (b c-a d)^2 (c+d x)^6}+\frac {b^2 \int \frac {(a+b x)^4}{(c+d x)^6} \, dx}{21 (b c-a d)^2} \\ & = \frac {(a+b x)^5}{7 (b c-a d) (c+d x)^7}+\frac {b (a+b x)^5}{21 (b c-a d)^2 (c+d x)^6}+\frac {b^2 (a+b x)^5}{105 (b c-a d)^3 (c+d x)^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.62 \[ \int \frac {(a+b x)^4}{(c+d x)^8} \, dx=-\frac {15 a^4 d^4+10 a^3 b d^3 (c+7 d x)+6 a^2 b^2 d^2 \left (c^2+7 c d x+21 d^2 x^2\right )+3 a b^3 d \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+b^4 \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )}{105 d^5 (c+d x)^7} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(83)=166\).
Time = 0.23 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.92
method | result | size |
risch | \(\frac {-\frac {b^{4} x^{4}}{3 d}-\frac {b^{3} \left (3 a d +b c \right ) x^{3}}{3 d^{2}}-\frac {b^{2} \left (6 a^{2} d^{2}+3 a b c d +b^{2} c^{2}\right ) x^{2}}{5 d^{3}}-\frac {b \left (10 a^{3} d^{3}+6 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{15 d^{4}}-\frac {15 a^{4} d^{4}+10 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+3 a \,b^{3} c^{3} d +b^{4} c^{4}}{105 d^{5}}}{\left (d x +c \right )^{7}}\) | \(171\) |
gosper | \(-\frac {35 d^{4} x^{4} b^{4}+105 a \,b^{3} d^{4} x^{3}+35 b^{4} c \,d^{3} x^{3}+126 a^{2} b^{2} d^{4} x^{2}+63 a \,b^{3} c \,d^{3} x^{2}+21 b^{4} c^{2} d^{2} x^{2}+70 a^{3} b \,d^{4} x +42 a^{2} b^{2} c \,d^{3} x +21 a \,b^{3} c^{2} d^{2} x +7 b^{4} c^{3} d x +15 a^{4} d^{4}+10 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+3 a \,b^{3} c^{3} d +b^{4} c^{4}}{105 d^{5} \left (d x +c \right )^{7}}\) | \(185\) |
default | \(-\frac {2 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 d^{5} \left (d x +c \right )^{6}}-\frac {a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}{7 d^{5} \left (d x +c \right )^{7}}-\frac {6 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{5 d^{5} \left (d x +c \right )^{5}}-\frac {b^{4}}{3 d^{5} \left (d x +c \right )^{3}}-\frac {b^{3} \left (a d -b c \right )}{d^{5} \left (d x +c \right )^{4}}\) | \(186\) |
parallelrisch | \(\frac {-35 b^{4} x^{4} d^{6}-105 a \,b^{3} d^{6} x^{3}-35 b^{4} c \,d^{5} x^{3}-126 a^{2} b^{2} d^{6} x^{2}-63 a \,b^{3} c \,d^{5} x^{2}-21 b^{4} c^{2} d^{4} x^{2}-70 a^{3} b \,d^{6} x -42 a^{2} b^{2} c \,d^{5} x -21 a \,b^{3} c^{2} d^{4} x -7 b^{4} c^{3} d^{3} x -15 a^{4} d^{6}-10 a^{3} b c \,d^{5}-6 a^{2} b^{2} c^{2} d^{4}-3 a \,b^{3} c^{3} d^{3}-b^{4} c^{4} d^{2}}{105 d^{7} \left (d x +c \right )^{7}}\) | \(193\) |
norman | \(\frac {-\frac {b^{4} x^{4}}{3 d}-\frac {\left (3 a \,b^{3} d^{3}+b^{4} c \,d^{2}\right ) x^{3}}{3 d^{4}}-\frac {\left (6 a^{2} b^{2} d^{4}+3 a \,b^{3} c \,d^{3}+b^{4} c^{2} d^{2}\right ) x^{2}}{5 d^{5}}-\frac {\left (10 a^{3} b \,d^{5}+6 a^{2} b^{2} c \,d^{4}+3 a \,c^{2} b^{3} d^{3}+b^{4} c^{3} d^{2}\right ) x}{15 d^{6}}-\frac {15 a^{4} d^{6}+10 a^{3} b c \,d^{5}+6 a^{2} b^{2} c^{2} d^{4}+3 a \,b^{3} c^{3} d^{3}+b^{4} c^{4} d^{2}}{105 d^{7}}}{\left (d x +c \right )^{7}}\) | \(197\) |
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Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (83) = 166\).
Time = 0.22 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.78 \[ \int \frac {(a+b x)^4}{(c+d x)^8} \, dx=-\frac {35 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 3 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4} + 35 \, {\left (b^{4} c d^{3} + 3 \, a b^{3} d^{4}\right )} x^{3} + 21 \, {\left (b^{4} c^{2} d^{2} + 3 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} + 7 \, {\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} + 10 \, a^{3} b d^{4}\right )} x}{105 \, {\left (d^{12} x^{7} + 7 \, c d^{11} x^{6} + 21 \, c^{2} d^{10} x^{5} + 35 \, c^{3} d^{9} x^{4} + 35 \, c^{4} d^{8} x^{3} + 21 \, c^{5} d^{7} x^{2} + 7 \, c^{6} d^{6} x + c^{7} d^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (73) = 146\).
Time = 114.48 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.00 \[ \int \frac {(a+b x)^4}{(c+d x)^8} \, dx=\frac {- 15 a^{4} d^{4} - 10 a^{3} b c d^{3} - 6 a^{2} b^{2} c^{2} d^{2} - 3 a b^{3} c^{3} d - b^{4} c^{4} - 35 b^{4} d^{4} x^{4} + x^{3} \left (- 105 a b^{3} d^{4} - 35 b^{4} c d^{3}\right ) + x^{2} \left (- 126 a^{2} b^{2} d^{4} - 63 a b^{3} c d^{3} - 21 b^{4} c^{2} d^{2}\right ) + x \left (- 70 a^{3} b d^{4} - 42 a^{2} b^{2} c d^{3} - 21 a b^{3} c^{2} d^{2} - 7 b^{4} c^{3} d\right )}{105 c^{7} d^{5} + 735 c^{6} d^{6} x + 2205 c^{5} d^{7} x^{2} + 3675 c^{4} d^{8} x^{3} + 3675 c^{3} d^{9} x^{4} + 2205 c^{2} d^{10} x^{5} + 735 c d^{11} x^{6} + 105 d^{12} x^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (83) = 166\).
Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.78 \[ \int \frac {(a+b x)^4}{(c+d x)^8} \, dx=-\frac {35 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 3 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4} + 35 \, {\left (b^{4} c d^{3} + 3 \, a b^{3} d^{4}\right )} x^{3} + 21 \, {\left (b^{4} c^{2} d^{2} + 3 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} + 7 \, {\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} + 10 \, a^{3} b d^{4}\right )} x}{105 \, {\left (d^{12} x^{7} + 7 \, c d^{11} x^{6} + 21 \, c^{2} d^{10} x^{5} + 35 \, c^{3} d^{9} x^{4} + 35 \, c^{4} d^{8} x^{3} + 21 \, c^{5} d^{7} x^{2} + 7 \, c^{6} d^{6} x + c^{7} d^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (83) = 166\).
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b x)^4}{(c+d x)^8} \, dx=-\frac {35 \, b^{4} d^{4} x^{4} + 35 \, b^{4} c d^{3} x^{3} + 105 \, a b^{3} d^{4} x^{3} + 21 \, b^{4} c^{2} d^{2} x^{2} + 63 \, a b^{3} c d^{3} x^{2} + 126 \, a^{2} b^{2} d^{4} x^{2} + 7 \, b^{4} c^{3} d x + 21 \, a b^{3} c^{2} d^{2} x + 42 \, a^{2} b^{2} c d^{3} x + 70 \, a^{3} b d^{4} x + b^{4} c^{4} + 3 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4}}{105 \, {\left (d x + c\right )}^{7} d^{5}} \]
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Time = 0.13 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.66 \[ \int \frac {(a+b x)^4}{(c+d x)^8} \, dx=-\frac {\frac {15\,a^4\,d^4+10\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+3\,a\,b^3\,c^3\,d+b^4\,c^4}{105\,d^5}+\frac {b^4\,x^4}{3\,d}+\frac {b^3\,x^3\,\left (3\,a\,d+b\,c\right )}{3\,d^2}+\frac {b\,x\,\left (10\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{15\,d^4}+\frac {b^2\,x^2\,\left (6\,a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{5\,d^3}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \]
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