Integrand size = 37, antiderivative size = 231 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^5} \]
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Time = 0.08 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \]
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Rule 664
Rule 672
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {(6 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^7} \, dx}{11 \left (c d^2-a e^2\right )} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {\left (8 c^2 d^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^6} \, dx}{33 \left (c d^2-a e^2\right )^2} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {\left (16 c^3 d^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx}{231 \left (c d^2-a e^2\right )^3} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^5} \\ \end{align*}
Time = 1.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2} \left (-105 a^3 e^6+35 a^2 c d e^4 (11 d+2 e x)-5 a c^2 d^2 e^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )+c^3 d^3 \left (231 d^3+198 d^2 e x+88 d e^2 x^2+16 e^3 x^3\right )\right )}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^8} \]
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Time = 4.93 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-16 x^{3} c^{3} d^{3} e^{3}+40 x^{2} a \,c^{2} d^{2} e^{4}-88 x^{2} c^{3} d^{4} e^{2}-70 x \,a^{2} c d \,e^{5}+220 x a \,c^{2} d^{3} e^{3}-198 x \,c^{3} d^{5} e +105 e^{6} a^{3}-385 d^{2} e^{4} a^{2} c +495 d^{4} e^{2} c^{2} a -231 c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 \left (e x +d \right )^{7} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\) | \(217\) |
default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{11 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {6 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{7}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (e^{2} a -c \,d^{2}\right )}\right )}{11 \left (e^{2} a -c \,d^{2}\right )}}{e^{8}}\) | \(293\) |
trager | \(-\frac {2 \left (-16 c^{5} d^{5} e^{3} x^{5}+8 a \,c^{4} d^{4} e^{4} x^{4}-88 c^{5} d^{6} e^{2} x^{4}-6 a^{2} c^{3} d^{3} e^{5} x^{3}+44 a \,c^{4} d^{5} e^{3} x^{3}-198 c^{5} d^{7} e \,x^{3}+5 a^{3} c^{2} d^{2} e^{6} x^{2}-33 a^{2} c^{3} d^{4} e^{4} x^{2}+99 a \,c^{4} d^{6} e^{2} x^{2}-231 c^{5} d^{8} x^{2}+140 d \,e^{7} c \,a^{4} x -550 a^{3} c^{2} d^{3} e^{5} x +792 a^{2} c^{3} d^{5} e^{3} x -462 a \,c^{4} d^{7} e x +105 a^{5} e^{8}-385 a^{4} c \,d^{2} e^{6}+495 a^{3} c^{2} d^{4} e^{4}-231 a^{2} c^{3} d^{6} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{1155 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{6}}\) | \(339\) |
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Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (215) = 430\).
Time = 16.98 (sec) , antiderivative size = 699, normalized size of antiderivative = 3.03 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {2 \, {\left (16 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 495 \, a^{3} c^{2} d^{4} e^{4} + 385 \, a^{4} c d^{2} e^{6} - 105 \, a^{5} e^{8} + 8 \, {\left (11 \, c^{5} d^{6} e^{2} - a c^{4} d^{4} e^{4}\right )} x^{4} + 2 \, {\left (99 \, c^{5} d^{7} e - 22 \, a c^{4} d^{5} e^{3} + 3 \, a^{2} c^{3} d^{3} e^{5}\right )} x^{3} + {\left (231 \, c^{5} d^{8} - 99 \, a c^{4} d^{6} e^{2} + 33 \, a^{2} c^{3} d^{4} e^{4} - 5 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (231 \, a c^{4} d^{7} e - 396 \, a^{2} c^{3} d^{5} e^{3} + 275 \, a^{3} c^{2} d^{3} e^{5} - 70 \, a^{4} c d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{1155 \, {\left (c^{4} d^{14} - 4 \, a c^{3} d^{12} e^{2} + 6 \, a^{2} c^{2} d^{10} e^{4} - 4 \, a^{3} c d^{8} e^{6} + a^{4} d^{6} e^{8} + {\left (c^{4} d^{8} e^{6} - 4 \, a c^{3} d^{6} e^{8} + 6 \, a^{2} c^{2} d^{4} e^{10} - 4 \, a^{3} c d^{2} e^{12} + a^{4} e^{14}\right )} x^{6} + 6 \, {\left (c^{4} d^{9} e^{5} - 4 \, a c^{3} d^{7} e^{7} + 6 \, a^{2} c^{2} d^{5} e^{9} - 4 \, a^{3} c d^{3} e^{11} + a^{4} d e^{13}\right )} x^{5} + 15 \, {\left (c^{4} d^{10} e^{4} - 4 \, a c^{3} d^{8} e^{6} + 6 \, a^{2} c^{2} d^{6} e^{8} - 4 \, a^{3} c d^{4} e^{10} + a^{4} d^{2} e^{12}\right )} x^{4} + 20 \, {\left (c^{4} d^{11} e^{3} - 4 \, a c^{3} d^{9} e^{5} + 6 \, a^{2} c^{2} d^{7} e^{7} - 4 \, a^{3} c d^{5} e^{9} + a^{4} d^{3} e^{11}\right )} x^{3} + 15 \, {\left (c^{4} d^{12} e^{2} - 4 \, a c^{3} d^{10} e^{4} + 6 \, a^{2} c^{2} d^{8} e^{6} - 4 \, a^{3} c d^{6} e^{8} + a^{4} d^{4} e^{10}\right )} x^{2} + 6 \, {\left (c^{4} d^{13} e - 4 \, a c^{3} d^{11} e^{3} + 6 \, a^{2} c^{2} d^{9} e^{5} - 4 \, a^{3} c d^{7} e^{7} + a^{4} d^{5} e^{9}\right )} x\right )}} \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Exception raised: TypeError} \]
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Time = 14.89 (sec) , antiderivative size = 2657, normalized size of antiderivative = 11.50 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Too large to display} \]
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