Integrand size = 37, antiderivative size = 119 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}{5 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{7/2}}{7 e^4}-\frac {2 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{9/2}}{3 e^4}+\frac {2 c^3 d^3 (d+e x)^{11/2}}{11 e^4} \]
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Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {640, 45} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx=-\frac {2 c^2 d^2 (d+e x)^{9/2} \left (c d^2-a e^2\right )}{3 e^4}+\frac {6 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^4}-\frac {2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3}{5 e^4}+\frac {2 c^3 d^3 (d+e x)^{11/2}}{11 e^4} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x)^3 (d+e x)^{3/2} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^{3/2}}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{e^3}+\frac {c^3 d^3 (d+e x)^{9/2}}{e^3}\right ) \, dx \\ & = -\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}{5 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{7/2}}{7 e^4}-\frac {2 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{9/2}}{3 e^4}+\frac {2 c^3 d^3 (d+e x)^{11/2}}{11 e^4} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 (d+e x)^{5/2} \left (231 a^3 e^6-99 a^2 c d e^4 (2 d-5 e x)+11 a c^2 d^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 e^4} \]
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Time = 2.73 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(97\) |
default | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(97\) |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (e^{6} a^{3}+\frac {15 x \,a^{2} c d \,e^{5}}{7}-\frac {6 \left (-\frac {35 c \,x^{2}}{18}+a \right ) c \,d^{2} a \,e^{4}}{7}-\frac {20 x \,c^{2} \left (-\frac {21 c \,x^{2}}{44}+a \right ) d^{3} e^{3}}{21}+\frac {8 c^{2} \left (-\frac {35 c \,x^{2}}{44}+a \right ) d^{4} e^{2}}{21}+\frac {40 x \,c^{3} d^{5} e}{231}-\frac {16 c^{3} d^{6}}{231}\right )}{5 e^{4}}\) | \(107\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (105 x^{3} c^{3} d^{3} e^{3}+385 x^{2} a \,c^{2} d^{2} e^{4}-70 x^{2} c^{3} d^{4} e^{2}+495 x \,a^{2} c d \,e^{5}-220 x a \,c^{2} d^{3} e^{3}+40 x \,c^{3} d^{5} e +231 e^{6} a^{3}-198 d^{2} e^{4} a^{2} c +88 d^{4} e^{2} c^{2} a -16 c^{3} d^{6}\right )}{1155 e^{4}}\) | \(131\) |
trager | \(\frac {2 \left (105 c^{3} d^{3} e^{5} x^{5}+385 a \,c^{2} d^{2} e^{6} x^{4}+140 c^{3} d^{4} e^{4} x^{4}+495 d \,e^{7} a^{2} c \,x^{3}+550 a \,c^{2} d^{3} e^{5} x^{3}+5 c^{3} d^{5} e^{3} x^{3}+231 a^{3} e^{8} x^{2}+792 a^{2} c \,d^{2} e^{6} x^{2}+33 a \,c^{2} d^{4} e^{4} x^{2}-6 c^{3} d^{6} e^{2} x^{2}+462 a^{3} d \,e^{7} x +99 a^{2} c \,d^{3} e^{5} x -44 a \,c^{2} d^{5} e^{3} x +8 c^{3} d^{7} e x +231 a^{3} e^{6} d^{2}-198 d^{4} e^{4} a^{2} c +88 a \,c^{2} d^{6} e^{2}-16 c^{3} d^{8}\right ) \sqrt {e x +d}}{1155 e^{4}}\) | \(243\) |
risch | \(\frac {2 \left (105 c^{3} d^{3} e^{5} x^{5}+385 a \,c^{2} d^{2} e^{6} x^{4}+140 c^{3} d^{4} e^{4} x^{4}+495 d \,e^{7} a^{2} c \,x^{3}+550 a \,c^{2} d^{3} e^{5} x^{3}+5 c^{3} d^{5} e^{3} x^{3}+231 a^{3} e^{8} x^{2}+792 a^{2} c \,d^{2} e^{6} x^{2}+33 a \,c^{2} d^{4} e^{4} x^{2}-6 c^{3} d^{6} e^{2} x^{2}+462 a^{3} d \,e^{7} x +99 a^{2} c \,d^{3} e^{5} x -44 a \,c^{2} d^{5} e^{3} x +8 c^{3} d^{7} e x +231 a^{3} e^{6} d^{2}-198 d^{4} e^{4} a^{2} c +88 a \,c^{2} d^{6} e^{2}-16 c^{3} d^{8}\right ) \sqrt {e x +d}}{1155 e^{4}}\) | \(243\) |
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Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (103) = 206\).
Time = 0.40 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (105 \, c^{3} d^{3} e^{5} x^{5} - 16 \, c^{3} d^{8} + 88 \, a c^{2} d^{6} e^{2} - 198 \, a^{2} c d^{4} e^{4} + 231 \, a^{3} d^{2} e^{6} + 35 \, {\left (4 \, c^{3} d^{4} e^{4} + 11 \, a c^{2} d^{2} e^{6}\right )} x^{4} + 5 \, {\left (c^{3} d^{5} e^{3} + 110 \, a c^{2} d^{3} e^{5} + 99 \, a^{2} c d e^{7}\right )} x^{3} - 3 \, {\left (2 \, c^{3} d^{6} e^{2} - 11 \, a c^{2} d^{4} e^{4} - 264 \, a^{2} c d^{2} e^{6} - 77 \, a^{3} e^{8}\right )} x^{2} + {\left (8 \, c^{3} d^{7} e - 44 \, a c^{2} d^{5} e^{3} + 99 \, a^{2} c d^{3} e^{5} + 462 \, a^{3} d e^{7}\right )} x\right )} \sqrt {e x + d}}{1155 \, e^{4}} \]
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Time = 1.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} d^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{3}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a c^{2} d^{2} e^{2} - 3 c^{3} d^{4}\right )}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a^{2} c d e^{4} - 6 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}\right )}{5 e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {9}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (105 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{3} d^{3} - 385 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 231 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{1155 \, e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 582 vs. \(2 (103) = 206\).
Time = 0.27 (sec) , antiderivative size = 582, normalized size of antiderivative = 4.89 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a^{3} d^{2} e^{3} + 3465 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} c d^{3} e + 2310 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{3} d e^{3} + \frac {693 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a c^{2} d^{4}}{e} + 1386 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} c d^{2} e + 231 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{3} e^{3} + \frac {99 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} c^{3} d^{5}}{e^{3}} + \frac {594 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a c^{2} d^{3}}{e} + 297 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{2} c d e + \frac {22 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{3} d^{4}}{e^{3}} + \frac {33 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a c^{2} d^{2}}{e} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} c^{3} d^{3}}{e^{3}}\right )}}{3465 \, e} \]
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Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \]
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