Integrand size = 37, antiderivative size = 119 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt {d+e x}} \, dx=-\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{7/2}}{7 e^4}+\frac {2 c d \left (c d^2-a e^2\right )^2 (d+e x)^{9/2}}{3 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{11/2}}{11 e^4}+\frac {2 c^3 d^3 (d+e x)^{13/2}}{13 e^4} \]
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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} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt {d+e x}} \, dx=-\frac {6 c^2 d^2 (d+e x)^{11/2} \left (c d^2-a e^2\right )}{11 e^4}+\frac {2 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )^2}{3 e^4}-\frac {2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^3}{7 e^4}+\frac {2 c^3 d^3 (d+e x)^{13/2}}{13 e^4} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x)^3 (d+e x)^{5/2} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^{5/2}}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{7/2}}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{9/2}}{e^3}+\frac {c^3 d^3 (d+e x)^{11/2}}{e^3}\right ) \, dx \\ & = -\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{7/2}}{7 e^4}+\frac {2 c d \left (c d^2-a e^2\right )^2 (d+e x)^{9/2}}{3 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{11/2}}{11 e^4}+\frac {2 c^3 d^3 (d+e x)^{13/2}}{13 e^4} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 (d+e x)^{7/2} \left (429 a^3 e^6-143 a^2 c d e^4 (2 d-7 e x)+13 a c^2 d^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 e^4} \]
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Time = 2.72 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\) | \(97\) |
default | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\) | \(97\) |
pseudoelliptic | \(\frac {2 \left (e^{6} a^{3}+\frac {7 x \,a^{2} c d \,e^{5}}{3}-\frac {2 c \left (-\frac {63 c \,x^{2}}{22}+a \right ) d^{2} a \,e^{4}}{3}-\frac {28 x \,c^{2} d^{3} \left (-\frac {33 c \,x^{2}}{52}+a \right ) e^{3}}{33}+\frac {8 c^{2} d^{4} \left (-\frac {63 c \,x^{2}}{52}+a \right ) e^{2}}{33}+\frac {56 x \,c^{3} d^{5} e}{429}-\frac {16 c^{3} d^{6}}{429}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{4}}\) | \(107\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (231 x^{3} c^{3} d^{3} e^{3}+819 x^{2} a \,c^{2} d^{2} e^{4}-126 x^{2} c^{3} d^{4} e^{2}+1001 x \,a^{2} c d \,e^{5}-364 x a \,c^{2} d^{3} e^{3}+56 x \,c^{3} d^{5} e +429 e^{6} a^{3}-286 d^{2} e^{4} a^{2} c +104 d^{4} e^{2} c^{2} a -16 c^{3} d^{6}\right )}{3003 e^{4}}\) | \(131\) |
trager | \(\frac {2 \left (231 c^{3} d^{3} e^{6} x^{6}+819 a \,c^{2} d^{2} e^{7} x^{5}+567 c^{3} d^{4} e^{5} x^{5}+1001 a^{2} c d \,e^{8} x^{4}+2093 a \,c^{2} d^{3} e^{6} x^{4}+371 c^{3} d^{5} e^{4} x^{4}+429 a^{3} e^{9} x^{3}+2717 a^{2} c \,d^{2} e^{7} x^{3}+1469 a \,c^{2} d^{4} e^{5} x^{3}+5 c^{3} d^{6} e^{3} x^{3}+1287 a^{3} d \,e^{8} x^{2}+2145 a^{2} c \,d^{3} e^{6} x^{2}+39 a \,c^{2} d^{5} e^{4} x^{2}-6 c^{3} d^{7} e^{2} x^{2}+1287 a^{3} d^{2} e^{7} x +143 a^{2} c \,d^{4} e^{5} x -52 a \,c^{2} d^{6} e^{3} x +8 c^{3} d^{8} e x +429 a^{3} d^{3} e^{6}-286 a^{2} c \,d^{5} e^{4}+104 a \,c^{2} d^{7} e^{2}-16 c^{3} d^{9}\right ) \sqrt {e x +d}}{3003 e^{4}}\) | \(301\) |
risch | \(\frac {2 \left (231 c^{3} d^{3} e^{6} x^{6}+819 a \,c^{2} d^{2} e^{7} x^{5}+567 c^{3} d^{4} e^{5} x^{5}+1001 a^{2} c d \,e^{8} x^{4}+2093 a \,c^{2} d^{3} e^{6} x^{4}+371 c^{3} d^{5} e^{4} x^{4}+429 a^{3} e^{9} x^{3}+2717 a^{2} c \,d^{2} e^{7} x^{3}+1469 a \,c^{2} d^{4} e^{5} x^{3}+5 c^{3} d^{6} e^{3} x^{3}+1287 a^{3} d \,e^{8} x^{2}+2145 a^{2} c \,d^{3} e^{6} x^{2}+39 a \,c^{2} d^{5} e^{4} x^{2}-6 c^{3} d^{7} e^{2} x^{2}+1287 a^{3} d^{2} e^{7} x +143 a^{2} c \,d^{4} e^{5} x -52 a \,c^{2} d^{6} e^{3} x +8 c^{3} d^{8} e x +429 a^{3} d^{3} e^{6}-286 a^{2} c \,d^{5} e^{4}+104 a \,c^{2} d^{7} e^{2}-16 c^{3} d^{9}\right ) \sqrt {e x +d}}{3003 e^{4}}\) | \(301\) |
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (103) = 206\).
Time = 0.42 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (231 \, c^{3} d^{3} e^{6} x^{6} - 16 \, c^{3} d^{9} + 104 \, a c^{2} d^{7} e^{2} - 286 \, a^{2} c d^{5} e^{4} + 429 \, a^{3} d^{3} e^{6} + 63 \, {\left (9 \, c^{3} d^{4} e^{5} + 13 \, a c^{2} d^{2} e^{7}\right )} x^{5} + 7 \, {\left (53 \, c^{3} d^{5} e^{4} + 299 \, a c^{2} d^{3} e^{6} + 143 \, a^{2} c d e^{8}\right )} x^{4} + {\left (5 \, c^{3} d^{6} e^{3} + 1469 \, a c^{2} d^{4} e^{5} + 2717 \, a^{2} c d^{2} e^{7} + 429 \, a^{3} e^{9}\right )} x^{3} - 3 \, {\left (2 \, c^{3} d^{7} e^{2} - 13 \, a c^{2} d^{5} e^{4} - 715 \, a^{2} c d^{3} e^{6} - 429 \, a^{3} d e^{8}\right )} x^{2} + {\left (8 \, c^{3} d^{8} e - 52 \, a c^{2} d^{6} e^{3} + 143 \, a^{2} c d^{4} e^{5} + 1287 \, a^{3} d^{2} e^{7}\right )} x\right )} \sqrt {e x + d}}{3003 \, e^{4}} \]
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Time = 1.06 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} d^{3} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{3}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 a c^{2} d^{2} e^{2} - 3 c^{3} d^{4}\right )}{11 e^{3}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a^{2} c d e^{4} - 6 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}\right )}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}\right )}{7 e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {11}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 611 vs. \(2 (103) = 206\).
Time = 0.21 (sec) , antiderivative size = 611, normalized size of antiderivative = 5.13 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{3} d^{3} e^{3} + 3003 \, {\left (\frac {{\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 )} c d}{e} + \frac {5 \, {\left (c d^{2} + a e^{2}\right )} {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )}}{e}\right )} a^{2} d^{2} e^{2} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3} d^{3}}{e^{3}} + 143 \, {\left (\frac {{\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^{2} d^{2}}{e^{2}} + \frac {18 \, {\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 )} {\left (c d^{2} + a e^{2}\right )} c d}{e^{2}} + \frac {21 \, {\left (c d^{2} + a e^{2}\right )}^{2} {\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 )}}{e^{2}}\right )} a d e + \frac {65 \, {\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 )} {\left (c d^{2} + a e^{2}\right )} c^{2} d^{2}}{e^{3}} + \frac {143 \, {\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 )} {\left (c d^{2} + a e^{2}\right )}^{2} c d}{e^{3}} + \frac {429 \, {\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 )} {\left (c d^{2} + a e^{2}\right )}^{3}}{e^{3}}\right )}}{15015 \, e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (103) = 206\).
Time = 0.31 (sec) , antiderivative size = 874, normalized size of antiderivative = 7.34 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt {d+e x}} \, dx=\text {Too large to display} \]
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Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^4} \]
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