Integrand size = 19, antiderivative size = 204 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{3 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 e^7}+\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \]
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Time = 0.06 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {711} \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {6 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac {8 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac {6 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {12 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{3 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right )^3 \sqrt {d+e x}}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{7/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^6}-\frac {6 c^3 d (d+e x)^{11/2}}{e^6}+\frac {c^3 (d+e x)^{13/2}}{e^6}\right ) \, dx \\ & = \frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{3 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 e^7}+\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{3/2} \left (15015 a^3 e^6+1287 a^2 c e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+39 a c^2 e^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+c^3 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \]
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Time = 1.97 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{5} e^{6} x^{6}-\frac {12}{65} d \,e^{5} x^{5}+\frac {24}{143} d^{2} e^{4} x^{4}-\frac {64}{429} x^{3} d^{3} e^{3}+\frac {128}{1001} d^{4} e^{2} x^{2}-\frac {512}{5005} d^{5} e x +\frac {1024}{15015} d^{6}\right ) c^{3}+\frac {128 \left (\frac {315}{128} e^{4} x^{4}-\frac {35}{16} d \,e^{3} x^{3}+\frac {15}{8} d^{2} e^{2} x^{2}-\frac {3}{2} d^{3} e x +d^{4}\right ) e^{2} a \,c^{2}}{385}+\frac {24 \left (\frac {15}{8} x^{2} e^{2}-\frac {3}{2} d e x +d^{2}\right ) e^{4} a^{2} c}{35}+e^{6} a^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{7}}\) | \(162\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 x^{6} c^{3} e^{6}-2772 x^{5} c^{3} d \,e^{5}+12285 x^{4} a \,c^{2} e^{6}+2520 x^{4} c^{3} d^{2} e^{4}-10920 x^{3} a \,c^{2} d \,e^{5}-2240 x^{3} c^{3} d^{3} e^{3}+19305 x^{2} a^{2} c \,e^{6}+9360 x^{2} a \,c^{2} d^{2} e^{4}+1920 x^{2} c^{3} d^{4} e^{2}-15444 x \,a^{2} c d \,e^{5}-7488 x a \,c^{2} d^{3} e^{3}-1536 x \,c^{3} d^{5} e +15015 e^{6} a^{3}+10296 d^{2} e^{4} a^{2} c +4992 d^{4} e^{2} c^{2} a +1024 c^{3} d^{6}\right )}{45045 e^{7}}\) | \(205\) |
trager | \(\frac {2 \left (3003 c^{3} e^{7} x^{7}+231 c^{3} d \,e^{6} x^{6}+12285 e^{7} c^{2} a \,x^{5}-252 c^{3} d^{2} e^{5} x^{5}+1365 d \,e^{6} c^{2} a \,x^{4}+280 c^{3} d^{3} e^{4} x^{4}+19305 a^{2} c \,e^{7} x^{3}-1560 a \,c^{2} d^{2} e^{5} x^{3}-320 c^{3} d^{4} e^{3} x^{3}+3861 a^{2} c d \,e^{6} x^{2}+1872 a \,c^{2} d^{3} e^{4} x^{2}+384 c^{3} d^{5} e^{2} x^{2}+15015 a^{3} e^{7} x -5148 a^{2} c \,d^{2} e^{5} x -2496 a \,c^{2} d^{4} e^{3} x -512 c^{3} d^{6} e x +15015 a^{3} d \,e^{6}+10296 a^{2} c \,d^{3} e^{4}+4992 a \,c^{2} d^{5} e^{2}+1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\) | \(259\) |
risch | \(\frac {2 \left (3003 c^{3} e^{7} x^{7}+231 c^{3} d \,e^{6} x^{6}+12285 e^{7} c^{2} a \,x^{5}-252 c^{3} d^{2} e^{5} x^{5}+1365 d \,e^{6} c^{2} a \,x^{4}+280 c^{3} d^{3} e^{4} x^{4}+19305 a^{2} c \,e^{7} x^{3}-1560 a \,c^{2} d^{2} e^{5} x^{3}-320 c^{3} d^{4} e^{3} x^{3}+3861 a^{2} c d \,e^{6} x^{2}+1872 a \,c^{2} d^{3} e^{4} x^{2}+384 c^{3} d^{5} e^{2} x^{2}+15015 a^{3} e^{7} x -5148 a^{2} c \,d^{2} e^{5} x -2496 a \,c^{2} d^{4} e^{3} x -512 c^{3} d^{6} e x +15015 a^{3} d \,e^{6}+10296 a^{2} c \,d^{3} e^{4}+4992 a \,c^{2} d^{5} e^{2}+1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\) | \(259\) |
derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\) | \(269\) |
default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\) | \(269\) |
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Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.24 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (3003 \, c^{3} e^{7} x^{7} + 231 \, c^{3} d e^{6} x^{6} + 1024 \, c^{3} d^{7} + 4992 \, a c^{2} d^{5} e^{2} + 10296 \, a^{2} c d^{3} e^{4} + 15015 \, a^{3} d e^{6} - 63 \, {\left (4 \, c^{3} d^{2} e^{5} - 195 \, a c^{2} e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{3} e^{4} + 39 \, a c^{2} d e^{6}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{4} e^{3} + 312 \, a c^{2} d^{2} e^{5} - 3861 \, a^{2} c e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{5} e^{2} + 624 \, a c^{2} d^{3} e^{4} + 1287 \, a^{2} c d e^{6}\right )} x^{2} - {\left (512 \, c^{3} d^{6} e + 2496 \, a c^{2} d^{4} e^{3} + 5148 \, a^{2} c d^{2} e^{5} - 15015 \, a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \]
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Time = 0.70 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.49 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (- \frac {6 c^{3} d \left (d + e x\right )^{\frac {13}{2}}}{13 e^{6}} + \frac {c^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 6 a^{2} c d e^{4} - 12 a c^{2} d^{3} e^{2} - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.02 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} - 20790 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{3} d + 12285 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 20020 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 19305 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 54054 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (176) = 352\).
Time = 0.27 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.30 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (45045 \, \sqrt {e x + d} a^{3} d + 15015 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{3} + \frac {9009 \, {\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}{e^{2}} + \frac {3861 \, {\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}{e^{2}} + \frac {429 \, {\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}{e^{4}} + \frac {195 \, {\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 )} a c^{2}}{e^{4}} + \frac {15 \, {\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}{e^{6}} + \frac {7 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} - 3465 \, {\left (e x + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (e x + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {e x + d} d^{7}\right )} c^{3}}{e^{6}}\right )}}{45045 \, e} \]
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Time = 0.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.92 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{7\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {12\,c^3\,d\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7} \]
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