Integrand size = 19, antiderivative size = 127 \[ \int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx=\frac {2 \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 c d \left (c d^2+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^5}-\frac {8 c^2 d (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]
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Time = 0.04 (sec) , antiderivative size = 127, 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} \[ \int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx=\frac {4 c (d+e x)^{11/2} \left (a e^2+3 c d^2\right )}{11 e^5}-\frac {8 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )}{9 e^5}+\frac {2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5}-\frac {8 c^2 d (d+e x)^{13/2}}{13 e^5} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{e^4}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^4}-\frac {4 c^2 d (d+e x)^{11/2}}{e^4}+\frac {c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx \\ & = \frac {2 \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 c d \left (c d^2+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^5}-\frac {8 c^2 d (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.76 \[ \int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (6435 a^2 e^4+130 a c e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \]
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Time = 2.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\frac {7}{15} x^{4} c^{2}+\frac {14}{11} a c \,x^{2}+a^{2}\right ) e^{4}-\frac {56 x c \left (\frac {33 c \,x^{2}}{65}+a \right ) d \,e^{3}}{99}+\frac {16 \left (\frac {63 c \,x^{2}}{65}+a \right ) c \,d^{2} e^{2}}{99}-\frac {448 x \,c^{2} d^{3} e}{6435}+\frac {128 c^{2} d^{4}}{6435}\right )}{7 e^{5}}\) | \(88\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 c^{2} x^{4} e^{4}-1848 x^{3} c^{2} d \,e^{3}+8190 x^{2} a c \,e^{4}+1008 x^{2} c^{2} d^{2} e^{2}-3640 x a c d \,e^{3}-448 x \,c^{2} d^{3} e +6435 a^{2} e^{4}+1040 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{45045 e^{5}}\) | \(106\) |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {8 \left (e^{2} a +c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\) | \(108\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {8 \left (e^{2} a +c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\) | \(108\) |
trager | \(\frac {2 \left (3003 c^{2} e^{7} x^{7}+7161 d \,e^{6} c^{2} x^{6}+8190 a \,e^{7} c \,x^{5}+4473 d^{2} e^{5} c^{2} x^{5}+20930 a c d \,e^{6} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 a^{2} e^{7} x^{3}+14690 a \,d^{2} e^{5} c \,x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+390 a c \,d^{3} e^{4} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x -520 a c \,d^{4} e^{3} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}+1040 a c \,d^{5} e^{2}+128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(225\) |
risch | \(\frac {2 \left (3003 c^{2} e^{7} x^{7}+7161 d \,e^{6} c^{2} x^{6}+8190 a \,e^{7} c \,x^{5}+4473 d^{2} e^{5} c^{2} x^{5}+20930 a c d \,e^{6} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 a^{2} e^{7} x^{3}+14690 a \,d^{2} e^{5} c \,x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+390 a c \,d^{3} e^{4} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x -520 a c \,d^{4} e^{3} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}+1040 a c \,d^{5} e^{2}+128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(225\) |
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Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (107) = 214\).
Time = 0.49 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.72 \[ \int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, c^{2} e^{7} x^{7} + 7161 \, c^{2} d e^{6} x^{6} + 128 \, c^{2} d^{7} + 1040 \, a c d^{5} e^{2} + 6435 \, a^{2} d^{3} e^{4} + 63 \, {\left (71 \, c^{2} d^{2} e^{5} + 130 \, a c e^{7}\right )} x^{5} + 35 \, {\left (c^{2} d^{3} e^{4} + 598 \, a c d e^{6}\right )} x^{4} - 5 \, {\left (8 \, c^{2} d^{4} e^{3} - 2938 \, a c d^{2} e^{5} - 1287 \, a^{2} e^{7}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{5} e^{2} + 130 \, a c d^{3} e^{4} + 6435 \, a^{2} d e^{6}\right )} x^{2} - {\left (64 \, c^{2} d^{6} e + 520 \, a c d^{4} e^{3} - 19305 \, a^{2} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]
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Time = 0.76 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.39 \[ \int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (- \frac {4 c^{2} d \left (d + e x\right )^{\frac {13}{2}}}{13 e^{4}} + \frac {c^{2} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{4}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (2 a c e^{2} + 6 c^{2} d^{2}\right )}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 4 a c d e^{2} - 4 c^{2} d^{3}\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{7 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{2} x + \frac {2 a c x^{3}}{3} + \frac {c^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.89 \[ \int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{2} - 13860 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{2} d + 8190 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 20020 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{45045 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (107) = 214\).
Time = 0.29 (sec) , antiderivative size = 704, normalized size of antiderivative = 5.54 \[ \int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx=\frac {2 \, {\left (45045 \, \sqrt {e x + d} a^{2} d^{3} + 45045 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} d^{2} + 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} d + \frac {6006 \, {\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 d^{3}}{e^{2}} + 1287 \, {\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} + \frac {7722 \, {\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 d^{2}}{e^{2}} + \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 )} c^{2} d^{3}}{e^{4}} + \frac {858 \, {\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 d}{e^{2}} + \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 )} c^{2} d^{2}}{e^{4}} + \frac {130 \, {\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}{e^{2}} + \frac {45 \, {\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^{2} d}{e^{4}} + \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^{2}}{e^{4}}\right )}}{45045 \, e} \]
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Time = 9.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}-\frac {\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5} \]
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