Integrand size = 19, antiderivative size = 204 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]
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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} \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {6 c^2 (d+e x)^{13/2} \left (a e^2+5 c d^2\right )}{13 e^7}-\frac {8 c^2 d (d+e x)^{11/2} \left (3 a e^2+5 c d^2\right )}{11 e^7}+\frac {2 c (d+e x)^{9/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {12 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^7}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^3}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{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)^{7/2}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{e^6}-\frac {6 c^3 d (d+e x)^{13/2}}{e^6}+\frac {c^3 (d+e x)^{15/2}}{e^6}\right ) \, dx \\ & = \frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{5/2} \left (51051 a^3 e^6+2431 a^2 c e^4 \left (8 d^2-20 d e x+35 e^2 x^2\right )+51 a c^2 e^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+c^3 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )}{255255 e^7} \]
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Time = 2.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (\frac {\left (5 e^{6} x^{6}-4 d \,e^{5} x^{5}+\frac {40}{13} d^{2} e^{4} x^{4}-\frac {320}{143} x^{3} d^{3} e^{3}+\frac {640}{429} d^{4} e^{2} x^{2}-\frac {2560}{3003} d^{5} e x +\frac {1024}{3003} d^{6}\right ) c^{3}}{17}+\frac {128 e^{2} a \left (\frac {1155}{128} e^{4} x^{4}-\frac {105}{16} d \,e^{3} x^{3}+\frac {35}{8} d^{2} e^{2} x^{2}-\frac {5}{2} d^{3} e x +d^{4}\right ) c^{2}}{1001}+\frac {8 e^{4} \left (\frac {35}{8} x^{2} e^{2}-\frac {5}{2} d e x +d^{2}\right ) a^{2} c}{21}+e^{6} a^{3}\right )}{5 e^{7}}\) | \(163\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (15015 x^{6} c^{3} e^{6}-12012 x^{5} c^{3} d \,e^{5}+58905 x^{4} a \,c^{2} e^{6}+9240 x^{4} c^{3} d^{2} e^{4}-42840 x^{3} a \,c^{2} d \,e^{5}-6720 x^{3} c^{3} d^{3} e^{3}+85085 x^{2} a^{2} c \,e^{6}+28560 x^{2} a \,c^{2} d^{2} e^{4}+4480 x^{2} c^{3} d^{4} e^{2}-48620 x \,a^{2} c d \,e^{5}-16320 x a \,c^{2} d^{3} e^{3}-2560 x \,c^{3} d^{5} e +51051 e^{6} a^{3}+19448 d^{2} e^{4} a^{2} c +6528 d^{4} e^{2} c^{2} a +1024 c^{3} d^{6}\right )}{255255 e^{7}}\) | \(205\) |
derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(269\) |
default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(269\) |
trager | \(\frac {2 \left (15015 e^{8} c^{3} x^{8}+18018 c^{3} d \,e^{7} x^{7}+58905 a \,c^{2} e^{8} x^{6}+231 c^{3} d^{2} e^{6} x^{6}+74970 a \,c^{2} d \,e^{7} x^{5}-252 c^{3} d^{3} e^{5} x^{5}+85085 a^{2} c \,e^{8} x^{4}+1785 a \,c^{2} d^{2} e^{6} x^{4}+280 c^{3} d^{4} e^{4} x^{4}+121550 a^{2} c d \,e^{7} x^{3}-2040 a \,c^{2} d^{3} e^{5} x^{3}-320 c^{3} d^{5} e^{3} x^{3}+51051 a^{3} e^{8} x^{2}+7293 a^{2} c \,d^{2} e^{6} x^{2}+2448 a \,c^{2} d^{4} e^{4} x^{2}+384 c^{3} d^{6} e^{2} x^{2}+102102 a^{3} d \,e^{7} x -9724 a^{2} c \,d^{3} e^{5} x -3264 a \,c^{2} d^{5} e^{3} x -512 c^{3} d^{7} e x +51051 a^{3} d^{2} e^{6}+19448 a^{2} c \,d^{4} e^{4}+6528 a \,c^{2} d^{6} e^{2}+1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(317\) |
risch | \(\frac {2 \left (15015 e^{8} c^{3} x^{8}+18018 c^{3} d \,e^{7} x^{7}+58905 a \,c^{2} e^{8} x^{6}+231 c^{3} d^{2} e^{6} x^{6}+74970 a \,c^{2} d \,e^{7} x^{5}-252 c^{3} d^{3} e^{5} x^{5}+85085 a^{2} c \,e^{8} x^{4}+1785 a \,c^{2} d^{2} e^{6} x^{4}+280 c^{3} d^{4} e^{4} x^{4}+121550 a^{2} c d \,e^{7} x^{3}-2040 a \,c^{2} d^{3} e^{5} x^{3}-320 c^{3} d^{5} e^{3} x^{3}+51051 a^{3} e^{8} x^{2}+7293 a^{2} c \,d^{2} e^{6} x^{2}+2448 a \,c^{2} d^{4} e^{4} x^{2}+384 c^{3} d^{6} e^{2} x^{2}+102102 a^{3} d \,e^{7} x -9724 a^{2} c \,d^{3} e^{5} x -3264 a \,c^{2} d^{5} e^{3} x -512 c^{3} d^{7} e x +51051 a^{3} d^{2} e^{6}+19448 a^{2} c \,d^{4} e^{4}+6528 a \,c^{2} d^{6} e^{2}+1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(317\) |
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Time = 0.27 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.49 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, c^{3} e^{8} x^{8} + 18018 \, c^{3} d e^{7} x^{7} + 1024 \, c^{3} d^{8} + 6528 \, a c^{2} d^{6} e^{2} + 19448 \, a^{2} c d^{4} e^{4} + 51051 \, a^{3} d^{2} e^{6} + 231 \, {\left (c^{3} d^{2} e^{6} + 255 \, a c^{2} e^{8}\right )} x^{6} - 126 \, {\left (2 \, c^{3} d^{3} e^{5} - 595 \, a c^{2} d e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{4} e^{4} + 51 \, a c^{2} d^{2} e^{6} + 2431 \, a^{2} c e^{8}\right )} x^{4} - 10 \, {\left (32 \, c^{3} d^{5} e^{3} + 204 \, a c^{2} d^{3} e^{5} - 12155 \, a^{2} c d e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{6} e^{2} + 816 \, a c^{2} d^{4} e^{4} + 2431 \, a^{2} c d^{2} e^{6} + 17017 \, a^{3} e^{8}\right )} x^{2} - 2 \, {\left (256 \, c^{3} d^{7} e + 1632 \, a c^{2} d^{5} e^{3} + 4862 \, a^{2} c d^{3} e^{5} - 51051 \, a^{3} d e^{7}\right )} x\right )} \sqrt {e x + d}}{255255 \, e^{7}} \]
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Time = 0.79 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.49 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (- \frac {2 c^{3} d \left (d + e x\right )^{\frac {15}{2}}}{5 e^{6}} + \frac {c^{3} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (- 6 a^{2} c d e^{4} - 12 a c^{2} d^{3} e^{2} - 6 c^{3} d^{5}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )}{5 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.02 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 102102 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} d + 58905 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 92820 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 218790 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{255255 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (176) = 352\).
Time = 0.29 (sec) , antiderivative size = 784, normalized size of antiderivative = 3.84 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \]
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Time = 9.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.92 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{9\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {4\,c^3\,d\,{\left (d+e\,x\right )}^{15/2}}{5\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7} \]
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