Integrand size = 25, antiderivative size = 201 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]
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Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {847, 794, 201, 223, 209} \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {3 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}+\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5} \]
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 847
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {\int x^3 \left (-4 d^2 e-9 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{9 e^2} \\ & = -\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {\int x^2 \left (27 d^3 e^2+32 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{72 e^4} \\ & = -\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {\int x \left (-64 d^4 e^3-189 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{504 e^6} \\ & = -\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {d^5 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{16 e^4} \\ & = \frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{64 e^4} \\ & = \frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{128 e^4} \\ & = \frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^9\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4} \\ & = \frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (1024 d^8+945 d^7 e x+512 d^6 e^2 x^2+630 d^5 e^3 x^3+384 d^4 e^4 x^4-7560 d^3 e^5 x^5-6400 d^2 e^6 x^6+5040 d e^7 x^7+4480 e^8 x^8\right )+1890 d^9 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{40320 e^5} \]
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Time = 0.37 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {\left (4480 e^{8} x^{8}+5040 d \,e^{7} x^{7}-6400 d^{2} e^{6} x^{6}-7560 d^{3} e^{5} x^{5}+384 d^{4} x^{4} e^{4}+630 d^{5} e^{3} x^{3}+512 d^{6} e^{2} x^{2}+945 d^{7} e x +1024 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{40320 e^{5}}+\frac {3 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{4} \sqrt {e^{2}}}\) | \(141\) |
default | \(e \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{9 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 e^{4}}\right )}{9 e^{2}}\right )+d \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6 e^{2}}\right )}{8 e^{2}}\right )\) | \(215\) |
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Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.69 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {1890 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (4480 \, e^{8} x^{8} + 5040 \, d e^{7} x^{7} - 6400 \, d^{2} e^{6} x^{6} - 7560 \, d^{3} e^{5} x^{5} + 384 \, d^{4} e^{4} x^{4} + 630 \, d^{5} e^{3} x^{3} + 512 \, d^{6} e^{2} x^{2} + 945 \, d^{7} e x + 1024 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{40320 \, e^{5}} \]
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Time = 0.57 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 d^{9} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{8}}{315 e^{5}} - \frac {3 d^{7} x}{128 e^{4}} - \frac {4 d^{6} x^{2}}{315 e^{3}} - \frac {d^{5} x^{3}}{64 e^{2}} - \frac {d^{4} x^{4}}{105 e} + \frac {3 d^{3} x^{5}}{16} + \frac {10 d^{2} e x^{6}}{63} - \frac {d e^{2} x^{7}}{8} - \frac {e^{3} x^{8}}{9}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d x^{5}}{5} + \frac {e x^{6}}{6}\right ) \left (d^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{4}}{9 \, e} + \frac {3 \, d^{9} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{128 \, \sqrt {e^{2}} e^{4}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x}{128 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{3}}{8 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x}{64 \, e^{4}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2}}{63 \, e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x}{16 \, e^{4}} - \frac {8 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{315 \, e^{5}} \]
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Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {3 \, d^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, e^{4} {\left | e \right |}} - \frac {1}{40320} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {1024 \, d^{8}}{e^{5}} + {\left (\frac {945 \, d^{7}}{e^{4}} + 2 \, {\left (\frac {256 \, d^{6}}{e^{3}} + {\left (\frac {315 \, d^{5}}{e^{2}} + 4 \, {\left (\frac {48 \, d^{4}}{e} - 5 \, {\left (189 \, d^{3} + 2 \, {\left (80 \, d^{2} e - 7 \, {\left (8 \, e^{3} x + 9 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
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Timed out. \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\int x^4\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right ) \,d x \]
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