Integrand size = 27, antiderivative size = 201 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]
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Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {864, 847, 794, 201, 223, 209} \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}+\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5} \]
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 847
Rule 864
Rubi steps \begin{align*} \text {integral}& = \int x^4 (d-e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx \\ & = \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 = 0.42 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (1024 d^8-945 d^7 e x+512 d^6 e^2 x^2-630 d^5 e^3 x^3+384 d^4 e^4 x^4+7560 d^3 e^5 x^5-6400 d^2 e^6 x^6-5040 d e^7 x^7+4480 e^8 x^8\right )-1890 d^9 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{40320 e^5} \]
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Time = 0.40 (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 | \(\frac {-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}}{e}-\frac {d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{5}}-\frac {d^{3} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{e^{4}}-\frac {d \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{e^{2}}+\frac {d^{4} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{e^{5}}\) | \(502\) |
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Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.69 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {1890 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (4480 \, e^{8} x^{8} - 5040 \, d e^{7} x^{7} - 6400 \, d^{2} e^{6} x^{6} + 7560 \, d^{3} e^{5} x^{5} + 384 \, d^{4} e^{4} x^{4} - 630 \, d^{5} e^{3} x^{3} + 512 \, d^{6} e^{2} x^{2} - 945 \, d^{7} e x + 1024 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{40320 \, e^{5}} \]
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Time = 1.47 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.12 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=d^{3} \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) - d^{2} e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {5 d^{8} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 e^{6}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {5 d^{6} x}{128 e^{6}} - \frac {5 d^{4} x^{3}}{192 e^{4}} - \frac {d^{2} x^{5}}{48 e^{2}} + \frac {x^{7}}{8}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{7} \sqrt {d^{2}}}{7} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {16 d^{8}}{315 e^{8}} - \frac {8 d^{6} x^{2}}{315 e^{6}} - \frac {2 d^{4} x^{4}}{105 e^{4}} - \frac {d^{2} x^{6}}{63 e^{2}} + \frac {x^{8}}{9}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{8} \sqrt {d^{2}}}{8} & \text {otherwise} \end {cases}\right ) \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.22 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {3 i \, d^{9} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e^{5}} - \frac {45 \, d^{9} \arcsin \left (\frac {e x}{d}\right )}{128 \, e^{5}} + \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{7} x}{8 \, e^{4}} - \frac {45 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x}{128 \, e^{4}} + \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{8}}{4 \, e^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x}{64 \, e^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x}{16 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, e^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{5 \, e^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x}{8 \, e^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}}{63 \, e^{5}} \]
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Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.66 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 \, d^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, e^{4} {\left | e \right |}} + \frac {1}{40320} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {1024 \, d^{8}}{e^{5}} - {\left (\frac {945 \, d^{7}}{e^{4}} - 2 \, {\left (\frac {256 \, d^{6}}{e^{3}} - {\left (\frac {315 \, d^{5}}{e^{2}} - 4 \, {\left (\frac {48 \, d^{4}}{e} + 5 \, {\left (189 \, d^{3} - 2 \, {\left (80 \, d^{2} e - 7 \, {\left (8 \, e^{3} x - 9 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
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Timed out. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{d+e\,x} \,d x \]
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