Integrand size = 24, antiderivative size = 188 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {55 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e} \]
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Time = 0.05 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {685, 655, 201, 223, 209} \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {55 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2} \]
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Rule 201
Rule 209
Rule 223
Rule 655
Rule 685
Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{9} (11 d) \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx \\ & = -\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{8} \left (11 d^2\right ) \int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx \\ & = -\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{8} \left (11 d^3\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx \\ & = \frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{48} \left (55 d^5\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx \\ & = \frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{64} \left (55 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx \\ & = \frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{128} \left (55 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{128} \left (55 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 ) \\ & = \frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {55 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.82 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-3712 d^8+4599 d^7 e x+10240 d^6 e^2 x^2+3066 d^5 e^3 x^3-8448 d^4 e^4 x^4-7224 d^3 e^5 x^5+1024 d^2 e^6 x^6+3024 d e^7 x^7+896 e^8 x^8\right )}{8064 e}-\frac {55 d^9 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{128 \sqrt {-e^2}} \]
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Time = 0.40 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-\frac {\left (-896 e^{8} x^{8}-3024 d \,e^{7} x^{7}-1024 d^{2} e^{6} x^{6}+7224 d^{3} e^{5} x^{5}+8448 d^{4} x^{4} e^{4}-3066 d^{5} e^{3} x^{3}-10240 d^{6} e^{2} x^{2}-4599 d^{7} e x +3712 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{8064 e}+\frac {55 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 \sqrt {e^{2}}}\) | \(138\) |
default | \(d^{3} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )+e^{3} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )+3 d \,e^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )-\frac {3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e}\) | \(304\) |
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Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.74 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {6930 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (896 \, e^{8} x^{8} + 3024 \, d e^{7} x^{7} + 1024 \, d^{2} e^{6} x^{6} - 7224 \, d^{3} e^{5} x^{5} - 8448 \, d^{4} e^{4} x^{4} + 3066 \, d^{5} e^{3} x^{3} + 10240 \, d^{6} e^{2} x^{2} + 4599 \, d^{7} e x - 3712 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8064 \, e} \]
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Time = 0.62 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.10 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {55 d^{9} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {29 d^{8}}{63 e} + \frac {73 d^{7} x}{128} + \frac {80 d^{6} e x^{2}}{63} + \frac {73 d^{5} e^{2} x^{3}}{192} - \frac {22 d^{4} e^{3} x^{4}}{21} - \frac {43 d^{3} e^{4} x^{5}}{48} + \frac {8 d^{2} e^{5} x^{6}}{63} + \frac {3 d e^{6} x^{7}}{8} + \frac {e^{7} x^{8}}{9}\right ) & \text {for}\: e^{2} \neq 0 \\\left (d^{2}\right )^{\frac {5}{2}} \left (\begin {cases} d^{3} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{4}}{4 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.77 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {55 \, d^{9} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{128 \, \sqrt {e^{2}}} + \frac {55}{128} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x + \frac {55}{192} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x + \frac {11}{48} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x - \frac {1}{9} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{2} - \frac {3}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x - \frac {29 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}}{63 \, e} \]
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Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.68 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {55 \, d^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, {\left | e \right |}} - \frac {1}{8064} \, {\left (\frac {3712 \, d^{8}}{e} - {\left (4599 \, d^{7} + 2 \, {\left (5120 \, d^{6} e + {\left (1533 \, d^{5} e^{2} - 4 \, {\left (1056 \, d^{4} e^{3} + {\left (903 \, d^{3} e^{4} - 2 \, {\left (64 \, d^{2} e^{5} + 7 \, {\left (8 \, e^{7} x + 27 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
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Timed out. \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int {\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]
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