Integrand size = 24, antiderivative size = 142 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=\frac {7}{8} d^3 x \sqrt {d^2-e^2 x^2}+\frac {7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac {7 d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e} \]
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Time = 0.04 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {669, 685, 655, 201, 223, 209} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=\frac {7 d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e}+\frac {7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac {7}{8} d^3 x \sqrt {d^2-e^2 x^2} \]
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Rule 201
Rule 209
Rule 223
Rule 655
Rule 669
Rule 685
Rubi steps \begin{align*} \text {integral}& = \int (d-e x)^3 \sqrt {d^2-e^2 x^2} \, dx \\ & = \frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac {1}{5} (7 d) \int (d-e x)^2 \sqrt {d^2-e^2 x^2} \, dx \\ & = \frac {7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac {1}{4} \left (7 d^2\right ) \int (d-e x) \sqrt {d^2-e^2 x^2} \, dx \\ & = \frac {7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac {1}{4} \left (7 d^3\right ) \int \sqrt {d^2-e^2 x^2} \, dx \\ & = \frac {7}{8} d^3 x \sqrt {d^2-e^2 x^2}+\frac {7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac {1}{8} \left (7 d^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {7}{8} d^3 x \sqrt {d^2-e^2 x^2}+\frac {7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac {1}{8} \left (7 d^5\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 {7}{8} d^3 x \sqrt {d^2-e^2 x^2}+\frac {7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac {7 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.78 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (136 d^4+15 d^3 e x-112 d^2 e^2 x^2+90 d e^3 x^3-24 e^4 x^4\right )}{120 e}-\frac {7 d^5 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{8 \sqrt {-e^2}} \]
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Time = 2.44 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {\left (-24 e^{4} x^{4}+90 d \,e^{3} x^{3}-112 d^{2} e^{2} x^{2}+15 d^{3} e x +136 d^{4}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{120 e}+\frac {7 d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}\) | \(94\) |
default | \(\frac {\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}\right )}{d}}{e^{3}}\) | \(352\) |
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Time = 0.39 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=-\frac {210 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (24 \, e^{4} x^{4} - 90 \, d e^{3} x^{3} + 112 \, d^{2} e^{2} x^{2} - 15 \, d^{3} e x - 136 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, e} \]
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Time = 3.33 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.23 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=d^{3} \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) - 3 d^{2} e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2}}{3 e^{2}} + \frac {x^{2}}{3}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{2} \sqrt {d^{2}}}{2} & \text {otherwise} \end {cases}\right ) + 3 d e^{2} \left (\begin {cases} \frac {d^{4} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=-\frac {7 i \, d^{5} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e} + \frac {7}{8} \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x + \frac {7 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{4 \, e} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{5 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{20 \, {\left (e^{2} x + d e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{12 \, e} \]
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Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.56 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=\frac {7 \, d^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, {\left | e \right |}} + \frac {1}{120} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {136 \, d^{4}}{e} + {\left (15 \, d^{3} - 2 \, {\left (56 \, d^{2} e + 3 \, {\left (4 \, e^{3} x - 15 \, d e^{2}\right )} x\right )} x\right )} x\right )} \]
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Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^3} \,d x \]
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