Integrand size = 24, antiderivative size = 143 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {14 (d+e x)^3}{3 e \sqrt {d^2-e^2 x^2}}-\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {35 (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}+\frac {35 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
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Time = 0.04 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {683, 685, 655, 223, 209} \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {35 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {14 (d+e x)^3}{3 e \sqrt {d^2-e^2 x^2}}-\frac {35 \sqrt {d^2-e^2 x^2} (d+e x)}{6 e}-\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e} \]
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Rule 209
Rule 223
Rule 655
Rule 683
Rule 685
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx \\ & = \frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {14 (d+e x)^3}{3 e \sqrt {d^2-e^2 x^2}}+\frac {35}{3} \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {14 (d+e x)^3}{3 e \sqrt {d^2-e^2 x^2}}-\frac {35 (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}+\frac {1}{2} (35 d) \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {14 (d+e x)^3}{3 e \sqrt {d^2-e^2 x^2}}-\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {35 (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}+\frac {1}{2} \left (35 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {14 (d+e x)^3}{3 e \sqrt {d^2-e^2 x^2}}-\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {35 (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}+\frac {1}{2} \left (35 d^2\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 {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {14 (d+e x)^3}{3 e \sqrt {d^2-e^2 x^2}}-\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {35 (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}+\frac {35 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.70 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {\frac {\sqrt {d^2-e^2 x^2} \left (164 d^3-229 d^2 e x+30 d e^2 x^2+3 e^3 x^3\right )}{(d-e x)^2}+210 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{6 e} \]
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Time = 2.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.09
method | result | size |
risch | \(-\frac {\left (e x +12 d \right ) \sqrt {-x^{2} e^{2}+d^{2}}}{2 e}+\frac {35 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}+\frac {80 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{3 e^{2} \left (x -\frac {d}{e}\right )}+\frac {16 d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{3 e^{3} \left (x -\frac {d}{e}\right )^{2}}\) | \(156\) |
default | \(d^{6} \left (\frac {x}{3 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}\right )+e^{6} \left (-\frac {x^{5}}{2 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {5 d^{2} \left (\frac {x^{3}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-x^{2} e^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}\right )}{2 e^{2}}\right )+6 d \,e^{5} \left (-\frac {x^{4}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 d^{2} \left (\frac {x^{2}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}\right )}{e^{2}}\right )+\frac {2 d^{5}}{e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+15 d^{2} e^{4} \left (\frac {x^{3}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-x^{2} e^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}\right )+20 d^{3} e^{3} \left (\frac {x^{2}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}\right )+15 d^{4} e^{2} \left (\frac {x}{2 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 e^{2}}\right )\) | \(483\) |
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Time = 0.37 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {164 \, d^{2} e^{2} x^{2} - 328 \, d^{3} e x + 164 \, d^{4} + 210 \, {\left (d^{2} e^{2} x^{2} - 2 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (3 \, e^{3} x^{3} + 30 \, d e^{2} x^{2} - 229 \, d^{2} e x + 164 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, {\left (e^{3} x^{2} - 2 \, d e^{2} x + d^{2} e\right )}} \]
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\[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{6}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {35}{6} \, d^{2} e^{4} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} - \frac {e^{4} x^{5}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {6 \, d e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {44 \, d^{3} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {16 \, d^{4} x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {82 \, d^{5}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} - \frac {61 \, d^{2} x}{6 \, \sqrt {-e^{2} x^{2} + d^{2}}} + \frac {35 \, d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {35 \, d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} - \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (x + \frac {12 \, d}{e}\right )} - \frac {32 \, {\left (4 \, d^{2} - \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2}}{e^{2} x} + \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2}}{e^{4} x^{2}}\right )}}{3 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{3} {\left | e \right |}} \]
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Timed out. \[ \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^6}{{\left (d^2-e^2\,x^2\right )}^{5/2}} \,d x \]
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