Integrand size = 24, antiderivative size = 81 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]
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Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {683, 667, 223, 209} \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}} \]
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Rule 209
Rule 223
Rule 667
Rule 683
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx \\ & = \frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\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)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {2 \left (-\frac {2 (d-2 e x) \sqrt {d^2-e^2 x^2}}{(d-e x)^2}-3 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )\right )}{3 e} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(73)=146\).
Time = 2.32 (sec) , antiderivative size = 281, normalized size of antiderivative = 3.47
method | result | size |
default | \(d^{4} \left (\frac {x}{3 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}\right )+e^{4} \left (\frac {x^{3}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-x^{2} e^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}\right )+4 d \,e^{3} \left (\frac {x^{2}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}\right )+6 d^{2} e^{2} \left (\frac {x}{2 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 e^{2}}\right )+\frac {4 d^{3}}{3 e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(281\) |
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Time = 0.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (2 \, e^{2} x^{2} - 4 \, d e x + 2 \, d^{2} + 3 \, {\left (e^{2} x^{2} - 2 \, d e x + d^{2}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (2 \, e x - d\right )}\right )}}{3 \, {\left (e^{3} x^{2} - 2 \, d e^{2} x + d^{2} e\right )}} \]
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\[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (73) = 146\).
Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {1}{3} \, e^{4} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {4 \, d e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {7 \, d^{2} x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {4 \, d^{3}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} - \frac {5 \, x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}}} + \frac {\arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {8 \, {\left (\frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} - 1\right )}}{3 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{3} {\left | e \right |}} \]
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Timed out. \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (d^2-e^2\,x^2\right )}^{5/2}} \,d x \]
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