Integrand size = 24, antiderivative size = 82 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197} \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 673
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d} \\ & = \frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^3} \\ & = \frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-3 d^4+12 d^3 e x+12 d^2 e^2 x^2-8 d e^3 x^3-8 e^4 x^4\right )}{15 d^5 e (d-e x)^2 (d+e x)^3} \]
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Time = 2.64 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (8 e^{4} x^{4}+8 d \,e^{3} x^{3}-12 d^{2} e^{2} x^{2}-12 d^{3} e x +3 d^{4}\right )}{15 d^{5} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(70\) |
trager | \(-\frac {\left (8 e^{4} x^{4}+8 d \,e^{3} x^{3}-12 d^{2} e^{2} x^{2}-12 d^{3} e x +3 d^{4}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{15 d^{5} \left (e x +d \right )^{3} \left (-e x +d \right )^{2} e}\) | \(79\) |
default | \(\frac {-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}}{e}\) | \(164\) |
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (70) = 140\).
Time = 0.30 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.05 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {3 \, e^{5} x^{5} + 3 \, d e^{4} x^{4} - 6 \, d^{2} e^{3} x^{3} - 6 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x + 3 \, d^{5} + {\left (8 \, e^{4} x^{4} + 8 \, d e^{3} x^{3} - 12 \, d^{2} e^{2} x^{2} - 12 \, d^{3} e x + 3 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{5} e^{6} x^{5} + d^{6} e^{5} x^{4} - 2 \, d^{7} e^{4} x^{3} - 2 \, d^{8} e^{3} x^{2} + d^{9} e^{2} x + d^{10} e\right )}} \]
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\[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e\right )}} + \frac {4 \, x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} + \frac {8 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5}} \]
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\[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}} \,d x } \]
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Time = 10.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^2}\,\left (3\,d^4-12\,d^3\,e\,x-12\,d^2\,e^2\,x^2+8\,d\,e^3\,x^3+8\,e^4\,x^4\right )}{15\,d^5\,e\,{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^2} \]
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