Integrand size = 24, antiderivative size = 148 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{63 d^7 \sqrt {d^2-e^2 x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197} \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{63 d^7 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 673
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{3 d} \\ & = -\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{21 d^2} \\ & = -\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 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 )^{5/2}} \, dx}{21 d^3} \\ & = \frac {8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{63 d^5} \\ & = \frac {8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{63 d^7 \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-19 d^6+6 d^5 e x+66 d^4 e^2 x^2+56 d^3 e^3 x^3-24 d^2 e^4 x^4-48 d e^5 x^5-16 e^6 x^6\right )}{63 d^7 e (d-e x)^2 (d+e x)^5} \]
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Time = 2.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (16 e^{6} x^{6}+48 d \,e^{5} x^{5}+24 d^{2} e^{4} x^{4}-56 x^{3} d^{3} e^{3}-66 d^{4} e^{2} x^{2}-6 d^{5} e x +19 d^{6}\right )}{63 \left (e x +d \right )^{2} d^{7} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(99\) |
trager | \(-\frac {\left (16 e^{6} x^{6}+48 d \,e^{5} x^{5}+24 d^{2} e^{4} x^{4}-56 x^{3} d^{3} e^{3}-66 d^{4} e^{2} x^{2}-6 d^{5} e x +19 d^{6}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{63 d^{7} \left (e x +d \right )^{5} \left (-e x +d \right )^{2} e}\) | \(101\) |
default | \(\frac {-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{3} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {2 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {5 e \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{7 d}\right )}{3 d}}{e^{3}}\) | \(268\) |
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Time = 0.35 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.58 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {19 \, e^{7} x^{7} + 57 \, d e^{6} x^{6} + 19 \, d^{2} e^{5} x^{5} - 95 \, d^{3} e^{4} x^{4} - 95 \, d^{4} e^{3} x^{3} + 19 \, d^{5} e^{2} x^{2} + 57 \, d^{6} e x + 19 \, d^{7} + {\left (16 \, e^{6} x^{6} + 48 \, d e^{5} x^{5} + 24 \, d^{2} e^{4} x^{4} - 56 \, d^{3} e^{3} x^{3} - 66 \, d^{4} e^{2} x^{2} - 6 \, d^{5} e x + 19 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{63 \, {\left (d^{7} e^{8} x^{7} + 3 \, d^{8} e^{7} x^{6} + d^{9} e^{6} x^{5} - 5 \, d^{10} e^{5} x^{4} - 5 \, d^{11} e^{4} x^{3} + d^{12} e^{3} x^{2} + 3 \, d^{13} e^{2} x + d^{14} e\right )}} \]
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\[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{3}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.70 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{9 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e\right )}} - \frac {2}{21 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e\right )}} - \frac {2}{21 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e\right )}} + \frac {8 \, x}{63 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} + \frac {16 \, x}{63 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7}} \]
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\[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{3}} \,d x } \]
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Time = 10.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {197\,x}{1008\,d^5}-\frac {155}{1008\,d^4\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{36\,d^3\,e\,{\left (d+e\,x\right )}^5}-\frac {13\,\sqrt {d^2-e^2\,x^2}}{252\,d^4\,e\,{\left (d+e\,x\right )}^4}-\frac {23\,\sqrt {d^2-e^2\,x^2}}{336\,d^5\,e\,{\left (d+e\,x\right )}^3}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{63\,d^7\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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