Integrand size = 24, antiderivative size = 103 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d-e x)} \]
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Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {669, 673, 665} \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d-e x)} \]
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Rule 665
Rule 669
Rule 673
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(d-e x)^3 \sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {\sqrt {d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac {2 \int \frac {1}{(d-e x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{5 d} \\ & = \frac {\sqrt {d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac {2 \int \frac {1}{(d-e x) \sqrt {d^2-e^2 x^2}} \, dx}{15 d^2} \\ & = \frac {\sqrt {d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d-e x)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.51 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (7 d^2-6 d e x+2 e^2 x^2\right )}{15 d^3 e (d-e x)^3} \]
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Time = 0.44 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.49
method | result | size |
trager | \(\frac {\left (2 e^{2} x^{2}-6 d e x +7 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} \left (-e x +d \right )^{3} e}\) | \(50\) |
gosper | \(\frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (2 e^{2} x^{2}-6 d e x +7 d^{2}\right )}{15 d^{3} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(55\) |
default | \(d^{3} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+e^{3} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+3 d \,e^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {3 d^{2}}{5 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(246\) |
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Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {7 \, e^{3} x^{3} - 21 \, d e^{2} x^{2} + 21 \, d^{2} e x - 7 \, d^{3} - {\left (2 \, e^{2} x^{2} - 6 \, d e x + 7 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{4} x^{3} - 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x - d^{6} e\right )}} \]
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\[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {7 \, d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {2 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (\frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} - \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} - 7\right )}}{15 \, d^{3} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \]
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Time = 12.66 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.48 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (7\,d^2-6\,d\,e\,x+2\,e^2\,x^2\right )}{15\,d^3\,e\,{\left (d-e\,x\right )}^3} \]
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