Integrand size = 24, antiderivative size = 77 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 77, 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 = {667, 198, 197} \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \]
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Rule 197
Rule 198
Rule 667
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {3}{5} \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx \\ & = \frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 d^2} \\ & = \frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (2 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3\right )}{5 d^4 e (d-e x)^3 (d+e x)} \]
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Time = 2.46 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{3} \left (-e x +d \right ) \left (2 e^{3} x^{3}-4 d \,e^{2} x^{2}+d^{2} e x +2 d^{3}\right )}{5 d^{4} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(65\) |
trager | \(\frac {\left (2 e^{3} x^{3}-4 d \,e^{2} x^{2}+d^{2} e x +2 d^{3}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{5 d^{4} \left (-e x +d \right )^{3} e \left (e x +d \right )}\) | \(67\) |
default | \(d^{2} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )+e^{2} \left (\frac {x}{4 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {2 d}{5 e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(193\) |
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Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \, e^{4} x^{4} - 4 \, d e^{3} x^{3} + 4 \, d^{3} e x - 2 \, d^{4} - {\left (2 \, e^{3} x^{3} - 4 \, d e^{2} x^{2} + d^{2} e x + 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{4} e^{5} x^{4} - 2 \, d^{5} e^{4} x^{3} + 2 \, d^{7} e^{2} x - d^{8} e\right )}} \]
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\[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \, x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} + \frac {2 \, x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}} \]
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\[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
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Time = 10.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^3+d^2\,e\,x-4\,d\,e^2\,x^2+2\,e^3\,x^3\right )}{5\,d^4\,e\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^3} \]
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