Integrand size = 25, antiderivative size = 89 \[ \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {803, 653, 197} \[ \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}} \]
[In]
[Out]
Rule 197
Rule 653
Rule 803
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 e} \\ & = \frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e} \\ & = \frac {(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (d^3-2 d^2 e x+8 d e^2 x^2-4 e^3 x^3\right )}{15 d^3 e^2 (d-e x)^3 (d+e x)} \]
[In]
[Out]
Time = 0.39 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {\left (-e x +d \right ) \left (e x +d \right )^{3} \left (-4 e^{3} x^{3}+8 d \,e^{2} x^{2}-2 d^{2} e x +d^{3}\right )}{15 d^{3} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(64\) |
trager | \(\frac {\left (-4 e^{3} x^{3}+8 d \,e^{2} x^{2}-2 d^{2} e x +d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} \left (-e x +d \right )^{3} e^{2} \left (e x +d \right )}\) | \(66\) |
default | \(e^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+\frac {d^{2}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+2 d e \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )\) | \(173\) |
[In]
[Out]
none
Time = 0.57 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.31 \[ \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e^{4} x^{4} - 2 \, d e^{3} x^{3} + 2 \, d^{3} e x - d^{4} + {\left (4 \, e^{3} x^{3} - 8 \, d e^{2} x^{2} + 2 \, d^{2} e x - d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{6} x^{4} - 2 \, d^{4} e^{5} x^{3} + 2 \, d^{6} e^{3} x - d^{7} e^{2}\right )}} \]
[In]
[Out]
\[ \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x \left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.22 \[ \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {2 \, x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e} - \frac {4 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e} \]
[In]
[Out]
\[ \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} x}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
Time = 11.33 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.73 \[ \int \frac {x (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (d^3-2\,d^2\,e\,x+8\,d\,e^2\,x^2-4\,e^3\,x^3\right )}{15\,d^3\,e^2\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^3} \]
[In]
[Out]