Integrand size = 26, antiderivative size = 88 \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 (1-a x)^4}+\frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 (1-a x)^3} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1653, 807, 673, 665} \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx=\frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 (1-a x)^3}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5} \]
[In]
[Out]
Rule 665
Rule 673
Rule 807
Rule 1653
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^4}+\frac {\int \frac {\left (4 a^2-3 a^3 x\right ) \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx}{a^4} \\ & = \frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^4}+\frac {23 \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx}{7 a^2} \\ & = \frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 (1-a x)^4}+\frac {23 \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^3} \, dx}{35 a^2} \\ & = \frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 (1-a x)^4}+\frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 (1-a x)^3} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.57 \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx=\frac {\sqrt {1-a^2 x^2} \left (2-8 a x+13 a^2 x^2+23 a^3 x^3\right )}{105 a^3 (-1+a x)^4} \]
[In]
[Out]
Time = 0.40 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \left (23 a^{2} x^{2}-10 a x +2\right ) \left (a x +1\right )}{105 \left (a x -1\right )^{4} a^{3}}\) | \(44\) |
trager | \(\frac {\left (23 a^{3} x^{3}+13 a^{2} x^{2}-8 a x +2\right ) \sqrt {-a^{2} x^{2}+1}}{105 \left (a x -1\right )^{4} a^{3}}\) | \(47\) |
default | \(-\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{7 a \left (x -\frac {1}{a}\right )^{5}}-\frac {2 a \left (\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x -\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x -\frac {1}{a}\right )^{3}}\right )}{7}}{a^{7}}-\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a^{6} \left (x -\frac {1}{a}\right )^{3}}-\frac {2 \left (\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x -\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x -\frac {1}{a}\right )^{3}}\right )}{a^{6}}\) | \(258\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.16 \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx=\frac {2 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 8 \, a x + {\left (23 \, a^{3} x^{3} + 13 \, a^{2} x^{2} - 8 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} + 2}{105 \, {\left (a^{7} x^{4} - 4 \, a^{6} x^{3} + 6 \, a^{5} x^{2} - 4 \, a^{4} x + a^{3}\right )}} \]
[In]
[Out]
\[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx=- \int \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} - 5 a^{4} x^{4} + 10 a^{3} x^{3} - 10 a^{2} x^{2} + 5 a x - 1}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (73) = 146\).
Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.74 \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx=\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{7} x^{4} - 4 \, a^{6} x^{3} + 6 \, a^{5} x^{2} - 4 \, a^{4} x + a^{3}\right )}} + \frac {29 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{6} x^{3} - 3 \, a^{5} x^{2} + 3 \, a^{4} x - a^{3}\right )}} + \frac {82 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{5} x^{2} - 2 \, a^{4} x + a^{3}\right )}} + \frac {23 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{4} x - a^{3}\right )}} \]
[In]
[Out]
Exception generated. \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.26 \[ \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx=\frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^7\,x^4-4\,a^6\,x^3+6\,a^5\,x^2-4\,a^4\,x+a^3\right )}+\frac {4\,\sqrt {1-a^2\,x^2}}{3\,\left (a^5\,x^2-2\,a^4\,x+a^3\right )}+\frac {4\,a\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,x^2-2\,a^5\,x+a^4\right )}+\frac {29\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (a\,\sqrt {-a^2}-3\,a^2\,x\,\sqrt {-a^2}+3\,a^3\,x^2\,\sqrt {-a^2}-a^4\,x^3\,\sqrt {-a^2}\right )}+\frac {23\,\sqrt {1-a^2\,x^2}}{105\,\left (a\,\sqrt {-a^2}-a^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {2\,a^2\,\sqrt {1-a^2\,x^2}}{3\,\left (a^7\,x^2-2\,a^6\,x+a^5\right )} \]
[In]
[Out]