Integrand size = 31, antiderivative size = 79 \[ \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \sqrt {1-d^2 x^2}}{6 d^4}+\frac {b \arcsin (d x)}{2 d^3} \]
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Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1623, 1823, 794, 222} \[ \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {\sqrt {1-d^2 x^2} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac {b \arcsin (d x)}{2 d^3}-\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2} \]
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Rule 222
Rule 794
Rule 1623
Rule 1823
Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d^2 x^2}} \, dx \\ & = -\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {\int \frac {x \left (-2 c-3 a d^2-3 b d^2 x\right )}{\sqrt {1-d^2 x^2}} \, dx}{3 d^2} \\ & = -\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \sqrt {1-d^2 x^2}}{6 d^4}+\frac {b \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{2 d^2} \\ & = -\frac {c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \sqrt {1-d^2 x^2}}{6 d^4}+\frac {b \sin ^{-1}(d x)}{2 d^3} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\frac {\sqrt {1-d^2 x^2} \left (-4 c-6 a d^2-3 b d^2 x-2 c d^2 x^2\right )}{6 d^4}+\frac {b \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{d^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.63 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.76
method | result | size |
default | \(-\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (2 \,\operatorname {csgn}\left (d \right ) c \,d^{2} x^{2} \sqrt {-d^{2} x^{2}+1}+3 \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) b \,d^{2} x +6 \,\operatorname {csgn}\left (d \right ) \sqrt {-d^{2} x^{2}+1}\, a \,d^{2}+4 \,\operatorname {csgn}\left (d \right ) \sqrt {-d^{2} x^{2}+1}\, c -3 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b d \right ) \operatorname {csgn}\left (d \right )}{6 d^{4} \sqrt {-d^{2} x^{2}+1}}\) | \(139\) |
risch | \(\frac {\left (2 c \,d^{2} x^{2}+3 b \,d^{2} x +6 a \,d^{2}+4 c \right ) \sqrt {d x +1}\, \left (d x -1\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{6 d^{4} \sqrt {-\left (d x +1\right ) \left (d x -1\right )}\, \sqrt {-d x +1}}+\frac {b \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{2 d^{2} \sqrt {d^{2}}\, \sqrt {-d x +1}\, \sqrt {d x +1}}\) | \(141\) |
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Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99 \[ \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {6 \, b d \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right ) + {\left (2 \, c d^{2} x^{2} + 3 \, b d^{2} x + 6 \, a d^{2} + 4 \, c\right )} \sqrt {d x + 1} \sqrt {-d x + 1}}{6 \, d^{4}} \]
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Timed out. \[ \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {\sqrt {-d^{2} x^{2} + 1} c x^{2}}{3 \, d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} b x}{2 \, d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} a}{d^{2}} + \frac {b \arcsin \left (d x\right )}{2 \, d^{3}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 1} c}{3 \, d^{4}} \]
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Time = 0.36 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\frac {6 \, b d \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right ) - {\left (6 \, a d^{2} + {\left (2 \, {\left (d x + 1\right )} c + 3 \, b d - 4 \, c\right )} {\left (d x + 1\right )} - 3 \, b d + 6 \, c\right )} \sqrt {d x + 1} \sqrt {-d x + 1}}{6 \, d^{4}} \]
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Time = 8.05 (sec) , antiderivative size = 244, normalized size of antiderivative = 3.09 \[ \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {\sqrt {1-d\,x}\,\left (\frac {a}{d^2}+\frac {a\,x}{d}\right )}{\sqrt {d\,x+1}}-\frac {2\,b\,\mathrm {atan}\left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )}{d^3}-\frac {\frac {14\,b\,{\left (\sqrt {1-d\,x}-1\right )}^3}{{\left (\sqrt {d\,x+1}-1\right )}^3}-\frac {14\,b\,{\left (\sqrt {1-d\,x}-1\right )}^5}{{\left (\sqrt {d\,x+1}-1\right )}^5}+\frac {2\,b\,{\left (\sqrt {1-d\,x}-1\right )}^7}{{\left (\sqrt {d\,x+1}-1\right )}^7}-\frac {2\,b\,\left (\sqrt {1-d\,x}-1\right )}{\sqrt {d\,x+1}-1}}{d^3\,{\left (\frac {{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+1\right )}^4}-\frac {\sqrt {1-d\,x}\,\left (\frac {2\,c}{3\,d^4}+\frac {c\,x^3}{3\,d}+\frac {c\,x^2}{3\,d^2}+\frac {2\,c\,x}{3\,d^3}\right )}{\sqrt {d\,x+1}} \]
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