Integrand size = 32, antiderivative size = 135 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}+\frac {2 b c \sqrt {1-d^2 x^2}}{d^4}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}-\frac {\left (2 b^2+c \left (4 a+\frac {3 c}{d^2}\right )\right ) \arcsin (d x)}{2 d^3} \]
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Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {913, 1828, 1829, 655, 222} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b d^2 \left (a+\frac {c}{d^2}\right )}{d^4 \sqrt {1-d^2 x^2}}-\frac {\arcsin (d x) \left (c \left (4 a+\frac {3 c}{d^2}\right )+2 b^2\right )}{2 d^3}+\frac {2 b c \sqrt {1-d^2 x^2}}{d^4}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4} \]
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Rule 222
Rule 655
Rule 913
Rule 1828
Rule 1829
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+b x+c x^2\right )^2}{\left (1-d^2 x^2\right )^{3/2}} \, dx \\ & = \frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}-\int \frac {\frac {c^2+b^2 d^2+2 a c d^2}{d^4}+\frac {2 b c x}{d^2}+\frac {c^2 x^2}{d^2}}{\sqrt {1-d^2 x^2}} \, dx \\ & = \frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}+\frac {\int \frac {-2 b^2-c \left (4 a+\frac {3 c}{d^2}\right )-4 b c x}{\sqrt {1-d^2 x^2}} \, dx}{2 d^2} \\ & = \frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}+\frac {2 b c \sqrt {1-d^2 x^2}}{d^4}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}-\frac {\left (2 b^2+c \left (4 a+\frac {3 c}{d^2}\right )\right ) \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{2 d^2} \\ & = \frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}+\frac {2 b c \sqrt {1-d^2 x^2}}{d^4}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}-\frac {\left (2 b^2+c \left (4 a+\frac {3 c}{d^2}\right )\right ) \sin ^{-1}(d x)}{2 d^3} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {\sqrt {1-d^2 x^2} \left (-8 b c-4 a b d^2-3 c^2 x-2 b^2 d^2 x-4 a c d^2 x-2 a^2 d^4 x+4 b c d^2 x^2+c^2 d^2 x^3\right )}{2 d^4 \left (-1+d^2 x^2\right )}+\frac {\left (-3 c^2-2 b^2 d^2-4 a c d^2\right ) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{d^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs. \(2(125)=250\).
Time = 0.62 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.56
method | result | size |
risch | \(-\frac {c \left (c x +4 b \right ) \left (d x -1\right ) \sqrt {d x +1}\, \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{2 d^{4} \sqrt {-\left (d x -1\right ) \left (d x +1\right )}\, \sqrt {-d x +1}}-\frac {\left (\frac {3 c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {2 b^{2} d^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {4 c \,d^{2} a \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {\left (a^{2} d^{4}+2 b a \,d^{3}+2 c \,d^{2} a +b^{2} d^{2}+2 b c d +c^{2}\right ) \sqrt {-d^{2} \left (x -\frac {1}{d}\right )^{2}-2 d \left (x -\frac {1}{d}\right )}}{d^{2} \left (x -\frac {1}{d}\right )}-\frac {\left (-a^{2} d^{4}+2 b a \,d^{3}-2 c \,d^{2} a -b^{2} d^{2}+2 b c d -c^{2}\right ) \sqrt {-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )}}{d^{2} \left (x +\frac {1}{d}\right )}\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{2 d^{4} \sqrt {-d x +1}\, \sqrt {d x +1}}\) | \(346\) |
default | \(-\frac {\sqrt {-d x +1}\, \left (2 \,\operatorname {csgn}\left (d \right ) d^{5} \sqrt {-d^{2} x^{2}+1}\, a^{2} x -\operatorname {csgn}\left (d \right ) c^{2} d^{3} x^{3} \sqrt {-d^{2} x^{2}+1}-4 \,\operatorname {csgn}\left (d \right ) b c \,d^{3} x^{2} \sqrt {-d^{2} x^{2}+1}+4 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a c \,d^{4} x^{2}+2 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} d^{4} x^{2}+4 \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} a c x +2 \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} b^{2} x +4 \,\operatorname {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, a b +3 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{2} d^{2} x^{2}+3 \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d \,c^{2} x +8 \,\operatorname {csgn}\left (d \right ) d \sqrt {-d^{2} x^{2}+1}\, b c -4 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a c \,d^{2}-2 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} d^{2}-3 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{2}\right ) \operatorname {csgn}\left (d \right )}{2 \left (d x -1\right ) \sqrt {-d^{2} x^{2}+1}\, d^{5} \sqrt {d x +1}}\) | \(381\) |
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Time = 0.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=-\frac {4 \, a b d^{3} + 8 \, b c d - 4 \, {\left (a b d^{5} + 2 \, b c d^{3}\right )} x^{2} - {\left (c^{2} d^{3} x^{3} + 4 \, b c d^{3} x^{2} - 4 \, a b d^{3} - 8 \, b c d - {\left (2 \, a^{2} d^{5} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{3} + 3 \, c^{2} d\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 2 \, {\left (2 \, {\left (b^{2} + 2 \, a c\right )} d^{2} - {\left (2 \, {\left (b^{2} + 2 \, a c\right )} d^{4} + 3 \, c^{2} d^{2}\right )} x^{2} + 3 \, c^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{2 \, {\left (d^{7} x^{2} - d^{5}\right )}} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{2}}{\left (- d x + 1\right )^{\frac {3}{2}} \left (d x + 1\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {a^{2} x}{\sqrt {-d^{2} x^{2} + 1}} - \frac {c^{2} x^{3}}{2 \, \sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {2 \, b c x^{2}}{\sqrt {-d^{2} x^{2} + 1} d^{2}} + \frac {2 \, a b}{\sqrt {-d^{2} x^{2} + 1} d^{2}} + \frac {{\left (b^{2} + 2 \, a c\right )} x}{\sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {{\left (b^{2} + 2 \, a c\right )} \arcsin \left (d x\right )}{d^{3}} + \frac {3 \, c^{2} x}{2 \, \sqrt {-d^{2} x^{2} + 1} d^{4}} - \frac {3 \, c^{2} \arcsin \left (d x\right )}{2 \, d^{5}} + \frac {4 \, b c}{\sqrt {-d^{2} x^{2} + 1} d^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (125) = 250\).
Time = 0.32 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.90 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {\frac {2 \, \sqrt {d x + 1} \sqrt {-d x + 1} {\left ({\left (d x + 1\right )} {\left (\frac {{\left (d x + 1\right )} c^{2}}{d^{4}} + \frac {4 \, b c d^{13} - 3 \, c^{2} d^{12}}{d^{16}}\right )} - \frac {a^{2} d^{16} + 2 \, a b d^{15} + b^{2} d^{14} + 2 \, a c d^{14} + 10 \, b c d^{13} - c^{2} d^{12}}{d^{16}}\right )}}{d x - 1} - \frac {4 \, {\left (2 \, b^{2} d^{2} + 4 \, a c d^{2} + 3 \, c^{2}\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{4}} + \frac {\frac {a^{2} d^{4} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {2 \, a b d^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {b^{2} d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {2 \, a c d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {2 \, b c d {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {c^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}}}{d^{4}} - \frac {{\left (a^{2} d^{4} - 2 \, a b d^{3} + b^{2} d^{2} + 2 \, a c d^{2} - 2 \, b c d + c^{2}\right )} \sqrt {d x + 1}}{d^{4} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}}{4 \, d} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^2}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}} \,d x \]
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