Integrand size = 29, antiderivative size = 107 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \arcsin (a x)}{d^2}+\frac {(a c-d)^2 \arctan \left (\frac {d+a^2 c x}{\sqrt {a^2 c^2-d^2} \sqrt {1-a^2 x^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}} \]
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Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1668, 858, 222, 739, 210} \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=\frac {(a c-d)^2 \arctan \left (\frac {a^2 c x+d}{\sqrt {1-a^2 x^2} \sqrt {a^2 c^2-d^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}}-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {\arcsin (a x) (a c-2 d)}{d^2} \]
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Rule 210
Rule 222
Rule 739
Rule 858
Rule 1668
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2}}{d}-\frac {\int \frac {-a^2 d^2+a^3 (a c-2 d) d x}{(c+d x) \sqrt {1-a^2 x^2}} \, dx}{a^2 d^2} \\ & = -\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a (a c-2 d)) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{d^2}+\frac {(a c-d)^2 \int \frac {1}{(c+d x) \sqrt {1-a^2 x^2}} \, dx}{d^2} \\ & = -\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2}-\frac {(a c-d)^2 \text {Subst}\left (\int \frac {1}{-a^2 c^2+d^2-x^2} \, dx,x,\frac {d+a^2 c x}{\sqrt {1-a^2 x^2}}\right )}{d^2} \\ & = -\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2}+\frac {(a c-d)^2 \tan ^{-1}\left (\frac {d+a^2 c x}{\sqrt {a^2 c^2-d^2} \sqrt {1-a^2 x^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.21 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=\frac {-d \sqrt {1-a^2 x^2}+(-2 a c+4 d) \arctan \left (\frac {a x}{-1+\sqrt {1-a^2 x^2}}\right )-\frac {2 (a c-d) \sqrt {a^2 c^2-d^2} \arctan \left (\frac {\sqrt {a^2 c^2-d^2} x}{c+d x-c \sqrt {1-a^2 x^2}}\right )}{a c+d}}{d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(232\) vs. \(2(99)=198\).
Time = 0.47 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.18
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{d \sqrt {-a^{2} x^{2}+1}}-\frac {\frac {a \left (a c -2 d \right ) \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{d \sqrt {a^{2}}}-\frac {\left (-a^{2} c^{2}+2 a c d -d^{2}\right ) \ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}}{d}\) | \(233\) |
default | \(-\frac {a \left (-\frac {2 d \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {a c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {d \sqrt {-a^{2} x^{2}+1}}{a}\right )}{d^{2}}-\frac {\left (a^{2} c^{2}-2 a c d +d^{2}\right ) \ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{3} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}\) | \(242\) |
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Time = 0.39 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.97 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=\left [-\frac {{\left (a c - d\right )} \sqrt {-\frac {a c - d}{a c + d}} \log \left (\frac {a^{2} c d x + d^{2} - {\left (a^{2} c^{2} - d^{2}\right )} \sqrt {-a^{2} x^{2} + 1} - {\left (a c d + d^{2} + {\left (a^{3} c^{2} + a^{2} c d\right )} x + \sqrt {-a^{2} x^{2} + 1} {\left (a c d + d^{2}\right )}\right )} \sqrt {-\frac {a c - d}{a c + d}}}{d x + c}\right ) - 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}, \frac {2 \, {\left (a c - d\right )} \sqrt {\frac {a c - d}{a c + d}} \arctan \left (\frac {{\left (d x - \sqrt {-a^{2} x^{2} + 1} c + c\right )} \sqrt {\frac {a c - d}{a c + d}}}{{\left (a c - d\right )} x}\right ) + 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}\right ] \]
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\[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=\int \frac {\left (a x + 1\right )^{2}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (c + d x\right )}\, dx \]
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Exception generated. \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.22 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=-\frac {{\left (a^{2} c - 2 \, a d\right )} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{d^{2} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{d} - \frac {2 \, {\left (a^{3} c^{2} - 2 \, a^{2} c d + a d^{2}\right )} \arctan \left (\frac {d + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{a x}}{\sqrt {a^{2} c^{2} - d^{2}}}\right )}{\sqrt {a^{2} c^{2} - d^{2}} d^{2} {\left | a \right |}} \]
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Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.38 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2\,x^2}}{d}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\left (2\,a\,\sqrt {-a^2}-\frac {a^2\,c\,\sqrt {-a^2}}{d}\right )}{a^2\,d}-\frac {\left (\ln \left (\sqrt {1-\frac {a^2\,c^2}{d^2}}\,\sqrt {1-a^2\,x^2}+\frac {a^2\,c\,x}{d}+1\right )-\ln \left (c+d\,x\right )\right )\,\left (a^2\,c^2-2\,a\,c\,d+d^2\right )}{d^3\,\sqrt {1-\frac {a^2\,c^2}{d^2}}} \]
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