Integrand size = 33, antiderivative size = 94 \[ \int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {2 \sqrt {-2 a-2 b-c} \arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 a+2 b+c} \]
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\[ \int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {2}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx \\ & = 2 \int \frac {1}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a} (-1+x)^2-\sqrt {x \left (b+c x+b x^2\right )+a \left (1+x^4\right )}}\right )}{\sqrt {-2 a-2 b-c}} \]
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Time = 1.36 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (4 a +b \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (x -1\right )^{2}}\right )}{\sqrt {2 a +2 b +c}}\) | \(80\) |
pseudoelliptic | \(-\frac {\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (4 a +b \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (x -1\right )^{2}}\right )}{\sqrt {2 a +2 b +c}}\) | \(80\) |
elliptic | \(\text {Expression too large to display}\) | \(2813\) |
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (82) = 164\).
Time = 0.43 (sec) , antiderivative size = 382, normalized size of antiderivative = 4.06 \[ \int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\left [\frac {\log \left (\frac {{\left (24 \, a^{2} + 16 \, a b + b^{2} + 4 \, a c\right )} x^{4} - 4 \, {\left (8 \, a^{2} - 4 \, a b - 3 \, b^{2} - 2 \, {\left (2 \, a + b\right )} c\right )} x^{3} + 2 \, {\left (24 \, a^{2} + 3 \, b^{2} - 4 \, {\left (a - 2 \, b\right )} c + 4 \, c^{2}\right )} x^{2} - 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left ({\left (4 \, a + b\right )} x^{2} - 2 \, {\left (2 \, a - b - c\right )} x + 4 \, a + b\right )} \sqrt {2 \, a + 2 \, b + c} + 24 \, a^{2} + 16 \, a b + b^{2} + 4 \, a c - 4 \, {\left (8 \, a^{2} - 4 \, a b - 3 \, b^{2} - 2 \, {\left (2 \, a + b\right )} c\right )} x}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )}{2 \, \sqrt {2 \, a + 2 \, b + c}}, \frac {\sqrt {-2 \, a - 2 \, b - c} \arctan \left (\frac {\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left ({\left (4 \, a + b\right )} x^{2} - 2 \, {\left (2 \, a - b - c\right )} x + 4 \, a + b\right )} \sqrt {-2 \, a - 2 \, b - c}}{2 \, {\left ({\left (2 \, a^{2} + 2 \, a b + a c\right )} x^{4} + {\left (2 \, a b + 2 \, b^{2} + b c\right )} x^{3} + {\left (2 \, {\left (a + b\right )} c + c^{2}\right )} x^{2} + 2 \, a^{2} + 2 \, a b + a c + {\left (2 \, a b + 2 \, b^{2} + b c\right )} x\right )}}\right )}{2 \, a + 2 \, b + c}\right ] \]
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\[ \int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {x + 1}{\left (x - 1\right ) \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \]
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\[ \int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { \frac {x + 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {x+1}{\left (x-1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \]
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