Integrand size = 52, antiderivative size = 88 \[ \int \frac {e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {e f \arctan \left (\frac {a b+\left (4 a^2+b^2-2 a c\right ) x+a b x^2}{2 a \sqrt {2 a-c} \sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{a \sqrt {2 a-c} d} \]
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Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2109} \[ \int \frac {e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {e f \arctan \left (\frac {x \left (4 a^2-2 a c+b^2\right )+a b x^2+a b}{2 a \sqrt {2 a-c} \sqrt {a x^4+a+b x^3+b x+c x^2}}\right )}{a d \sqrt {2 a-c}} \]
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Rule 2109
Rubi steps \begin{align*} \text {integral}& = \frac {e f \tan ^{-1}\left (\frac {a b+\left (4 a^2+b^2-2 a c\right ) x+a b x^2}{2 a \sqrt {2 a-c} \sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{a \sqrt {2 a-c} d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(246\) vs. \(2(88)=176\).
Time = 0.94 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.80 \[ \int \frac {e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {e f \left (-2 \sqrt {-2 a+c} \arctan \left (\frac {b \left (-\sqrt {-2 a+c} x+\sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{2 a \sqrt {2 a-c} \left (1+x^2\right )}\right )+\sqrt {2 a-c} \left (2 \log \left (-\sqrt {-2 a+c} x+\sqrt {a+b x+c x^2+b x^3+a x^4}\right )-\log \left (a b^2 \left (-1+x^2\right )^2+8 a^3 \left (1+x^2\right )^2-4 a^2 c \left (1+x^2\right )^2+b^2 x \left (b+2 c x+b x^2-2 \sqrt {-2 a+c} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )\right )\right )\right )}{2 a \sqrt {-(-2 a+c)^2} d} \]
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Time = 3.96 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {e f \ln \left (\frac {2 \sqrt {-2 a +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}\, a -4 a^{2} x +\left (-b \,x^{2}+2 c x -b \right ) a -b^{2} x}{a \,x^{2}+b x +a}\right )}{d \sqrt {-2 a +c}\, a}\) | \(92\) |
pseudoelliptic | \(\frac {e f \ln \left (\frac {2 \sqrt {-2 a +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}\, a -4 a^{2} x +\left (-b \,x^{2}+2 c x -b \right ) a -b^{2} x}{a \,x^{2}+b x +a}\right )}{d \sqrt {-2 a +c}\, a}\) | \(92\) |
elliptic | \(\text {Expression too large to display}\) | \(254498\) |
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Time = 2.16 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.68 \[ \int \frac {e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\left [-\frac {\sqrt {-2 \, a + c} e f \log \left (\frac {2 \, a b^{3} x^{3} + 2 \, a b^{3} x - {\left (8 \, a^{4} - a^{2} b^{2} - 4 \, a^{3} c\right )} x^{4} - 8 \, a^{4} + a^{2} b^{2} + 4 \, a^{3} c + {\left (16 \, a^{4} + 10 \, a^{2} b^{2} + b^{4} + 8 \, a^{2} c^{2} - 4 \, {\left (6 \, a^{3} + a b^{2}\right )} c\right )} x^{2} - 4 \, {\left (a^{2} b x^{2} + a^{2} b + {\left (4 \, a^{3} + a b^{2} - 2 \, a^{2} c\right )} x\right )} \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt {-2 \, a + c}}{a^{2} x^{4} + 2 \, a b x^{3} + 2 \, a b x + {\left (2 \, a^{2} + b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (2 \, a^{2} - a c\right )} d}, -\frac {\sqrt {2 \, a - c} e f \arctan \left (\frac {2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt {2 \, a - c} a}{a b x^{2} + a b + {\left (4 \, a^{2} + b^{2} - 2 \, a c\right )} x}\right )}{{\left (2 \, a^{2} - a c\right )} d}\right ] \]
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\[ \int \frac {e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=- \frac {e f \left (\int \frac {x^{2}}{a x^{2} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} + a \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} + b x \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx + \int \left (- \frac {1}{a x^{2} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} + a \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} + b x \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\right )\, dx\right )}{d} \]
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\[ \int \frac {e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {e f x^{2} - e f}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (a d x^{2} + b d x + a d\right )}} \,d x } \]
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\[ \int \frac {e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {e f x^{2} - e f}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (a d x^{2} + b d x + a d\right )}} \,d x } \]
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Timed out. \[ \int \frac {e f-e f x^2}{\left (a d+b d x+a d x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {e\,f-e\,f\,x^2}{\left (a\,d\,x^2+b\,d\,x+a\,d\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \]
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