Integrand size = 31, antiderivative size = 287 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=-\frac {4 c \text {arctanh}\left (\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}+\frac {4 c \text {arctanh}\left (\frac {\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \]
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Time = 0.19 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {925, 95, 214} \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\frac {4 c \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {f+g x} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {4 c \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {f+g x} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}} \]
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Rule 95
Rule 214
Rule 925
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 c}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}}-\frac {2 c}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}}\right ) \, dx \\ & = \frac {(2 c) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{\sqrt {b^2-4 a c}} \\ & = \frac {(4 c) \text {Subst}\left (\int \frac {1}{-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e-\left (-2 c f+\left (b-\sqrt {b^2-4 a c}\right ) g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c}}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e-\left (-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c}} \\ & = -\frac {4 c \tanh ^{-1}\left (\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}+\frac {4 c \tanh ^{-1}\left (\frac {\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \\ \end{align*}
Time = 2.52 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\frac {\sqrt {2} \sqrt {c d^2+e (-b d+a e)} \left (\frac {\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c d^2-b d e+a e^2} \sqrt {f+g x}}{\sqrt {-2 c d f+b e f+\sqrt {b^2-4 a c} e f+b d g-\sqrt {b^2-4 a c} d g-2 a e g} \sqrt {d+e x}}\right )}{\sqrt {-2 c d f+b e f+\sqrt {b^2-4 a c} e f+b d g-\sqrt {b^2-4 a c} d g-2 a e g}}+\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c d^2-b d e+a e^2} \sqrt {f+g x}}{\sqrt {-2 c d f+b e f-\sqrt {b^2-4 a c} e f+b d g+\sqrt {b^2-4 a c} d g-2 a e g} \sqrt {d+e x}}\right )}{\sqrt {-2 c d f+b e f-\sqrt {b^2-4 a c} e f+b d g+\sqrt {b^2-4 a c} d g-2 a e g}}\right )}{\sqrt {b^2-4 a c} \left (-c d^2+e (b d-a e)\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(5506\) vs. \(2(243)=486\).
Time = 0.66 (sec) , antiderivative size = 5507, normalized size of antiderivative = 19.19
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Leaf count of result is larger than twice the leaf count of optimal. 19727 vs. \(2 (243) = 486\).
Time = 118.80 (sec) , antiderivative size = 19727, normalized size of antiderivative = 68.74 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\int \frac {1}{\sqrt {d + e x} \sqrt {f + g x} \left (a + b x + c x^{2}\right )}\, dx \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\text {Hanged} \]
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