Integrand size = 28, antiderivative size = 110 \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=-\frac {2 \sqrt {\frac {\sqrt {-c} (f+g x)}{\sqrt {-c} f+g}} \operatorname {EllipticPi}\left (\frac {2 e}{\sqrt {-c} d+e},\arcsin \left (\frac {\sqrt {1-\sqrt {-c} x}}{\sqrt {2}}\right ),\frac {2 g}{\sqrt {-c} f+g}\right )}{\left (\sqrt {-c} d+e\right ) \sqrt {f+g x}} \]
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Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {946, 174, 552, 551} \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=-\frac {2 \sqrt {\frac {\sqrt {-c} (f+g x)}{\sqrt {-c} f+g}} \operatorname {EllipticPi}\left (\frac {2 e}{\sqrt {-c} d+e},\arcsin \left (\frac {\sqrt {1-\sqrt {-c} x}}{\sqrt {2}}\right ),\frac {2 g}{\sqrt {-c} f+g}\right )}{\left (\sqrt {-c} d+e\right ) \sqrt {f+g x}} \]
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Rule 174
Rule 551
Rule 552
Rule 946
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {1-\sqrt {-c} x} \sqrt {1+\sqrt {-c} x} (d+e x) \sqrt {f+g x}} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\sqrt {-c} d+e-e x^2\right ) \sqrt {f+\frac {g}{\sqrt {-c}}-\frac {g x^2}{\sqrt {-c}}}} \, dx,x,\sqrt {1-\sqrt {-c} x}\right )\right ) \\ & = -\frac {\left (2 \sqrt {1+\frac {g \left (-1+\sqrt {-c} x\right )}{\sqrt {-c} f+g}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\sqrt {-c} d+e-e x^2\right ) \sqrt {1-\frac {g x^2}{\sqrt {-c} \left (f+\frac {g}{\sqrt {-c}}\right )}}} \, dx,x,\sqrt {1-\sqrt {-c} x}\right )}{\sqrt {f+g x}} \\ & = -\frac {2 \sqrt {1-\frac {g \left (1-\sqrt {-c} x\right )}{\sqrt {-c} f+g}} \Pi \left (\frac {2 e}{\sqrt {-c} d+e};\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c} x}}{\sqrt {2}}\right )|\frac {2 g}{\sqrt {-c} f+g}\right )}{\left (\sqrt {-c} d+e\right ) \sqrt {f+g x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 22.00 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.37 \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=-\frac {2 i \sqrt {\frac {g \left (\frac {i}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i g}{\sqrt {c}}-g x}{f+g x}} (f+g x) \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i g}{\sqrt {c} f+i g}\right )-\operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i g}{\sqrt {c} f+i g}\right )\right )}{\sqrt {-f-\frac {i g}{\sqrt {c}}} (e f-d g) \sqrt {1+c x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(214\) vs. \(2(95)=190\).
Time = 1.81 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.95
method | result | size |
default | \(\frac {2 \left (g +f \sqrt {-c}\right ) \Pi \left (\sqrt {\frac {\left (g x +f \right ) \sqrt {-c}}{g +f \sqrt {-c}}}, -\frac {\left (g +f \sqrt {-c}\right ) e}{\sqrt {-c}\, \left (d g -e f \right )}, \sqrt {\frac {g +f \sqrt {-c}}{f \sqrt {-c}-g}}\right ) \sqrt {-\frac {\left (x \sqrt {-c}-1\right ) g}{g +f \sqrt {-c}}}\, \sqrt {-\frac {\left (x \sqrt {-c}+1\right ) g}{f \sqrt {-c}-g}}\, \sqrt {\frac {\left (g x +f \right ) \sqrt {-c}}{g +f \sqrt {-c}}}\, \sqrt {c \,x^{2}+1}\, \sqrt {g x +f}}{\sqrt {-c}\, \left (d g -e f \right ) \left (c g \,x^{3}+c f \,x^{2}+g x +f \right )}\) | \(215\) |
elliptic | \(\frac {2 \sqrt {\left (g x +f \right ) \left (c \,x^{2}+1\right )}\, \left (\frac {f}{g}+\frac {1}{\sqrt {-c}}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}+\frac {1}{\sqrt {-c}}}}\, \sqrt {\frac {x +\frac {1}{\sqrt {-c}}}{-\frac {f}{g}+\frac {1}{\sqrt {-c}}}}\, \sqrt {\frac {x -\frac {1}{\sqrt {-c}}}{-\frac {f}{g}-\frac {1}{\sqrt {-c}}}}\, \Pi \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}+\frac {1}{\sqrt {-c}}}}, \frac {-\frac {f}{g}-\frac {1}{\sqrt {-c}}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}-\frac {1}{\sqrt {-c}}}{-\frac {f}{g}+\frac {1}{\sqrt {-c}}}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+1}\, e \sqrt {c g \,x^{3}+c f \,x^{2}+g x +f}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\) | \(240\) |
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Timed out. \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\int \frac {1}{\left (d + e x\right ) \sqrt {f + g x} \sqrt {c x^{2} + 1}}\, dx \]
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\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + 1} {\left (e x + d\right )} \sqrt {g x + f}} \,d x } \]
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\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + 1} {\left (e x + d\right )} \sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+1}\,\left (d+e\,x\right )} \,d x \]
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