Integrand size = 21, antiderivative size = 136 \[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {733, 435} \[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]
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Rule 435
Rule 733
Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}} \\ & = -\frac {2 \sqrt {-a} \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 20.38 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {2 i \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {\frac {g \left (\sqrt {a}+i \sqrt {c} x\right )}{-i \sqrt {c} f+\sqrt {a} g}} \sqrt {f+g x} \left (E\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{\sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{g \left (i \sqrt {a}+\sqrt {c} x\right )}} \sqrt {a+c x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(395\) vs. \(2(108)=216\).
Time = 0.44 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.91
method | result | size |
default | \(\frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, \left (c f -g \sqrt {-a c}\right ) \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}+c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}-c f}}\, \left (\sqrt {-a c}\, F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) g -\sqrt {-a c}\, E\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) g +f F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c -E\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c f \right )}{g \left (c g \,x^{3}+c f \,x^{2}+a g x +f a \right ) c^{2}}\) | \(396\) |
elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 f \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 g \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(556\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {c g} f {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) - 3 \, \sqrt {c g} g {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right )\right )}}{3 \, c g} \]
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\[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {f + g x}}{\sqrt {a + c x^{2}}}\, dx \]
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\[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a}} \,d x } \]
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\[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {f+g\,x}}{\sqrt {c\,x^2+a}} \,d x \]
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