Integrand size = 21, antiderivative size = 322 \[ \int \frac {\sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 g}+\frac {4 \sqrt {-a} \sqrt {c} f \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 g^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} g^2 \sqrt {f+g x} \sqrt {a+c x^2}} \]
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Time = 0.36 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {749, 858, 733, 435, 430} \[ \int \frac {\sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=-\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} g^2 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {4 \sqrt {-a} \sqrt {c} f \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 g^2 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 g} \]
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Rule 430
Rule 435
Rule 733
Rule 749
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 g}+\frac {2 \int \frac {a g-c f x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 g} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 g}+\frac {1}{3} \left (2 \left (a+\frac {c f^2}{g^2}\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx-\frac {(2 c f) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3 g^2} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 g}-\frac {\left (4 a \sqrt {c} f \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 )}{3 \sqrt {-a} g^2 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (4 a \left (a+\frac {c f^2}{g^2}\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} \sqrt {c} \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 g}+\frac {4 \sqrt {-a} \sqrt {c} f \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 )}{3 g^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (a+\frac {c f^2}{g^2}\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\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 )}{3 \sqrt {c} \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 22.50 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (g^2 \left (a+c x^2\right )-\frac {2 \left (f g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (a+c x^2\right )+\sqrt {c} f \left (-i \sqrt {c} f+\sqrt {a} g\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-\sqrt {a} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{3 g^3 \sqrt {a+c x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(583\) vs. \(2(256)=512\).
Time = 0.98 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.81
method | result | size |
risch | \(\frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{3 g}+\frac {2 \left (\frac {2 a g \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {2 c f \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right ) \sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}}{3 g \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(584\) |
elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 g}+\frac {4 a \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{3 \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {4 c f \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{3 g \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(587\) |
default | \(-\frac {2 \sqrt {c \,x^{2}+a}\, \sqrt {g x +f}\, \left (2 \sqrt {-a c}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}+c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}-c f}}\, F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) a \,g^{3}+2 \sqrt {-a c}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}+c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}-c f}}\, F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c \,f^{2} g -2 a c \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}+c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}-c f}}\, E\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) f \,g^{2}-2 \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}+c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}-c f}}\, E\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c^{2} f^{3}-c^{2} g^{3} x^{3}-c^{2} f \,g^{2} x^{2}-a c \,g^{3} x -a c f \,g^{2}\right )}{3 c \left (c g \,x^{3}+c f \,x^{2}+a g x +f a \right ) g^{3}}\) | \(688\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (6 \, \sqrt {c g} c f g {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, \sqrt {c x^{2} + a} \sqrt {g x + f} c g^{2} + 2 \, {\left (c f^{2} + 3 \, a g^{2}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right )}}{9 \, c g^{3}} \]
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\[ \int \frac {\sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {a + c x^{2}}}{\sqrt {f + g x}}\, dx \]
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\[ \int \frac {\sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{\sqrt {g x + f}} \,d x } \]
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\[ \int \frac {\sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{\sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {c\,x^2+a}}{\sqrt {f+g\,x}} \,d x \]
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