Integrand size = 30, antiderivative size = 454 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {\sqrt [4]{c f^2+a g^2} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}} \left (1+\frac {\sqrt {c d^2+a e^2} (f+g x)}{\sqrt {c f^2+a g^2} (d+e x)}\right ) \sqrt {\frac {1-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}{\left (1+\frac {\sqrt {c d^2+a e^2} (f+g x)}{\sqrt {c f^2+a g^2} (d+e x)}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (1+\frac {c d f+a e g}{\sqrt {c d^2+a e^2} \sqrt {c f^2+a g^2}}\right )\right )}{\sqrt [4]{c d^2+a e^2} (e f-d g) \sqrt {a+c x^2} \sqrt {1-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}} \]
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Time = 0.36 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {950, 1117} \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {(d+e x) \sqrt [4]{a g^2+c f^2} \sqrt {\frac {\left (a+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2+c d^2}}{(d+e x) \sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac {2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2+c d^2}}{(d+e x) \sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f+a e g}{\sqrt {c d^2+a e^2} \sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt {a+c x^2} \sqrt [4]{a e^2+c d^2} (e f-d g) \sqrt {\frac {(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac {2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}} \]
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Rule 950
Rule 1117
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (2 (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {(2 c d f+2 a e g) x^2}{c f^2+a g^2}+\frac {\left (c d^2+a e^2\right ) x^4}{c f^2+a g^2}}} \, dx,x,\frac {\sqrt {f+g x}}{\sqrt {d+e x}}\right )}{(e f-d g) \sqrt {a+c x^2}} \\ & = -\frac {\sqrt [4]{c f^2+a g^2} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}} \left (1+\frac {\sqrt {c d^2+a e^2} (f+g x)}{\sqrt {c f^2+a g^2} (d+e x)}\right ) \sqrt {\frac {1-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}{\left (1+\frac {\sqrt {c d^2+a e^2} (f+g x)}{\sqrt {c f^2+a g^2} (d+e x)}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c d^2+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt {d+e x}}\right )|\frac {1}{2} \left (1+\frac {c d f+a e g}{\sqrt {c d^2+a e^2} \sqrt {c f^2+a g^2}}\right )\right )}{\sqrt [4]{c d^2+a e^2} (e f-d g) \sqrt {a+c x^2} \sqrt {1-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 24.18 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {\sqrt {2} \left (i \sqrt {a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {\frac {d-\frac {i \sqrt {a} e}{\sqrt {c}}+\frac {i \sqrt {c} d x}{\sqrt {a}}+e x}{d+e x}} \sqrt {\frac {\left (i \sqrt {c} d+\sqrt {a} e\right ) (f+g x)}{\left (i \sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(e f-d g) \left (i \sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-i \sqrt {a} g\right ) (d+e x)}}\right ),-\frac {\frac {i \sqrt {c} d f}{\sqrt {a}}-e f+d g+\frac {i \sqrt {a} e g}{\sqrt {c}}}{2 e f-2 d g}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right ) \sqrt {\frac {(e f-d g) \left (i \sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-i \sqrt {a} g\right ) (d+e x)}} \sqrt {f+g x} \sqrt {a+c x^2}} \]
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Time = 5.08 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {2 \left (c \,e^{2} f \,x^{2}-\sqrt {-a c}\, e^{2} g \,x^{2}+2 c d e f x -2 \sqrt {-a c}\, d e g x +c \,d^{2} f -\sqrt {-a c}\, d^{2} g \right ) F\left (\sqrt {\frac {\left (e \sqrt {-a c}-c d \right ) \left (g x +f \right )}{\left (g \sqrt {-a c}-c f \right ) \left (e x +d \right )}}, \sqrt {\frac {\left (e \sqrt {-a c}+c d \right ) \left (g \sqrt {-a c}-c f \right )}{\left (g \sqrt {-a c}+c f \right ) \left (e \sqrt {-a c}-c d \right )}}\right ) \sqrt {\frac {\left (d g -e f \right ) \left (c x +\sqrt {-a c}\right )}{\left (g \sqrt {-a c}-c f \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (d g -e f \right ) \left (-c x +\sqrt {-a c}\right )}{\left (g \sqrt {-a c}+c f \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (e \sqrt {-a c}-c d \right ) \left (g x +f \right )}{\left (g \sqrt {-a c}-c f \right ) \left (e x +d \right )}}\, \sqrt {c \,x^{2}+a}\, \sqrt {g x +f}\, \sqrt {e x +d}}{\sqrt {-\frac {\left (g x +f \right ) \left (e x +d \right ) \left (-c x +\sqrt {-a c}\right ) \left (c x +\sqrt {-a c}\right )}{c}}\, \left (-e \sqrt {-a c}+c d \right ) \left (d g -e f \right ) \sqrt {\left (g x +f \right ) \left (e x +d \right ) \left (c \,x^{2}+a \right )}}\) | \(401\) |
elliptic | \(\frac {2 \sqrt {\left (g x +f \right ) \left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {f}{g}+\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {\left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {f}{g}\right )}{\left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {d}{e}\right )}}\, \left (x +\frac {d}{e}\right )^{2} \sqrt {\frac {\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}{\left (\frac {f}{g}+\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {d}{e}\right )}}\, \sqrt {\frac {\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}{\left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {d}{e}\right )}}\, F\left (\sqrt {\frac {\left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {f}{g}\right )}{\left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {d}{e}\right )}}, \sqrt {\frac {\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \left (-\frac {f}{g}+\frac {\sqrt {-a c}}{c}\right )}{\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \left (-\frac {d}{e}+\frac {\sqrt {-a c}}{c}\right )}}\right )}{\sqrt {g x +f}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}\, \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \left (-\frac {d}{e}+\frac {f}{g}\right ) \sqrt {c e g \left (x +\frac {f}{g}\right ) \left (x +\frac {d}{e}\right ) \left (x -\frac {\sqrt {-a c}}{c}\right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}}\) | \(448\) |
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \sqrt {d + e x} \sqrt {f + g x}}\, dx \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}\,\sqrt {d+e\,x}} \,d x \]
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