Integrand size = 33, antiderivative size = 588 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {\sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+b x+c x^2\right )}{\left (c f^2-b f g+a g^2\right ) (d+e x)^2}} \left (1+\frac {\sqrt {c d^2-b d e+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}\right ) \sqrt {\frac {1-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b f g+a g^2\right ) (d+e x)}+\frac {\left (c d^2-b d e+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}}{\left (1+\frac {\sqrt {c d^2-b d e+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b d e+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b f g+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (2+\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}\right )\right )}{\sqrt [4]{c d^2-b d e+a e^2} (e f-d g) \sqrt {a+b x+c x^2} \sqrt {1-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b f g+a g^2\right ) (d+e x)}+\frac {\left (c d^2-b d e+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}}} \]
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Time = 0.56 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {949, 1117} \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {(d+e x) \sqrt [4]{c f^2-g (b f-a g)} \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt {a+b x+c x^2} (e f-d g) \sqrt [4]{a e^2-b d e+c d^2} \sqrt {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}} \]
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Rule 949
Rule 1117
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (2 (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+b x+c x^2\right )}{\left (c f^2-b f g+a g^2\right ) (d+e x)^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {(2 c d f-b e f-b d g+2 a e g) x^2}{c f^2-b f g+a g^2}+\frac {\left (c d^2-b d e+a e^2\right ) x^4}{c f^2-b f g+a g^2}}} \, dx,x,\frac {\sqrt {f+g x}}{\sqrt {d+e x}}\right )}{(e f-d g) \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+b x+c x^2\right )}{\left (c f^2-b f g+a g^2\right ) (d+e x)^2}} \left (1+\frac {\sqrt {c d^2-b d e+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}\right ) \sqrt {\frac {1-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b f g+a g^2\right ) (d+e x)}+\frac {\left (c d^2-b d e+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}}{\left (1+\frac {\sqrt {c d^2-b d e+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c d^2-b d e+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b f g+a g^2} \sqrt {d+e x}}\right )|\frac {1}{4} \left (2+\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}\right )\right )}{\sqrt [4]{c d^2-b d e+a e^2} (e f-d g) \sqrt {a+b x+c x^2} \sqrt {1-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b f g+a g^2\right ) (d+e x)}+\frac {\left (c d^2-b d e+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}}} \\ \end{align*}
Time = 26.55 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {2} e \sqrt {-\frac {e \left (c d^2+e (-b d+a e)\right ) (f+g x)}{\left (-2 c d e f+e \sqrt {\left (b^2-4 a c\right ) e^2} f-2 a e^2 g-d \sqrt {\left (b^2-4 a c\right ) e^2} g+b e (e f+d g)\right ) (d+e x)}} \sqrt {a+x (b+c x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2 a e^2-2 c d e x+b e (-d+e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}}}{\sqrt {2}}\right ),\frac {2 \sqrt {\left (b^2-4 a c\right ) e^2} (e f-d g)}{-2 c d e f+e \sqrt {\left (b^2-4 a c\right ) e^2} f-2 a e^2 g-d \sqrt {\left (b^2-4 a c\right ) e^2} g+b e (e f+d g)}\right )}{\sqrt {\left (b^2-4 a c\right ) e^2} \sqrt {d+e x} \sqrt {f+g x} \sqrt {-\frac {\left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))}{\left (b^2-4 a c\right ) (d+e x)^2}}} \]
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Time = 6.06 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {8 \left (-2 c^{2} d g \,x^{2}+2 c^{2} e f \,x^{2}+2 \sqrt {-4 a c +b^{2}}\, c d g x -2 \sqrt {-4 a c +b^{2}}\, c e f x -2 b c d g x +2 b c e f x +\sqrt {-4 a c +b^{2}}\, b d g -\sqrt {-4 a c +b^{2}}\, b e f +2 a c d g -2 a c e f -b^{2} d g +b^{2} e f \right ) F\left (\sqrt {-\frac {\left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right ) \left (g x +f \right )}{\left (d g -e f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}}, 2 \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, \left (d g -e f \right ) c}{\left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right ) \left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right )}}\right ) \sqrt {\frac {\left (2 c f -b g +g \sqrt {-4 a c +b^{2}}\right ) \left (e x +d \right )}{\left (d g -e f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {\frac {\left (2 c f -b g +g \sqrt {-4 a c +b^{2}}\right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {-\frac {\left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right ) \left (g x +f \right )}{\left (d g -e f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {g x +f}\, \sqrt {e x +d}}{\sqrt {-\frac {\left (g x +f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (e x +d \right )}{c}}\, \left (b g -2 c f -g \sqrt {-4 a c +b^{2}}\right ) \left (-e \sqrt {-4 a c +b^{2}}+b e -2 c d \right ) \sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right ) \left (e x +d \right )}}\) | \(601\) |
elliptic | \(\frac {2 \sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right ) \left (e x +d \right )}\, \left (-\frac {f}{g}+\frac {d}{e}\right ) \sqrt {\frac {\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {f}{g}\right )}{\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, {\left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2} \sqrt {\frac {\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{\left (\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, \sqrt {\frac {\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \left (x +\frac {d}{e}\right )}{\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {f}{g}\right )}{\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}, \sqrt {\frac {\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (-\frac {f}{g}+\frac {d}{e}\right )}{\left (-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}\right )}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {e x +d}\, \left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {c e g \left (x +\frac {f}{g}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {d}{e}\right )}}\) | \(621\) |
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\sqrt {d + e x} \sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {d+e\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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