Integrand size = 33, antiderivative size = 475 \[ \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {2} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {\frac {(e f-d g) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b f+\sqrt {b^2-4 a c} f-2 a g\right ) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g},\arcsin \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right ),\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} g \sqrt {\frac {2 a c}{b+\sqrt {b^2-4 a c}}+c x} \sqrt {a+b x+c x^2}} \]
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Time = 0.24 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {940} \[ \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {2} (d+e x) \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )} \sqrt {\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) (e f-d g)}{(d+e x) \left (2 c f-g \left (\sqrt {b^2-4 a c}+b\right )\right )}} \sqrt {\frac {\left (x \left (\sqrt {b^2-4 a c}+b\right )+2 a\right ) (e f-d g)}{(d+e x) \left (f \sqrt {b^2-4 a c}-2 a g+b f\right )}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g},\arcsin \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right ),\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{g \sqrt {\frac {2 a c}{\sqrt {b^2-4 a c}+b}+c x} \sqrt {a+b x+c x^2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \]
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Rule 940
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {\frac {(e f-d g) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b f+\sqrt {b^2-4 a c} f-2 a g\right ) (d+e x)}} (d+e x) \Pi \left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right )|\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} g \sqrt {\frac {2 a c}{b+\sqrt {b^2-4 a c}}+c x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1118\) vs. \(2(475)=950\).
Time = 28.87 (sec) , antiderivative size = 1118, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {\sqrt {2} \sqrt {-\frac {g \left (c f^2+g (-b f+a g)\right ) (d+e x)}{\left (-2 c d f g-2 a e g^2+e f \sqrt {\left (b^2-4 a c\right ) g^2}-d g \sqrt {\left (b^2-4 a c\right ) g^2}+b g (e f+d g)\right ) (f+g x)}} (f+g x)^{3/2} \left (\frac {2 e f \sqrt {\left (b^2-4 a c\right ) g^2} \sqrt {-\frac {\left (c f^2+g (-b f+a g)\right ) (a+x (b+c x))}{\left (b^2-4 a c\right ) (f+g x)^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {-2 a g^2+2 c f g x+b g (f-g x)+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}}}{\sqrt {2}}\right ),\frac {2 \sqrt {\left (b^2-4 a c\right ) g^2} (-e f+d g)}{2 c d f g+2 a e g^2-e f \sqrt {\left (b^2-4 a c\right ) g^2}+d g \sqrt {\left (b^2-4 a c\right ) g^2}-b g (e f+d g)}\right )}{c f^2+g (-b f+a g)}+\frac {d g \left (2 a g^2-f \sqrt {\left (b^2-4 a c\right ) g^2}-2 c f g x-g \sqrt {\left (b^2-4 a c\right ) g^2} x+b g (-f+g x)\right ) \sqrt {\frac {2 a g^2-2 c f g x+b g (-f+g x)+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {-2 a g^2+2 c f g x+b g (f-g x)+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}}}{\sqrt {2}}\right ),\frac {2 \sqrt {\left (b^2-4 a c\right ) g^2} (-e f+d g)}{2 c d f g+2 a e g^2-e f \sqrt {\left (b^2-4 a c\right ) g^2}+d g \sqrt {\left (b^2-4 a c\right ) g^2}-b g (e f+d g)}\right )}{\left (c f^2+g (-b f+a g)\right ) (f+g x) \sqrt {\frac {-2 a g^2+2 c f g x+b g (f-g x)+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}}}-\frac {4 e \sqrt {\left (b^2-4 a c\right ) g^2} \sqrt {-\frac {\left (c f^2+g (-b f+a g)\right ) (a+x (b+c x))}{\left (b^2-4 a c\right ) (f+g x)^2}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}},\arcsin \left (\frac {\sqrt {\frac {-2 a g^2+2 c f g x+b g (f-g x)+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}}}{\sqrt {2}}\right ),\frac {2 \sqrt {\left (b^2-4 a c\right ) g^2} (-e f+d g)}{2 c d f g+2 a e g^2-e f \sqrt {\left (b^2-4 a c\right ) g^2}+d g \sqrt {\left (b^2-4 a c\right ) g^2}-b g (e f+d g)}\right )}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )}{g^2 \sqrt {d+e x} \sqrt {a+x (b+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1462\) vs. \(2(420)=840\).
Time = 5.12 (sec) , antiderivative size = 1463, normalized size of antiderivative = 3.08
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1463\) |
default | \(\text {Expression too large to display}\) | \(10161\) |
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Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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