Integrand size = 31, antiderivative size = 764 \[ \int \frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 e}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^2 g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (e g (b e f-3 b d g+2 a e g)+c \left (-e^2 f^2+3 d^2 g^2\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} g}{-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^3 g \sqrt {f+g x} \sqrt {a+x (b+c x)}}-\frac {\sqrt {2} \left (c d^2-b d e+a e^2\right ) \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} e^3 \sqrt {a+b x+c x^2}} \]
[Out]
Time = 2.52 (sec) , antiderivative size = 969, normalized size of antiderivative = 1.27, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {932, 6874, 732, 430, 857, 435, 948, 175, 552, 551} \[ \int \frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {\sqrt {2} \sqrt {b^2-4 a c} (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^2 g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} f (c e f-3 c d g+b e g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^2 g \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (3 c d (e f-d g)-e (2 b e f-3 b d g+2 a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^3 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \left (c d^2-b e d+a e^2\right ) \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} e^3 \sqrt {c x^2+b x+a}}+\frac {2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{3 e} \]
[In]
[Out]
Rule 175
Rule 430
Rule 435
Rule 551
Rule 552
Rule 732
Rule 857
Rule 932
Rule 948
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 e}-\frac {\int \frac {b d f-3 a e f+a d g+2 (c d f-b e f+b d g-a e g) x-(c e f-3 c d g+b e g) x^2}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3 e} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 e}-\frac {\int \left (\frac {3 c d (e f-d g)-e (2 b e f-3 b d g+2 a e g)}{e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {(c e f-3 c d g+b e g) x}{e \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {3 \left (c d^2-b d e+a e^2\right ) (e f-d g)}{e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}\right ) \, dx}{3 e} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 e}+\frac {\left (\left (c d^2-b d e+a e^2\right ) (e f-d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{e^3}+\frac {(c e f-3 c d g+b e g) \int \frac {x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3 e^2}-\frac {(3 c d (e f-d g)-e (2 b e f-3 b d g+2 a e g)) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3 e^3} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 e}+\frac {(c e f-3 c d g+b e g) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{3 e^2 g}-\frac {(f (c e f-3 c d g+b e g)) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3 e^2 g}+\frac {\left (\left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \int \frac {1}{\sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x} (d+e x) \sqrt {f+g x}} \, dx}{e^3 \sqrt {a+b x+c x^2}}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (3 c d (e f-d g)-e (2 b e f-3 b d g+2 a e g)) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 c e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (3 c d (e f-d g)-e (2 b e f-3 b d g+2 a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\left (2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}}} \, dx,x,\sqrt {f+g x}\right )}{e^3 \sqrt {a+b x+c x^2}}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 c e^2 g \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} f (c e f-3 c d g+b e g) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 c e^2 g \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 e}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^2 g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} f (c e f-3 c d g+b e g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^2 g \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (3 c d (e f-d g)-e (2 b e f-3 b d g+2 a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\left (2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{e^3 \sqrt {a+b x+c x^2}} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 e}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^2 g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} f (c e f-3 c d g+b e g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^2 g \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (3 c d (e f-d g)-e (2 b e f-3 b d g+2 a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\left (2 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c (f+g x)}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c x^2}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{e^3 \sqrt {a+b x+c x^2}} \\ & = \frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 e}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^2 g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} f (c e f-3 c d g+b e g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^2 g \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (3 c d (e f-d g)-e (2 b e f-3 b d g+2 a e g)) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c e^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \left (c d^2-b d e+a e^2\right ) \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} e^3 \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 36.44 (sec) , antiderivative size = 1470, normalized size of antiderivative = 1.92 \[ \int \frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {2 \sqrt {f+g x} \sqrt {a+x (b+c x)}}{3 e}+\frac {(f+g x)^{3/2} \sqrt {a+x (b+c x)} \left (4 e (c e f-3 c d g+b e g) \sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \left (c \left (-1+\frac {f}{f+g x}\right )^2+\frac {g \left (b-\frac {b f}{f+g x}+\frac {a g}{f+g x}\right )}{f+g x}\right )-\frac {i \sqrt {2} e (c e f-3 c d g+b e g) \left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) g^2}-\frac {2 a g^2}{f+g x}-2 c f \left (-1+\frac {f}{f+g x}\right )+b g \left (-1+\frac {2 f}{f+g x}\right )}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) g^2}+\frac {2 a g^2}{f+g x}+2 c f \left (-1+\frac {f}{f+g x}\right )+b \left (g-\frac {2 f g}{f+g x}\right )}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\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}}\right )}{\sqrt {f+g x}}+\frac {i \sqrt {2} e \left (-6 c^2 d f g+b e g \left (-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right )+c \left (-2 a e g^2+\sqrt {\left (b^2-4 a c\right ) g^2} (e f-3 d g)+3 b g (e f+d g)\right )\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) g^2}-\frac {2 a g^2}{f+g x}-2 c f \left (-1+\frac {f}{f+g x}\right )+b g \left (-1+\frac {2 f}{f+g x}\right )}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) g^2}+\frac {2 a g^2}{f+g x}+2 c f \left (-1+\frac {f}{f+g x}\right )+b \left (g-\frac {2 f g}{f+g x}\right )}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right ),-\frac {-2 c f+b g+\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}}\right )}{\sqrt {f+g x}}-\frac {6 i \sqrt {2} c \left (-c d^2+e (b d-a e)\right ) g^2 \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) g^2}-\frac {2 a g^2}{f+g x}-2 c f \left (-1+\frac {f}{f+g x}\right )+b g \left (-1+\frac {2 f}{f+g x}\right )}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) g^2}+\frac {2 a g^2}{f+g x}+2 c f \left (-1+\frac {f}{f+g x}\right )+b \left (g-\frac {2 f g}{f+g x}\right )}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \operatorname {EllipticPi}\left (\frac {(e f-d g) \left (2 c f-b g-\sqrt {\left (b^2-4 a c\right ) g^2}\right )}{2 e \left (c f^2+g (-b f+a g)\right )},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right ),-\frac {-2 c f+b g+\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}}\right )}{\sqrt {f+g x}}\right )}{6 c e^3 g^2 \sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {a+b x+c x^2} \sqrt {\frac {(f+g x)^2 \left (c \left (-1+\frac {f}{f+g x}\right )^2+\frac {g \left (b-\frac {b f}{f+g x}+\frac {a g}{f+g x}\right )}{f+g x}\right )}{g^2}}} \]
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Time = 3.88 (sec) , antiderivative size = 1245, normalized size of antiderivative = 1.63
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1245\) |
risch | \(\text {Expression too large to display}\) | \(2338\) |
default | \(\text {Expression too large to display}\) | \(6812\) |
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Timed out. \[ \int \frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{d+e x} \, dx=\int \frac {\sqrt {f + g x} \sqrt {a + b x + c x^{2}}}{d + e x}\, dx \]
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\[ \int \frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{d+e x} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} \sqrt {g x + f}}{e x + d} \,d x } \]
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\[ \int \frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{d+e x} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} \sqrt {g x + f}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{d+e x} \, dx=\int \frac {\sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a}}{d+e\,x} \,d x \]
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