Integrand size = 31, antiderivative size = 567 \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} e (c e f-5 c d g+2 b e g) \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
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Time = 0.54 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {955, 1667, 857, 732, 435, 430} \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {4 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} (2 b e g-5 c d g+c e f) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e g (9 a e g+20 b d g+3 b e f)+8 b^2 e^2 g^2-\left (c^2 \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right )\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e \sqrt {f+g x} \sqrt {a+b x+c x^2} (-4 b e g+7 c d g+c e f)}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c} \]
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Rule 430
Rule 435
Rule 732
Rule 857
Rule 955
Rule 1667
Rubi steps \begin{align*} \text {integral}& = \frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\int \frac {-5 c d^2 f+e (b d f+2 a e f+a d g)-(c d (8 e f+5 d g)-e (3 b e f+2 b d g+3 a e g)) x-e (c e f+7 c d g-4 b e g) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{5 c} \\ & = \frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}-\frac {2 \int \frac {-\frac {1}{2} g \left (4 b^2 e^2 f g+b e \left (4 a e g^2-c f (e f+10 d g)\right )+c g \left (15 c d^2 f-a e (7 e f+10 d g)\right )\right )-\frac {1}{2} g \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^2} \\ & = \frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (2 e (c e f-5 c d g+2 b e g) \left (c f^2-b f g+a g^2\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^2}+\frac {\left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^2} \\ & = \frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \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 )}{15 c^3 g^2 \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 (4 \sqrt {2} \sqrt {b^2-4 a c} e (c e f-5 c d g+2 b e g) \left (c f^2-b f g+a g^2\right ) \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 )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = \frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \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 )}{15 c^3 g^2 \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 {4 \sqrt {2} \sqrt {b^2-4 a c} e (c e f-5 c d g+2 b e g) \left (c f^2-b f g+a g^2\right ) \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 )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 31.19 (sec) , antiderivative size = 1002, normalized size of antiderivative = 1.77 \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {\left (-\frac {2 e (-c e f-10 c d g+4 b e g)}{15 c^2 g}+\frac {2 e^2 x}{5 c}\right ) \sqrt {f+g x} \left (a+b x+c x^2\right )}{\sqrt {a+x (b+c x)}}-\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2} \left (\left (-8 b^2 e^2 g^2+c e g (3 b e f+20 b d g+9 a e g)+c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \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 {1-\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {1+\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \left (\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)+c^2 \left (-2 e^2 f^2+10 d e f g+15 d^2 g^2\right )\right ) 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 )+\left (-30 c^3 d^2 f g^2+8 b^2 e^2 g^2 \left (b g-\sqrt {\left (b^2-4 a c\right ) g^2}\right )+c e g \left (-17 a b e g^2+9 a e g \sqrt {\left (b^2-4 a c\right ) g^2}+b \sqrt {\left (b^2-4 a c\right ) g^2} (3 e f+20 d g)-b^2 g (11 e f+20 d g)\right )-c^2 \left (-15 b d g^2 (2 e f+d g)-2 a e g^2 (7 e f+10 d g)+\sqrt {\left (b^2-4 a c\right ) g^2} \left (-2 e^2 f^2+10 d e f g+15 d^2 g^2\right )\right )\right ) \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 )\right )}{2 \sqrt {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 {f+g x}}\right )}{15 c^3 g^3 \sqrt {a+x (b+c x)} \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 = 4.75 (sec) , antiderivative size = 985, normalized size of antiderivative = 1.74
method | result | size |
elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 e^{2} x \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{5 c}+\frac {2 \left (2 d e g +e^{2} f -\frac {2 e^{2} \left (2 b g +2 c f \right )}{5 c}\right ) \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{3 c g}+\frac {2 \left (d^{2} f -\frac {2 e^{2} f a}{5 c}-\frac {2 \left (2 d e g +e^{2} f -\frac {2 e^{2} \left (2 b g +2 c f \right )}{5 c}\right ) \left (\frac {a g}{2}+\frac {b f}{2}\right )}{3 c g}\right ) \left (\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}+\frac {2 \left (d^{2} g +2 d e f -\frac {2 e^{2} \left (\frac {3 a g}{2}+\frac {3 b f}{2}\right )}{5 c}-\frac {2 \left (2 d e g +e^{2} f -\frac {2 e^{2} \left (2 b g +2 c f \right )}{5 c}\right ) \left (b g +c f \right )}{3 c g}\right ) \left (\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}\) | \(985\) |
risch | \(\text {Expression too large to display}\) | \(2623\) |
default | \(\text {Expression too large to display}\) | \(8248\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (2 \, c^{3} e^{2} f^{3} - 2 \, {\left (5 \, c^{3} d e - b c^{2} e^{2}\right )} f^{2} g + {\left (30 \, c^{3} d^{2} - 20 \, b c^{2} d e + {\left (7 \, b^{2} c - 12 \, a c^{2}\right )} e^{2}\right )} f g^{2} - {\left (15 \, b c^{2} d^{2} - 10 \, {\left (2 \, b^{2} c - 3 \, a c^{2}\right )} d e + {\left (8 \, b^{3} - 21 \, a b c\right )} e^{2}\right )} g^{3}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) + 3 \, {\left (2 \, c^{3} e^{2} f^{2} g - {\left (10 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f g^{2} - {\left (15 \, c^{3} d^{2} - 20 \, b c^{2} d e + {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} e^{2}\right )} g^{3}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) + 3 \, {\left (3 \, c^{3} e^{2} g^{3} x + c^{3} e^{2} f g^{2} + 2 \, {\left (5 \, c^{3} d e - 2 \, b c^{2} e^{2}\right )} g^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{45 \, c^{4} g^{3}} \]
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\[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{2} \sqrt {f + g x}}{\sqrt {a + b x + c x^{2}}}\, dx \]
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\[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}} \,d x } \]
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\[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^2}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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