Integrand size = 31, antiderivative size = 631 \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {8 e^2 (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}+\frac {\sqrt {2} \sqrt {b^2-4 a c} e \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^3 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (4 b e^3 g^2 (b f-a g)+c^2 \left (8 e^3 f^3-30 d e^2 f^2 g+45 d^2 e f g^2-15 d^3 g^3\right )-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
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Time = 0.64 (sec) , antiderivative size = 631, 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 = {944, 1667, 857, 732, 435, 430} \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c e g (-9 a e g-30 b d g+7 b e f)+8 b^2 e^2 g^2+c^2 \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))+4 b e^3 g^2 (b f-a g)+c^2 \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {8 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (b e g-3 c d g+c e f)}{15 c^2 g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g} \]
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Rule 430
Rule 435
Rule 732
Rule 857
Rule 944
Rule 1667
Rubi steps \begin{align*} \text {integral}& = \frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}-\frac {\int \frac {b d e^2 f-5 c d^3 g+a e^2 (2 e f+d g)+e (c d (2 e f-15 d g)+e (3 b e f+2 b d g+3 a e g)) x+4 e^2 (c e f-3 c d g+b e g) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{5 c g} \\ & = -\frac {8 e^2 (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}-\frac {2 \int \frac {-\frac {1}{2} g \left (4 b^2 e^3 f g+b e^2 \left (4 a e g^2+c f (4 e f-15 d g)\right )+c g \left (15 c d^3 g-a e^2 (2 e f+15 d g)\right )\right )-\frac {1}{2} e g \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^3} \\ & = -\frac {8 e^2 (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}+\frac {\left (e \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^3}-\frac {\left (4 b e^3 g^2 (b f-a g)+c^2 \left (8 e^3 f^3-30 d e^2 f^2 g+45 d^2 e f g^2-15 d^3 g^3\right )-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^3} \\ & = -\frac {8 e^2 (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} e \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 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^3 \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} \left (4 b e^3 g^2 (b f-a g)+c^2 \left (8 e^3 f^3-30 d e^2 f^2 g+45 d^2 e f g^2-15 d^3 g^3\right )-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))\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^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = -\frac {8 e^2 (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}+\frac {\sqrt {2} \sqrt {b^2-4 a c} e \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 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^3 \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 (4 b e^3 g^2 (b f-a g)+c^2 \left (8 e^3 f^3-30 d e^2 f^2 g+45 d^2 e f g^2-15 d^3 g^3\right )-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))\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^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 34.49 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {4 e g^2 \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) (a+x (b+c x))}{\sqrt {f+g x}}+4 c e^2 g^2 \sqrt {f+g x} (a+x (b+c x)) (-4 b e g+c (-4 e f+15 d g+3 e g x))-\frac {i (f+g x) \sqrt {1-\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {2+\frac {4 \left (c f^2+g (-b f+a g)\right )}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \left (e \left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )-\left (-8 b^3 e^3 g^3+b^2 e^2 g^2 \left (c e f+30 c d g+8 e \sqrt {\left (b^2-4 a c\right ) g^2}\right )+b c e g \left (-45 c d^2 g^2+e \left (17 a e g^2+\sqrt {\left (b^2-4 a c\right ) g^2} (7 e f-30 d g)\right )\right )+c \left (-a e^2 g^2 \left (4 c e f+30 c d g+9 e \sqrt {\left (b^2-4 a c\right ) g^2}\right )+c \left (30 c d^3 g^3+e \sqrt {\left (b^2-4 a c\right ) g^2} \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right ),-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )\right )}{\sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}}{30 c^3 g^4 \sqrt {a+x (b+c x)}} \]
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Time = 6.15 (sec) , antiderivative size = 985, normalized size of antiderivative = 1.56
method | result | size |
elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 e^{3} x \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{5 c g}+\frac {2 \left (3 d \,e^{2}-\frac {2 \left (2 b g +2 c f \right ) e^{3}}{5 c g}\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^{3}-\frac {2 f a \,e^{3}}{5 c g}-\frac {2 \left (3 d \,e^{2}-\frac {2 \left (2 b g +2 c f \right ) e^{3}}{5 c g}\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 (3 d^{2} e -\frac {2 e^{3} \left (\frac {3 a g}{2}+\frac {3 b f}{2}\right )}{5 c g}-\frac {2 \left (3 d \,e^{2}-\frac {2 \left (2 b g +2 c f \right ) e^{3}}{5 c g}\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 | \(\text {Expression too large to display}\) | \(2635\) |
default | \(\text {Expression too large to display}\) | \(8755\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{3} e^{3} f^{3} - 3 \, {\left (10 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} f^{2} g + 3 \, {\left (15 \, c^{3} d^{2} e - 5 \, b c^{2} d e^{2} + {\left (b^{2} c - a c^{2}\right )} e^{3}\right )} f g^{2} - {\left (45 \, c^{3} d^{3} - 45 \, b c^{2} d^{2} e + 15 \, {\left (2 \, b^{2} c - 3 \, a c^{2}\right )} d e^{2} - {\left (8 \, b^{3} - 21 \, a b c\right )} e^{3}\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 (8 \, c^{3} e^{3} f^{2} g - {\left (30 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} f g^{2} + {\left (45 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} e^{3}\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^{3} g^{3} x - 4 \, c^{3} e^{3} f g^{2} + {\left (15 \, c^{3} d e^{2} - 4 \, b c^{2} e^{3}\right )} g^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{45 \, c^{4} g^{4}} \]
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\[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \]
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\[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}} \,d x } \]
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\[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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