Integrand size = 31, antiderivative size = 755 \[ \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=-\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
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Time = 1.19 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {934, 1667, 857, 732, 435, 430} \[ \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (35 d^2 g^2-42 d e f g+16 e^2 f^2\right )\right )+b c e g^2 (-29 a e g-28 b d g+9 b e f)+8 b^3 e^2 g^3-2 c^3 f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2-\left (c^2 \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )\right )}{105 c^2 g^3}-\frac {2 e (f+g x)^{3/2} \sqrt {a+b x+c x^2} (-b e g-4 c d g+6 c e f)}{35 c g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g} \]
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Rule 430
Rule 435
Rule 732
Rule 857
Rule 934
Rule 1667
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {\int \frac {(d+e x) \left (b d f+4 a e f-6 a d g+(2 c d f+5 b e f-5 b d g-2 a e g) x+(6 c e f-4 c d g-b e g) x^2\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{7 g} \\ & = \frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}-\frac {2 \int \frac {\frac {1}{2} g \left (b^2 e^2 f^2 g-2 a c g \left (9 e^2 f^2-16 d e f g+15 d^2 g^2\right )+b f \left (3 a e^2 g^2-c \left (6 e^2 f^2-4 d e f g-5 d^2 g^2\right )\right )\right )+\frac {1}{2} g \left (b e^2 g^2 (5 b f+3 a g)-2 c^2 f \left (6 e^2 f^2-4 d e f g-5 d^2 g^2\right )+c g \left (2 a e g (e f-14 d g)-b \left (28 e^2 f^2-50 d e f g+25 d^2 g^2\right )\right )\right ) x+g^2 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{35 c g^4} \\ & = -\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}-\frac {4 \int \frac {-\frac {1}{4} g^3 \left (4 b^3 e^2 f g^2+b^2 e g \left (4 a e g^2+c f (5 e f-14 d g)\right )-b c \left (a e g^2 (11 e f+14 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )-2 a c g \left (5 a e^2 g^2-c \left (6 e^2 f^2-14 d e f g+35 d^2 g^2\right )\right )\right )-\frac {1}{4} g^3 \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^6} \\ & = -\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\left (2 \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^4}+\frac {\left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^4} \\ & = -\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\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 )}{105 c^3 g^4 \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} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\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 )}{105 c^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = -\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\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 )}{105 c^3 g^4 \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} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 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 \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 )}{105 c^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 35.65 (sec) , antiderivative size = 10030, normalized size of antiderivative = 13.28 \[ \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\text {Result too large to show} \]
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Time = 2.48 (sec) , antiderivative size = 1272, normalized size of antiderivative = 1.68
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1272\) |
risch | \(\text {Expression too large to display}\) | \(4566\) |
default | \(\text {Expression too large to display}\) | \(12922\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 828, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left ({\left (48 \, c^{4} e^{2} f^{4} - 8 \, {\left (14 \, c^{4} d e + 5 \, b c^{3} e^{2}\right )} f^{3} g + 2 \, {\left (35 \, c^{4} d^{2} + 49 \, b c^{3} d e - {\left (5 \, b^{2} c^{2} - 31 \, a c^{3}\right )} e^{2}\right )} f^{2} g^{2} - {\left (70 \, b c^{3} d^{2} - 28 \, {\left (b^{2} c^{2} - 6 \, a c^{3}\right )} d e + {\left (5 \, b^{3} c - 22 \, a b c^{2}\right )} e^{2}\right )} f g^{3} - {\left (35 \, {\left (b^{2} c^{2} - 6 \, a c^{3}\right )} d^{2} - 14 \, {\left (2 \, b^{3} c - 9 \, a b c^{2}\right )} d e + {\left (8 \, b^{4} - 41 \, a b^{2} c + 30 \, a^{2} c^{2}\right )} e^{2}\right )} g^{4}\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 (48 \, c^{4} e^{2} f^{3} g - 16 \, {\left (7 \, c^{4} d e + b c^{3} e^{2}\right )} f^{2} g^{2} + {\left (70 \, c^{4} d^{2} + 42 \, b c^{3} d e - {\left (9 \, b^{2} c^{2} - 26 \, a c^{3}\right )} e^{2}\right )} f g^{3} - {\left (35 \, b c^{3} d^{2} - 28 \, {\left (b^{2} c^{2} - 3 \, a c^{3}\right )} d e + {\left (8 \, b^{3} c - 29 \, a b c^{2}\right )} e^{2}\right )} g^{4}\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 (15 \, c^{4} e^{2} g^{4} x^{2} + 24 \, c^{4} e^{2} f^{2} g^{2} - {\left (56 \, c^{4} d e + 5 \, b c^{3} e^{2}\right )} f g^{3} + {\left (35 \, c^{4} d^{2} + 14 \, b c^{3} d e - 2 \, {\left (2 \, b^{2} c^{2} - 5 \, a c^{3}\right )} e^{2}\right )} g^{4} - 3 \, {\left (6 \, c^{4} e^{2} f g^{3} - {\left (14 \, c^{4} d e + b c^{3} e^{2}\right )} g^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{315 \, c^{4} g^{5}} \]
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\[ \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\left (d + e x\right )^{2} \sqrt {a + b x + c x^{2}}}{\sqrt {f + g x}}\, dx \]
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\[ \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{2}}{\sqrt {g x + f}} \,d x } \]
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\[ \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{2}}{\sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {{\left (d+e\,x\right )}^2\,\sqrt {c\,x^2+b\,x+a}}{\sqrt {f+g\,x}} \,d x \]
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