Integrand size = 31, antiderivative size = 1098 \[ \int \frac {(d+e x)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\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 )}{315 c^4 g^5 \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 (c f^2-b f g+a g^2\right ) \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)+2 c^3 \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\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 )}{315 c^4 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
[Out]
Time = 2.38 (sec) , antiderivative size = 1098, normalized size of antiderivative = 1.00, number of steps used = 9, 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)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \sqrt {c x^2+b x+a} (d+e x)^3}{9 g}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (-2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right ) c^4-g \left (6 a e g \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 g f^2+189 d^2 e g^2 f-105 d^3 g^3\right )\right ) c^3+3 e g^2 \left (\left (7 e^2 f^2-27 d e g f+42 d^2 g^2\right ) b^2-a e g (19 e f-87 d g) b+14 a^2 e^2 g^2\right ) c^2+8 b^2 e^2 g^3 (2 b e f-9 b d g-9 a e g) c+16 b^4 e^3 g^4\right ) \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 )}{315 c^4 g^5 \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} \left (c f^2-b g f+a g^2\right ) \left (2 \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right ) c^3-3 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) c^2+3 b e^2 g^2 (5 b e f-12 b d g-9 a e g) c+8 b^3 e^3 g^3\right ) \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 )}{315 c^4 g^5 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{63 c g^4}-\frac {2 e \left (-2 \left (64 e^2 f^2-111 d e g f+42 d^2 g^2\right ) c^2+e g (17 b e f-27 b d g-14 a e g) c+6 b^2 e^2 g^2\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{315 c^2 g^4}+\frac {2 \left (-\left (\left (152 e^3 f^3-408 d e^2 g f^2+336 d^2 e g^2 f-70 d^3 g^3\right ) c^3\right )-3 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) c^2+3 b e^2 g^2 (5 b e f-12 b d g-9 a e g) c+8 b^3 e^3 g^3\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}{315 c^3 g^4} \]
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 857
Rule 934
Rule 1667
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {\int \frac {(d+e x)^2 \left (b d f+6 a e f-8 a d g+(2 c d f+7 b e f-7 b d g-2 a e g) x+(8 c e f-6 c d g-b e g) x^2\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{9 g} \\ & = \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {2 \int \frac {\frac {1}{2} g \left (b^2 e^3 f^3 g-2 a c g \left (20 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+28 d^3 g^3\right )+b f \left (5 a e^3 f g^2-c \left (8 e^3 f^3-6 d e^2 f^2 g-7 d^3 g^3\right )\right )\right )+\frac {1}{2} g \left (2 b e^3 f g^2 (4 b f+5 a g)-2 c^2 \left (8 e^3 f^4-6 d e^2 f^3 g-7 d^3 f g^3\right )-c g \left (2 a e g \left (40 e^2 f^2-72 d e f g+63 d^2 g^2\right )+b \left (62 e^3 f^3-48 d e^2 f^2 g-63 d^2 e f g^2+49 d^3 g^3\right )\right )\right ) x+\frac {1}{2} g^2 \left (b e^3 g^2 (13 b f+5 a g)-c^2 \left (88 e^3 f^3-66 d e^2 f^2 g-84 d^2 e f g^2+42 d^3 g^3\right )+c e g \left (2 a e g (e f-27 d g)-3 b \left (31 e^2 f^2-61 d e f g+35 d^2 g^2\right )\right )\right ) x^2+\frac {1}{2} e g^3 \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) x^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{63 c g^5} \\ & = \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {4 \int \frac {-\frac {1}{4} g^4 \left (6 b^3 e^3 f^2 g^2+3 b^2 e^2 f g \left (6 a e g^2+c f (4 e f-9 d g)\right )-2 a c g \left (21 a e^3 f g^2+c \left (92 e^3 f^3-258 d e^2 f^2 g+231 d^2 e f g^2-140 d^3 g^3\right )\right )+b c f \left (3 a e^2 g^2 (4 e f-27 d g)-c \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )\right )\right )-\frac {1}{4} g^4 \left (6 b^2 e^3 g^3 (5 b f+3 a g)-2 c^3 f \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )-3 c e^2 g^2 \left (14 a^2 e g^2-b^2 f (19 e f-45 d g)+a b g (23 e f+27 d g)\right )-c^2 g \left (6 a e g \left (2 e^2 f^2+9 d e f g-63 d^2 g^2\right )+b \left (296 e^3 f^3-816 d e^2 f^2 g+735 d^2 e f g^2-245 d^3 g^3\right )\right )\right ) x-\frac {3}{4} g^5 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-2 c^3 \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{315 c^2 g^8} \\ & = \frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {8 \int \frac {\frac {3}{8} g^6 \left (8 b^4 e^3 f g^3+b^3 e^2 g^2 \left (8 a e g^2+9 c f (e f-4 d g)\right )-3 b^2 c e g \left (2 a e g^2 (5 e f+6 d g)-c f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )+2 a c^2 g \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-b c \left (27 a^2 e^3 g^4+3 a c e g^2 \left (8 e^2 f^2-33 d e f g-21 d^2 g^2\right )+c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right )+\frac {3}{8} g^6 \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{945 c^3 g^{10}} \\ & = \frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {\left (\left (c f^2-b f g+a g^2\right ) \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)+2 c^3 \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{315 c^3 g^5}-\frac {\left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{315 c^3 g^5} \\ & = \frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\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 )}{315 c^4 g^5 \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 (c f^2-b f g+a g^2\right ) \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)+2 c^3 \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\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 )}{315 c^4 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = \frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)-c^3 \left (152 e^3 f^3-408 d e^2 f^2 g+336 d^2 e f g^2-70 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 g}-\frac {2 e \left (6 b^2 e^2 g^2+c e g (17 b e f-27 b d g-14 a e g)-2 c^2 \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^4}-\frac {2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^4}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 b^4 e^3 g^4+8 b^2 c e^2 g^3 (2 b e f-9 b d g-9 a e g)-2 c^4 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )+3 c^2 e g^2 \left (14 a^2 e^2 g^2-a b e g (19 e f-87 d g)+b^2 \left (7 e^2 f^2-27 d e f g+42 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-b \left (40 e^3 f^3-144 d e^2 f^2 g+189 d^2 e f g^2-105 d^3 g^3\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 )}{315 c^4 g^5 \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 (c f^2-b f g+a g^2\right ) \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (5 b e f-12 b d g-9 a e g)+2 c^3 \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )-3 c^2 e g \left (6 a e g (2 e f-5 d g)-b \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\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 )}{315 c^4 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 37.00 (sec) , antiderivative size = 17771, normalized size of antiderivative = 16.18 \[ \int \frac {(d+e x)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\text {Result too large to show} \]
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Time = 4.70 (sec) , antiderivative size = 1845, normalized size of antiderivative = 1.68
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1845\) |
risch | \(\text {Expression too large to display}\) | \(7892\) |
default | \(\text {Expression too large to display}\) | \(22215\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 1241, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\text {Too large to display} \]
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\[ \int \frac {(d+e x)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\left (d + e x\right )^{3} \sqrt {a + b x + c x^{2}}}{\sqrt {f + g x}}\, dx \]
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\[ \int \frac {(d+e x)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{3}}{\sqrt {g x + f}} \,d x } \]
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\[ \int \frac {(d+e x)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{3}}{\sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^3 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {{\left (d+e\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a}}{\sqrt {f+g\,x}} \,d x \]
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