Integrand size = 31, antiderivative size = 1551 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=-\frac {2 \left (64 b^4 e^4 g^4+4 b^2 c e^3 g^3 (7 b e f-66 b d g-69 a e g)+c^4 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )+3 c^2 e^2 g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 e g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3465 c^4 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}+\frac {2 \left (48 b^3 e^3 g^3+b c e^2 g^2 (67 b e f-198 b d g-157 a e g)+c^3 \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )-c^2 e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e f g+99 d^2 g^2\right )\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{3465 c^3 g^4}-\frac {2 e \left (8 b^2 e^2 g^2+c e g (19 b e f-33 b d g-18 a e g)+c^2 \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{693 c^2 g^4}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {a+b x+c x^2}}{99 c g^4}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (128 b^5 e^3 g^5-8 b^3 c e^2 g^4 (7 b e f+66 b d g+87 a e g)+2 c^5 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+b c^2 e g^3 \left (771 a^2 e^2 g^2+6 a b e g (43 e f+396 d g)-b^2 \left (37 e^2 f^2-264 d e f g-792 d^2 g^2\right )\right )-c^4 g \left (b f \left (56 e^3 f^3-264 d e^2 f^2 g+495 d^2 e f g^2-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right )-c^3 g^2 \left (6 a^2 e^2 g^2 (26 e f+231 d g)-9 a b e g \left (15 e^2 f^2-110 d e f g-319 d^2 g^2\right )+b^2 \left (37 e^3 f^3-198 d e^2 f^2 g+495 d^2 e f g^2+462 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 )}{3465 c^5 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 (64 b^4 e^3 g^4+4 b^2 c e^2 g^3 (7 b e f-66 b d g-69 a e g)-2 c^4 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+3 c^2 e g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\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 )}{3465 c^5 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
[Out]
Time = 5.02 (sec) , antiderivative size = 1551, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {932, 1667, 857, 732, 435, 430} \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\frac {2 \sqrt {f+g x} \sqrt {c x^2+b x+a} (d+e x)^4}{11 e}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^5-g \left (b f \left (56 e^3 f^3-264 d e^2 g f^2+495 d^2 e g^2 f-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right )\right ) c^4-g^2 \left (\left (37 e^3 f^3-198 d e^2 g f^2+495 d^2 e g^2 f+462 d^3 g^3\right ) b^2-9 a e g \left (15 e^2 f^2-110 d e g f-319 d^2 g^2\right ) b+6 a^2 e^2 g^2 (26 e f+231 d g)\right ) c^3+b e g^3 \left (-\left (\left (37 e^2 f^2-264 d e g f-792 d^2 g^2\right ) b^2\right )+6 a e g (43 e f+396 d g) b+771 a^2 e^2 g^2\right ) c^2-8 b^3 e^2 g^4 (7 b e f+66 b d g+87 a e g) c+128 b^5 e^3 g^5\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 )}{3465 c^5 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 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^4-g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^2 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^3 g^4\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 )}{3465 c^5 g^5 \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{99 c g^4}-\frac {2 e \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{693 c^2 g^4}+\frac {2 \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{3465 c^3 g^4}-\frac {2 \left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}{3465 c^4 e g^4} \]
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 857
Rule 932
Rule 1667
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}-\frac {\int \frac {(d+e x)^3 \left (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\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{11 e} \\ & = \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {a+b x+c x^2}}{99 c g^4}-\frac {2 \int \frac {\frac {1}{2} g \left (b^2 e^4 f^4 g+7 a b e^4 f^3 g^2+a c g \left (7 e^4 f^4-21 d e^3 f^3 g-27 d^3 e f g^3+9 d^4 g^4\right )+b c \left (e^4 f^5-3 d e^3 f^4 g+9 d^4 f g^4\right )\right )+\frac {1}{2} g \left (b e^4 f^2 g^2 (11 b f+21 a g)+2 c^2 \left (e^4 f^5-3 d e^3 f^4 g+9 d^4 f g^4\right )+c g \left (3 a e g \left (7 e^3 f^3-21 d e^2 f^2 g-27 d^2 e f g^2+3 d^3 g^3\right )+b \left (13 e^4 f^4-33 d e^3 f^3 g+9 d^3 e f g^3+18 d^4 g^4\right )\right )\right ) x+\frac {3}{2} g^2 \left (b e^4 f g^2 (9 b f+7 a g)+c^2 \left (5 e^4 f^4-15 d e^3 f^3 g+15 d^3 e f g^3+9 d^4 g^4\right )+c e g \left (a e g \left (7 e^2 f^2-48 d e f g-9 d^2 g^2\right )+b \left (14 e^3 f^3-27 d e^2 f^2 g-9 d^2 e f g^2+15 d^3 g^3\right )\right )\right ) x^2+\frac {1}{2} e g^3 \left (b e^3 g^2 (25 b f+7 a g)+3 c^2 \left (11 e^3 f^3-33 d e^2 f^2 g+9 d^2 e f g^2+27 d^3 g^3\right )-c e g \left (2 a e g (10 e f+33 d g)-b \left (58 e^2 f^2-120 d e f g+27 d^2 g^2\right )\right )\right ) x^3+\frac {1}{2} e^2 g^4 \left (8 b^2 e^2 g^2+c e g (19 b e f-33 b d g-18 a e g)+c^2 \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) x^4}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{99 c e g^5} \\ & = \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}-\frac {2 e \left (8 b^2 e^2 g^2+c e g (19 b e f-33 b d g-18 a e g)+c^2 \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{693 c^2 g^4}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {a+b x+c x^2}}{99 c g^4}-\frac {4 \int \frac {-\frac {1}{4} g^5 \left (8 b^3 e^4 f^3 g^2+b^2 e^3 f^2 g \left (40 a e g^2+3 c f (4 e f-11 d g)\right )+b c f \left (a e^3 f g^2 (28 e f-165 d g)+c \left (22 e^4 f^4-75 d e^3 f^3 g+81 d^2 e^2 f^2 g^2-63 d^4 g^4\right )\right )-3 a c g \left (30 a e^4 f^2 g^2-c \left (32 e^4 f^4-111 d e^3 f^3 g+135 d^2 e^2 f^2 g^2+63 d^3 e f g^3-21 d^4 g^4\right )\right )\right )-\frac {1}{4} g^5 \left (16 b^2 e^4 f g^3 (4 b f+5 a g)+2 c^3 \left (22 e^4 f^5-75 d e^3 f^4 g+81 d^2 e^2 f^3 g^2-63 d^4 f g^4\right )-c e^3 f g^2 \left (180 a^2 e g^2-b^2 f (91 e f-264 d g)+a b g (101 e f+330 d g)\right )+c^2 g \left (a e g \left (107 e^3 f^3-519 d e^2 f^2 g+1377 d^2 e f g^2-63 d^3 g^3\right )+b \left (179 e^4 f^4-603 d e^3 f^3 g+648 d^2 e^2 f^2 g^2-63 d^3 e f g^3-126 d^4 g^4\right )\right )\right ) x-\frac {1}{4} g^6 \left (8 b^2 e^4 g^3 (13 b f+5 a g)+c^3 \left (214 e^4 f^4-741 d e^3 f^3 g+891 d^2 e^2 f^2 g^2-315 d^3 e f g^3-189 d^4 g^4\right )-c e^3 g^2 \left (90 a^2 e g^2-b^2 f (146 e f-429 d g)+11 a b g (26 e f+15 d g)\right )-c^2 e g \left (2 a e g \left (100 e^2 f^2-264 d e f g-297 d^2 g^2\right )-b \left (292 e^3 f^3-1044 d e^2 f^2 g+1242 d^2 e f g^2-315 d^3 g^3\right )\right )\right ) x^2-\frac {1}{4} e g^7 \left (48 b^3 e^3 g^3+b c e^2 g^2 (67 b e f-198 b d g-157 a e g)+c^3 \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )-c^2 e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e f g+99 d^2 g^2\right )\right )\right ) x^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{693 c^2 e g^9} \\ & = \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}+\frac {2 \left (48 b^3 e^3 g^3+b c e^2 g^2 (67 b e f-198 b d g-157 a e g)+c^3 \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )-c^2 e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e f g+99 d^2 g^2\right )\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{3465 c^3 g^4}-\frac {2 e \left (8 b^2 e^2 g^2+c e g (19 b e f-33 b d g-18 a e g)+c^2 \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{693 c^2 g^4}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {a+b x+c x^2}}{99 c g^4}-\frac {8 \int \frac {\frac {3}{8} g^8 \left (16 b^4 e^4 f^2 g^3+3 b^3 e^3 f g^2 \left (16 a e g^2+c f (3 e f-22 d g)\right )-b^2 c e^2 f g \left (2 a e g^2 (26 e f+99 d g)-c f \left (4 e^2 f^2-33 d e f g+99 d^2 g^2\right )\right )+a c^2 g \left (2 a e^3 f g^2 (e f+231 d g)+c \left (73 e^4 f^4-288 d e^3 f^3 g+432 d^2 e^2 f^2 g^2-882 d^3 e f g^3+105 d^4 g^4\right )\right )-b c f \left (157 a^2 e^4 g^4+3 a c e^2 g^2 \left (8 e^2 f^2-55 d e f g-99 d^2 g^2\right )-c^2 \left (41 e^4 f^4-156 d e^3 f^3 g+234 d^2 e^2 f^2 g^2-189 d^3 e f g^3+105 d^4 g^4\right )\right )\right )+\frac {3}{8} g^8 \left (16 b^3 e^4 g^4 (5 b f+3 a g)+2 c^4 f \left (41 e^4 f^4-156 d e^3 f^3 g+234 d^2 e^2 f^2 g^2-189 d^3 e f g^3+105 d^4 g^4\right )-b c e^3 g^3 \left (157 a^2 e g^2-b^2 f (37 e f-330 d g)+2 a b g (164 e f+99 d g)\right )+c^2 e^2 g^2 \left (2 a^2 e g^2 (76 e f+231 d g)-3 a b g \left (37 e^2 f^2-352 d e f g-99 d^2 g^2\right )+b^2 f \left (13 e^2 f^2-132 d e f g+495 d^2 g^2\right )\right )-c^3 g \left (22 a e g \left (2 e^3 f^3-15 d e^2 f^2 g+54 d^2 e f g^2+21 d^3 g^3\right )-3 b \left (46 e^4 f^4-192 d e^3 f^3 g+321 d^2 e^2 f^2 g^2-280 d^3 e f g^3+70 d^4 g^4\right )\right )\right ) x+\frac {3}{8} g^9 \left (64 b^4 e^4 g^4+4 b^2 c e^3 g^3 (7 b e f-66 b d g-69 a e g)+c^4 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )+3 c^2 e^2 g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 e g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\right )\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3465 c^3 e g^{12}} \\ & = -\frac {2 \left (64 b^4 e^4 g^4+4 b^2 c e^3 g^3 (7 b e f-66 b d g-69 a e g)+c^4 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )+3 c^2 e^2 g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 e g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3465 c^4 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}+\frac {2 \left (48 b^3 e^3 g^3+b c e^2 g^2 (67 b e f-198 b d g-157 a e g)+c^3 \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )-c^2 e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e f g+99 d^2 g^2\right )\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{3465 c^3 g^4}-\frac {2 e \left (8 b^2 e^2 g^2+c e g (19 b e f-33 b d g-18 a e g)+c^2 \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{693 c^2 g^4}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {a+b x+c x^2}}{99 c g^4}-\frac {16 \int \frac {-\frac {3}{16} e g^{10} \left (64 b^5 e^3 f g^4+4 b^4 e^2 g^3 \left (16 a e g^2-c f (5 e f+66 d g)\right )-b^3 c e g^2 \left (8 a e g^2 (49 e f+33 d g)+9 c f \left (2 e^2 f^2-11 d e f g-44 d^2 g^2\right )\right )+2 a c^2 g \left (75 a^2 e^3 g^4-9 a c e g^2 \left (e^2 f^2+66 d e f g+55 d^2 g^2\right )-c^2 f \left (16 e^3 f^3-66 d e^2 f^2 g+99 d^2 e f g^2-924 d^3 g^3\right )\right )+b c^2 \left (3 a^2 e^2 g^4 (178 e f+297 d g)+a c g^2 \left (52 e^3 f^3-297 d e^2 f^2 g-1782 d^2 e f g^2-231 d^3 g^3\right )+c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )\right )-b^2 c g \left (276 a^2 e^3 g^4-6 a c e g^2 \left (13 e^2 f^2+231 d e f g+66 d^2 g^2\right )+c^2 f \left (20 e^3 f^3-99 d e^2 f^2 g+198 d^2 e f g^2+231 d^3 g^3\right )\right )\right )-\frac {3}{16} e g^{10} \left (128 b^5 e^3 g^5-8 b^3 c e^2 g^4 (7 b e f+66 b d g+87 a e g)+2 c^5 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+b c^2 e g^3 \left (771 a^2 e^2 g^2+6 a b e g (43 e f+396 d g)-b^2 \left (37 e^2 f^2-264 d e f g-792 d^2 g^2\right )\right )-c^4 g \left (b f \left (56 e^3 f^3-264 d e^2 f^2 g+495 d^2 e f g^2-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right )-c^3 g^2 \left (6 a^2 e^2 g^2 (26 e f+231 d g)-9 a b e g \left (15 e^2 f^2-110 d e f g-319 d^2 g^2\right )+b^2 \left (37 e^3 f^3-198 d e^2 f^2 g+495 d^2 e f g^2+462 d^3 g^3\right )\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{10395 c^4 e g^{14}} \\ & = -\frac {2 \left (64 b^4 e^4 g^4+4 b^2 c e^3 g^3 (7 b e f-66 b d g-69 a e g)+c^4 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )+3 c^2 e^2 g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 e g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3465 c^4 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}+\frac {2 \left (48 b^3 e^3 g^3+b c e^2 g^2 (67 b e f-198 b d g-157 a e g)+c^3 \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )-c^2 e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e f g+99 d^2 g^2\right )\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{3465 c^3 g^4}-\frac {2 e \left (8 b^2 e^2 g^2+c e g (19 b e f-33 b d g-18 a e g)+c^2 \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{693 c^2 g^4}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {a+b x+c x^2}}{99 c g^4}+\frac {\left (\left (c f^2-b f g+a g^2\right ) \left (64 b^4 e^3 g^4+4 b^2 c e^2 g^3 (7 b e f-66 b d g-69 a e g)-2 c^4 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+3 c^2 e g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3465 c^4 g^5}+\frac {\left (128 b^5 e^3 g^5-8 b^3 c e^2 g^4 (7 b e f+66 b d g+87 a e g)+2 c^5 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+b c^2 e g^3 \left (771 a^2 e^2 g^2+6 a b e g (43 e f+396 d g)-b^2 \left (37 e^2 f^2-264 d e f g-792 d^2 g^2\right )\right )-c^4 g \left (b f \left (56 e^3 f^3-264 d e^2 f^2 g+495 d^2 e f g^2-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right )-c^3 g^2 \left (6 a^2 e^2 g^2 (26 e f+231 d g)-9 a b e g \left (15 e^2 f^2-110 d e f g-319 d^2 g^2\right )+b^2 \left (37 e^3 f^3-198 d e^2 f^2 g+495 d^2 e f g^2+462 d^3 g^3\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{3465 c^4 g^5} \\ & = -\frac {2 \left (64 b^4 e^4 g^4+4 b^2 c e^3 g^3 (7 b e f-66 b d g-69 a e g)+c^4 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )+3 c^2 e^2 g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 e g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3465 c^4 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}+\frac {2 \left (48 b^3 e^3 g^3+b c e^2 g^2 (67 b e f-198 b d g-157 a e g)+c^3 \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )-c^2 e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e f g+99 d^2 g^2\right )\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{3465 c^3 g^4}-\frac {2 e \left (8 b^2 e^2 g^2+c e g (19 b e f-33 b d g-18 a e g)+c^2 \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{693 c^2 g^4}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {a+b x+c x^2}}{99 c g^4}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (128 b^5 e^3 g^5-8 b^3 c e^2 g^4 (7 b e f+66 b d g+87 a e g)+2 c^5 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+b c^2 e g^3 \left (771 a^2 e^2 g^2+6 a b e g (43 e f+396 d g)-b^2 \left (37 e^2 f^2-264 d e f g-792 d^2 g^2\right )\right )-c^4 g \left (b f \left (56 e^3 f^3-264 d e^2 f^2 g+495 d^2 e f g^2-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right )-c^3 g^2 \left (6 a^2 e^2 g^2 (26 e f+231 d g)-9 a b e g \left (15 e^2 f^2-110 d e f g-319 d^2 g^2\right )+b^2 \left (37 e^3 f^3-198 d e^2 f^2 g+495 d^2 e f g^2+462 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 )}{3465 c^5 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 (64 b^4 e^3 g^4+4 b^2 c e^2 g^3 (7 b e f-66 b d g-69 a e g)-2 c^4 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+3 c^2 e g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\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 )}{3465 c^5 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \left (64 b^4 e^4 g^4+4 b^2 c e^3 g^3 (7 b e f-66 b d g-69 a e g)+c^4 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )+3 c^2 e^2 g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 e g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3465 c^4 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}+\frac {2 \left (48 b^3 e^3 g^3+b c e^2 g^2 (67 b e f-198 b d g-157 a e g)+c^3 \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )-c^2 e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e f g+99 d^2 g^2\right )\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{3465 c^3 g^4}-\frac {2 e \left (8 b^2 e^2 g^2+c e g (19 b e f-33 b d g-18 a e g)+c^2 \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{693 c^2 g^4}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {a+b x+c x^2}}{99 c g^4}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (128 b^5 e^3 g^5-8 b^3 c e^2 g^4 (7 b e f+66 b d g+87 a e g)+2 c^5 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+b c^2 e g^3 \left (771 a^2 e^2 g^2+6 a b e g (43 e f+396 d g)-b^2 \left (37 e^2 f^2-264 d e f g-792 d^2 g^2\right )\right )-c^4 g \left (b f \left (56 e^3 f^3-264 d e^2 f^2 g+495 d^2 e f g^2-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right )-c^3 g^2 \left (6 a^2 e^2 g^2 (26 e f+231 d g)-9 a b e g \left (15 e^2 f^2-110 d e f g-319 d^2 g^2\right )+b^2 \left (37 e^3 f^3-198 d e^2 f^2 g+495 d^2 e f g^2+462 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 )}{3465 c^5 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 (64 b^4 e^3 g^4+4 b^2 c e^2 g^3 (7 b e f-66 b d g-69 a e g)-2 c^4 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+3 c^2 e g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\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 )}{3465 c^5 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 37.07 (sec) , antiderivative size = 26600, normalized size of antiderivative = 17.15 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(3253\) vs. \(2(1469)=2938\).
Time = 3.12 (sec) , antiderivative size = 3254, normalized size of antiderivative = 2.10
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3254\) |
risch | \(\text {Expression too large to display}\) | \(11966\) |
default | \(\text {Expression too large to display}\) | \(32647\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 1741, normalized size of antiderivative = 1.12 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\text {Too large to display} \]
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\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\int \left (d + e x\right )^{3} \sqrt {f + g x} \sqrt {a + b x + c x^{2}}\, dx \]
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\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{3} \sqrt {g x + f} \,d x } \]
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\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{3} \sqrt {g x + f} \,d x } \]
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Timed out. \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\int \sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a} \,d x \]
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