Optimal. Leaf size=1363 \[ -\frac {2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{7 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (c^2 \left (24 C d^2+e (4 B d+3 A e)\right ) d^3-c e \left (b d \left (43 C d^2+6 B e d+15 A e^2\right )-a e \left (33 C d^2+9 B e d+19 A e^2\right )\right ) d-e^2 \left (-d \left (15 C d^2+6 B e d+8 A e^2\right ) b^2+a e \left (12 C d^2+23 B e d+12 A e^2\right ) b+7 a^2 e^2 (C d-3 B e)\right )+e \left (7 c^2 \left (6 C d^2+e (B d-3 A e)\right ) d^2+e^2 \left (\left (45 C d^2-3 B e d-4 A e^2\right ) b^2-7 a e (12 C d-B e) b+35 a^2 C e^2\right )+c e \left (a e \left (93 C d^2-9 B e d-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{105 e^3 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 \left (2 c^3 \left (24 C d^2+e (4 B d+3 A e)\right ) d^3+c^2 e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b d \left (128 C d^2+e (19 B d+9 A e)\right )\right ) d-b e^3 \left (\left (15 C d^2+6 B e d+8 A e^2\right ) b^2-14 a e (3 C d+B e) b+35 a^2 C e^2\right )+c e^2 \left (d \left (103 C d^2+e (9 B d+19 A e)\right ) b^2-a e \left (237 C d^2+e (B d-29 A e)\right ) b+14 a^2 e^2 (11 C d-3 B e)\right )\right ) \sqrt {c x^2+b x+a}}{105 e^3 \left (c d^2-b e d+a e^2\right )^3 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 c^3 \left (24 C d^2+e (4 B d+3 A e)\right ) d^3+c^2 e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b d \left (128 C d^2+e (19 B d+9 A e)\right )\right ) d-b e^3 \left (\left (15 C d^2+6 B e d+8 A e^2\right ) b^2-14 a e (3 C d+B e) b+35 a^2 C e^2\right )+c e^2 \left (d \left (103 C d^2+e (9 B d+19 A e)\right ) b^2-a e \left (237 C d^2+e (B d-29 A e)\right ) b+14 a^2 e^2 (11 C d-3 B e)\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\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} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 e^4 \left (c d^2-b e d+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (2 c^2 \left (24 C d^2+e (4 B d+3 A e)\right ) d^2+c e \left (2 a e \left (51 C d^2+e (12 B d-5 A e)\right )-b d \left (104 C d^2+3 e (5 B d+2 A e)\right )\right )+e^2 \left (\left (60 C d^2+e (3 B d+4 A e)\right ) b^2-7 a e (18 C d+B e) b+70 a^2 C e^2\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\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} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 e^4 \left (c d^2-b e d+a e^2\right )^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 4.20, antiderivative size = 1363, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1650, 810, 834, 843, 718, 424, 419} \[ -\frac {2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{7 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (\left (24 C d^5+e (4 B d+3 A e) d^3\right ) c^2-d e \left (b d \left (43 C d^2+6 B e d+15 A e^2\right )-a e \left (33 C d^2+9 B e d+19 A e^2\right )\right ) c-e^2 \left (-d \left (15 C d^2+6 B e d+8 A e^2\right ) b^2+a e \left (12 C d^2+23 B e d+12 A e^2\right ) b+7 a^2 e^2 (C d-3 B e)\right )+e \left (7 \left (6 C d^4+e (B d-3 A e) d^2\right ) c^2+e \left (a e \left (93 C d^2-9 B e d-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\right )\right ) c+e^2 \left (\left (45 C d^2-3 B e d-4 A e^2\right ) b^2-7 a e (12 C d-B e) b+35 a^2 C e^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{105 e^3 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 \left (\left (48 C d^5+2 e (4 B d+3 A e) d^3\right ) c^3+d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+e (19 B d+9 A e) d\right )\right ) c^2+e^2 \left (\left (103 C d^3+e (9 B d+19 A e) d\right ) b^2-a e \left (237 C d^2+e (B d-29 A e)\right ) b+14 a^2 e^2 (11 C d-3 B e)\right ) c-b e^3 \left (\left (15 C d^2+6 B e d+8 A e^2\right ) b^2-14 a e (3 C d+B e) b+35 a^2 C e^2\right )\right ) \sqrt {c x^2+b x+a}}{105 e^3 \left (c d^2-b e d+a e^2\right )^3 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 \left (24 C d^5+e (4 B d+3 A e) d^3\right ) c^3+d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+e (19 B d+9 A e) d\right )\right ) c^2+e^2 \left (\left (103 C d^3+e (9 B d+19 A e) d\right ) b^2-a e \left (237 C d^2+e (B d-29 A e)\right ) b+14 a^2 e^2 (11 C d-3 B e)\right ) c-b e^3 \left (\left (15 C d^2+6 B e d+8 A e^2\right ) b^2-14 a e (3 C d+B e) b+35 a^2 C e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\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} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 e^4 \left (c d^2-b e d+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (\left (48 C d^4+2 e (4 B d+3 A e) d^2\right ) c^2+e \left (2 a e \left (51 C d^2+e (12 B d-5 A e)\right )-b \left (104 C d^3+3 e (5 B d+2 A e) d\right )\right ) c+e^2 \left (\left (60 C d^2+e (3 B d+4 A e)\right ) b^2-7 a e (18 C d+B e) b+70 a^2 C e^2\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\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} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 e^4 \left (c d^2-b e d+a e^2\right )^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 419
Rule 424
Rule 718
Rule 810
Rule 834
Rule 843
Rule 1650
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \int \frac {\left (-\frac {3 b C d^2-b e (3 B d+4 A e)+7 e (A c d-a C d+a B e)}{2 e}-\frac {1}{2} \left (B c d-7 b C d+\frac {6 c C d^2}{e}-A c e+7 a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx}{7 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}+\frac {4 \int \frac {\frac {b^3 e^2 \left (15 C d^2+6 B d e+8 A e^2\right )-6 a c e \left (7 a e^2 (2 C d-B e)+c d \left (6 C d^2+B d e-8 A e^2\right )\right )+b \left (35 a^2 C e^4+a c e^2 \left (111 C d^2-6 B d e-29 A e^2\right )+c^2 d^2 \left (24 C d^2+4 B d e+3 A e^2\right )\right )-b^2 \left (14 a e^3 (3 C d+B e)+c d e \left (43 C d^2+6 B d e+15 A e^2\right )\right )}{4 e}+\frac {c \left (c^2 \left (48 C d^4+2 d^2 e (4 B d+3 A e)\right )+e^2 \left (70 a^2 C e^2-7 a b e (18 C d+B e)+b^2 \left (60 C d^2+3 B d e+4 A e^2\right )\right )+c e \left (2 a e \left (51 C d^2+12 B d e-5 A e^2\right )-b d \left (104 C d^2+15 B d e+6 A e^2\right )\right )\right ) x}{4 e}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{105 e^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {2 \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {8 \int \frac {\frac {c \left (b^3 d e^2 \left (45 C d^2-e (3 B d+4 A e)\right )-b^2 \left (c d^2 e \left (61 C d^2+9 B d e-9 A e^2\right )+4 a e^3 \left (36 C d^2-B d e+A e^2\right )\right )+b \left (7 a^2 e^4 (23 C d+B e)+c^2 d^3 \left (24 C d^2+4 B d e+3 A e^2\right )+5 a c d e^2 \left (19 C d^2+9 B d e+5 A e^2\right )\right )-2 a e \left (35 a^2 C e^4+a c e^2 \left (9 C d^2+33 B d e-5 A e^2\right )+c^2 d^2 \left (6 C d^2+B d e+27 A e^2\right )\right )\right )}{8 e}+\frac {c \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+15 B d e-29 A e^2\right )-b d \left (128 C d^2+19 B d e+9 A e^2\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+B d e-29 A e^2\right )+b^2 d \left (103 C d^2+9 B d e+19 A e^2\right )\right )\right ) x}{8 e}}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 e^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {2 \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {\left (c \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{105 e^4 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (8 \left (-\frac {c d \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+15 B d e-29 A e^2\right )-b d \left (128 C d^2+19 B d e+9 A e^2\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+B d e-29 A e^2\right )+b^2 d \left (103 C d^2+9 B d e+19 A e^2\right )\right )\right )}{8 e}+\frac {1}{8} c \left (b^3 d e^2 \left (45 C d^2-e (3 B d+4 A e)\right )-b^2 \left (c d^2 e \left (61 C d^2+9 B d e-9 A e^2\right )+4 a e^3 \left (36 C d^2-B d e+A e^2\right )\right )+b \left (7 a^2 e^4 (23 C d+B e)+c^2 d^3 \left (24 C d^2+4 B d e+3 A e^2\right )+5 a c d e^2 \left (19 C d^2+9 B d e+5 A e^2\right )\right )-2 a e \left (35 a^2 C e^4+a c e^2 \left (9 C d^2+33 B d e-5 A e^2\right )+c^2 d^2 \left (6 C d^2+B d e+27 A e^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 e^3 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {2 \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\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 e^4 \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}-\frac {\left (16 \sqrt {2} \sqrt {b^2-4 a c} \left (-\frac {c d \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+15 B d e-29 A e^2\right )-b d \left (128 C d^2+19 B d e+9 A e^2\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+B d e-29 A e^2\right )+b^2 d \left (103 C d^2+9 B d e+19 A e^2\right )\right )\right )}{8 e}+\frac {1}{8} c \left (b^3 d e^2 \left (45 C d^2-e (3 B d+4 A e)\right )-b^2 \left (c d^2 e \left (61 C d^2+9 B d e-9 A e^2\right )+4 a e^3 \left (36 C d^2-B d e+A e^2\right )\right )+b \left (7 a^2 e^4 (23 C d+B e)+c^2 d^3 \left (24 C d^2+4 B d e+3 A e^2\right )+5 a c d e^2 \left (19 C d^2+9 B d e+5 A e^2\right )\right )-2 a e \left (35 a^2 C e^4+a c e^2 \left (9 C d^2+33 B d e-5 A e^2\right )+c^2 d^2 \left (6 C d^2+B d e+27 A e^2\right )\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, 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 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt {d+e 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} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 e^4 \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 \left (48 C d^4+2 d^2 e (4 B d+3 A e)\right )+e^2 \left (70 a^2 C e^2-7 a b e (18 C d+B e)+b^2 \left (60 C d^2+3 B d e+4 A e^2\right )\right )+c e \left (2 a e \left (51 C d^2+12 B d e-5 A e^2\right )-b d \left (104 C d^2+15 B d e+6 A e^2\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \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} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 e^4 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 16.02, size = 19853, normalized size = 14.57 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 88790, normalized size = 65.14 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x + C x^{2}\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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