Optimal. Leaf size=1904 \[ -\frac {2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{9 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{9/2}}-\frac {2 \left (c^2 \left (8 C d^2+e (4 B d+5 A e)\right ) d^3-c e \left (3 b d \left (5 C d^2+2 B e d+5 A e^2\right )-a e \left (7 C d^2+11 B e d+13 A e^2\right )\right ) d-e^2 \left (-d \left (5 C d^2+4 B e d+8 A e^2\right ) b^2-a e \left (2 C d^2-17 B e d-10 A e^2\right ) b+3 a^2 e^2 (3 C d-5 B e)\right )+e \left (3 c^2 \left (6 C d^2+e (3 B d-5 A e)\right ) d^2+c e \left (a e \left (47 C d^2+B e d-7 A e^2\right )-3 b d \left (15 C d^2+2 B e d-5 A e^2\right )\right )+e^2 \left (\left (25 C d^2-e (B d+2 A e)\right ) b^2-3 a e (16 C d-B e) b+21 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)^{7/2}}+\frac {2 \left (2 c^4 \left (8 C d^2+e (4 B d+5 A e)\right ) d^4+c^3 e \left (6 a e \left (11 C d^2+8 B e d-34 A e^2\right )-b d \left (56 C d^2+5 e (5 B d+4 A e)\right )\right ) d^2+2 b^2 e^4 \left (\left (5 C d^2+4 B e d+8 A e^2\right ) b^2-6 a e (3 C d+2 B e) b+21 a^2 C e^2\right )-6 c^2 e^2 \left (-b^2 \left (11 C d^2+3 B e d+11 A e^2\right ) d^2+a b e \left (30 C d^2-5 B e d-34 A e^2\right ) d-a^2 e^2 \left (30 C d^2-36 B e d+7 A e^2\right )\right )-c e^3 \left (d \left (20 C d^2+25 B e d+56 A e^2\right ) b^3-6 a e \left (5 C d^2+7 B e d-12 A e^2\right ) b^2-3 a^2 e^2 (12 C d+29 B e) b+126 a^3 C e^3\right )\right ) \sqrt {c x^2+b x+a}}{315 e^3 \left (c d^2-b e d+a e^2\right )^4 \sqrt {d+e x}}+\frac {2 \left (2 c^3 \left (8 C d^2+e (4 B d+5 A e)\right ) d^3+3 c^2 e \left (2 a e \left (9 C d^2+7 B e d-9 A e^2\right )-b d \left (16 C d^2+7 B e d+5 A e^2\right )\right ) d+3 c e^2 \left (d \left (15 C d^2+3 B e d+7 A e^2\right ) b^2-a e \left (41 C d^2+5 B e d-9 A e^2\right ) b+2 a^2 e^2 (17 C d-5 B e)\right )-b e^3 \left (\left (5 C d^2+4 B e d+8 A e^2\right ) b^2-6 a e (3 C d+2 B e) b+21 a^2 C e^2\right )\right ) \sqrt {c x^2+b x+a}}{315 e^3 \left (c d^2-b e d+a e^2\right )^3 (d+e x)^{3/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 c^4 \left (8 C d^2+e (4 B d+5 A e)\right ) d^4+c^3 e \left (6 a e \left (11 C d^2+8 B e d-34 A e^2\right )-b d \left (56 C d^2+5 e (5 B d+4 A e)\right )\right ) d^2+2 b^2 e^4 \left (\left (5 C d^2+4 B e d+8 A e^2\right ) b^2-6 a e (3 C d+2 B e) b+21 a^2 C e^2\right )-6 c^2 e^2 \left (-b^2 \left (11 C d^2+3 B e d+11 A e^2\right ) d^2+a b e \left (30 C d^2-5 B e d-34 A e^2\right ) d-a^2 e^2 \left (30 C d^2-36 B e d+7 A e^2\right )\right )-c e^3 \left (d \left (20 C d^2+25 B e d+56 A e^2\right ) b^3-6 a e \left (5 C d^2+7 B e d-12 A e^2\right ) b^2-3 a^2 e^2 (12 C d+29 B e) b+126 a^3 C e^3\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 )}{315 e^4 \left (c d^2-b e d+a e^2\right )^4 \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^3 \left (8 C d^2+e (4 B d+5 A e)\right ) d^3+3 c^2 e \left (2 a e \left (9 C d^2+7 B e d-9 A e^2\right )-b d \left (16 C d^2+7 B e d+5 A e^2\right )\right ) d+3 c e^2 \left (d \left (15 C d^2+3 B e d+7 A e^2\right ) b^2-a e \left (41 C d^2+5 B e d-9 A e^2\right ) b+2 a^2 e^2 (17 C d-5 B e)\right )-b e^3 \left (\left (5 C d^2+4 B e d+8 A e^2\right ) b^2-6 a e (3 C d+2 B e) b+21 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 )}{315 e^4 \left (c d^2-b e d+a e^2\right )^3 \sqrt {d+e x} \sqrt {c x^2+b x+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 6.24, antiderivative size = 1904, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \text {result too large to display} \]
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)^{11/2}} \, dx &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}-\frac {2 \int \frac {\left (-\frac {3 \left (b C d^2-b e (B d+2 A e)+3 e (A c d-a C d+a B e)\right )}{2 e}-\frac {3}{2} \left (B c d-3 b C d+\frac {2 c C d^2}{e}-A c e+3 a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^{9/2}} \, dx}{9 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac {3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\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)^{7/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}+\frac {4 \int \frac {\frac {3 \left (b^3 e^2 \left (5 C d^2+4 e (B d+2 A e)\right )-10 a c e \left (3 a e^2 (2 C d-B e)+c d \left (2 C d^2+B d e-4 A e^2\right )\right )-3 b^2 \left (2 a e^3 (3 C d+2 B e)+c d e \left (5 C d^2+2 B d e+5 A e^2\right )\right )+b \left (21 a^2 C e^4+3 a c e^2 \left (19 C d^2+2 B d e-9 A e^2\right )+c^2 d^2 \left (8 C d^2+4 B d e+5 A e^2\right )\right )\right )}{4 e}+\frac {3 c \left (2 c^2 \left (8 C d^4+d^2 e (4 B d+5 A e)\right )+3 e^2 \left (14 a^2 C e^2-a b e (22 C d+3 B e)+b^2 \left (10 C d^2+B d e+2 A e^2\right )\right )+c e \left (2 a e \left (17 C d^2+16 B d e-7 A e^2\right )-b d \left (40 C d^2+17 B d e+10 A e^2\right )\right )\right ) x}{4 e}}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{315 e^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {2 \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}-\frac {2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac {3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\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)^{7/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}-\frac {8 \int \frac {\frac {3 \left (2 b^4 e^3 \left (5 C d^2+4 e (B d+2 A e)\right )+b c \left (3 a^2 e^4 (19 C d+29 B e)-3 a c d e^2 \left (19 C d^2-15 B d e-59 A e^2\right )-c^2 d^3 \left (8 C d^2+4 B d e+5 A e^2\right )\right )-6 a c e \left (21 a^2 C e^4-c^2 d^2 \left (2 C d^2+B d e-25 A e^2\right )-a c e^2 \left (13 C d^2-31 B d e+7 A e^2\right )\right )-3 b^3 \left (4 a e^4 (3 C d+2 B e)+c d e^2 \left (5 C d^2+7 B d e+16 A e^2\right )\right )+3 b^2 \left (14 a^2 C e^5+2 a c e^3 \left (2 C d^2+e (5 B d-12 A e)\right )+c^2 d^2 e \left (7 C d^2+3 e (B d+5 A e)\right )\right )\right )}{8 e}-\frac {3 c \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) x}{8 e}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{945 e^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {2 \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^4 \sqrt {d+e x}}-\frac {2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac {3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\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)^{7/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}+\frac {16 \int \frac {-\frac {3 c \left (b^4 d e^3 \left (5 C d^2+4 e (B d+2 A e)\right )+3 b^2 \left (a^2 e^5 (C d-4 B e)-c^2 d^3 e \left (9 C d^2+4 B d e-10 A e^2\right )-2 a c d e^3 \left (26 C d^2-B d e+11 A e^2\right )\right )+b \left (21 a^3 C e^6+3 a^2 c e^4 \left (94 C d^2+24 B d e-9 A e^2\right )+c^3 d^4 \left (8 C d^2+4 B d e+5 A e^2\right )+3 a c^2 d^2 e^2 \left (15 C d^2+36 B d e+46 A e^2\right )\right )-2 a c e \left (3 a^2 e^4 (38 C d-5 B e)-6 a c d e^2 \left (2 C d^2-19 B d e+8 A e^2\right )+c^2 d^3 \left (2 C d^2+B d e+80 A e^2\right )\right )+b^3 e^2 \left (30 c C d^4-3 c d^2 e (4 B d+9 A e)-a e^2 \left (13 C d^2+8 e (B d-A e)\right )\right )\right )}{16 e}-\frac {3 c \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) x}{16 e}}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{945 e^2 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac {2 \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^4 \sqrt {d+e x}}-\frac {2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac {3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\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)^{7/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}+\frac {\left (c \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{315 e^4 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (c \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{315 e^4 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac {2 \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^4 \sqrt {d+e x}}-\frac {2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac {3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\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)^{7/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\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 )}{315 e^4 \left (c d^2-b d e+a e^2\right )^4 \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 (2 \sqrt {2} \sqrt {b^2-4 a c} \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\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 )}{315 e^4 \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 (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^4 \sqrt {d+e x}}-\frac {2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac {3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\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)^{7/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\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 )}{315 e^4 \left (c d^2-b d e+a e^2\right )^4 \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 (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 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 )}{315 e^4 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 18.39, size = 29140, normalized size = 15.30 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{e^{6} x^{6} + 6 \, d e^{5} x^{5} + 15 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x + d^{6}}, x\right ) \]
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 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.80, size = 153623, normalized size = 80.68 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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