Integrand size = 34, antiderivative size = 1904 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx =\text {Too large to display} \]
-2/9*(C*d^2-e*(-A*e+B*d))*(c*x^2+b*x+a)^(3/2)/e/(a*e^2-b*d*e+c*d^2)/(e*x+d )^(9/2)+2/315*(2*c^3*d^3*(8*C*d^2+e*(5*A*e+4*B*d))+3*c^2*d*e*(2*a*e*(-9*A* e^2+7*B*d*e+9*C*d^2)-b*d*(5*A*e^2+7*B*d*e+16*C*d^2))+3*c*e^2*(2*a^2*e^2*(- 5*B*e+17*C*d)-a*b*e*(-9*A*e^2+5*B*d*e+41*C*d^2)+b^2*d*(7*A*e^2+3*B*d*e+15* C*d^2))-b*e^3*(21*a^2*C*e^2-6*a*b*e*(2*B*e+3*C*d)+b^2*(8*A*e^2+4*B*d*e+5*C *d^2)))*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^(3/2)-2/105* (c^2*d^3*(8*C*d^2+e*(5*A*e+4*B*d))-e^2*(3*a^2*e^2*(-5*B*e+3*C*d)-a*b*e*(-1 0*A*e^2-17*B*d*e+2*C*d^2)-b^2*d*(8*A*e^2+4*B*d*e+5*C*d^2))-c*d*e*(3*b*d*(5 *A*e^2+2*B*d*e+5*C*d^2)-a*e*(13*A*e^2+11*B*d*e+7*C*d^2))+e*(3*c^2*d^2*(6*C *d^2+e*(-5*A*e+3*B*d))+c*e*(a*e*(-7*A*e^2+B*d*e+47*C*d^2)-3*b*d*(-5*A*e^2+ 2*B*d*e+15*C*d^2))+e^2*(21*a^2*C*e^2-3*a*b*e*(-B*e+16*C*d)+b^2*(25*C*d^2-e *(2*A*e+B*d))))*x)*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^( 7/2)+2/315*(2*c^4*d^4*(8*C*d^2+e*(5*A*e+4*B*d))+2*b^2*e^4*(21*a^2*C*e^2-6* a*b*e*(2*B*e+3*C*d)+b^2*(8*A*e^2+4*B*d*e+5*C*d^2))-6*c^2*e^2*(a*b*d*e*(-34 *A*e^2-5*B*d*e+30*C*d^2)-a^2*e^2*(7*A*e^2-36*B*d*e+30*C*d^2)-b^2*d^2*(11*A *e^2+3*B*d*e+11*C*d^2))-c*e^3*(126*a^3*C*e^3-3*a^2*b*e^2*(29*B*e+12*C*d)-6 *a*b^2*e*(-12*A*e^2+7*B*d*e+5*C*d^2)+b^3*d*(56*A*e^2+25*B*d*e+20*C*d^2))+c ^3*d^2*e*(6*a*e*(-34*A*e^2+8*B*d*e+11*C*d^2)-b*d*(56*C*d^2+5*e*(4*A*e+5*B* d))))*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c*d^2)^4/(e*x+d)^(1/2)-1/315*(2 *c^4*d^4*(8*C*d^2+e*(5*A*e+4*B*d))+2*b^2*e^4*(21*a^2*C*e^2-6*a*b*e*(2*B...
Result contains complex when optimal does not.
Time = 38.31 (sec) , antiderivative size = 29140, normalized size of antiderivative = 15.30 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\text {Result too large to show} \]
Time = 4.55 (sec) , antiderivative size = 1966, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2181, 27, 1229, 27, 1237, 27, 1237, 27, 1269, 1172, 321, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {2 \int -\frac {3 \left (b C d^2-b e (B d+2 A e)+3 e (A c d-a C d+a B e)+e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) x\right ) \sqrt {c x^2+b x+a}}{2 e (d+e x)^{9/2}}dx}{9 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{9 e (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b C d^2-b e (B d+2 A e)+3 e (A c d-a C d+a B e)+e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^{9/2}}dx}{3 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{9 e (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {e^2 \left (5 C d^2+4 e (B d+2 A e)\right ) b^3-3 \left (2 a (3 C d+2 B e) e^3+c d \left (5 C d^2+e (2 B d+5 A e)\right ) e\right ) b^2+\left (21 a^2 C e^4+3 a c \left (19 C d^2+e (2 B d-9 A e)\right ) e^2+c^2 \left (8 C d^4+e (4 B d+5 A e) d^2\right )\right ) b-10 a c e \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)\right )+c \left (2 \left (8 C d^4+e (4 B d+5 A e) d^2\right ) c^2+e \left (2 a e \left (17 C d^2+e (16 B d-7 A e)\right )-b \left (40 C d^3+e (17 B d+10 A e) d\right )\right ) c+3 e^2 \left (\left (10 C d^2+e (B d+2 A e)\right ) b^2-a e (22 C d+3 B e) b+14 a^2 C e^2\right )\right ) x}{2 (d+e x)^{5/2} \sqrt {c x^2+b x+a}}dx}{35 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (-10 A e^2-17 B d e+2 C d^2\right )+b^2 (-d) \left (8 A e^2+4 B d e+5 C d^2\right )\right )-c d e \left (3 b d \left (5 A e^2+2 B d e+5 C d^2\right )-a e \left (13 A e^2+11 B d e+7 C d^2\right )\right )+e x \left (7 e \left (c d^2-e (b d-a e)\right ) \left (3 a C e-A c e-3 b C d+B c d+\frac {2 c C d^2}{e}\right )+(2 c d-b e) \left (3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (2 A e+B d)\right )+c d e (B d-4 A e)+2 c C d^3\right )\right )+c^2 \left (d^3 e (5 A e+4 B d)+8 C d^5\right )\right )}{35 e^2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}}{3 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{9 e (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {e^2 \left (5 C d^2+4 e (B d+2 A e)\right ) b^3-3 \left (2 a (3 C d+2 B e) e^3+c d \left (5 C d^2+e (2 B d+5 A e)\right ) e\right ) b^2+\left (21 a^2 C e^4+3 a c \left (19 C d^2+e (2 B d-9 A e)\right ) e^2+c^2 \left (8 C d^4+e (4 B d+5 A e) d^2\right )\right ) b-10 a c e \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)\right )+c \left (2 \left (8 C d^4+e (4 B d+5 A e) d^2\right ) c^2+e \left (2 a e \left (17 C d^2+e (16 B d-7 A e)\right )-b \left (40 C d^3+e (17 B d+10 A e) d\right )\right ) c+3 e^2 \left (\left (10 C d^2+e (B d+2 A e)\right ) b^2-a e (22 C d+3 B e) b+14 a^2 C e^2\right )\right ) x}{(d+e x)^{5/2} \sqrt {c x^2+b x+a}}dx}{35 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (-10 A e^2-17 B d e+2 C d^2\right )+b^2 (-d) \left (8 A e^2+4 B d e+5 C d^2\right )\right )-c d e \left (3 b d \left (5 A e^2+2 B d e+5 C d^2\right )-a e \left (13 A e^2+11 B d e+7 C d^2\right )\right )+e x \left (7 e \left (c d^2-e (b d-a e)\right ) \left (3 a C e-A c e-3 b C d+B c d+\frac {2 c C d^2}{e}\right )+(2 c d-b e) \left (3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (2 A e+B d)\right )+c d e (B d-4 A e)+2 c C d^3\right )\right )+c^2 \left (d^3 e (5 A e+4 B d)+8 C d^5\right )\right )}{35 e^2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}}{3 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{9 e (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \int \frac {2 e^3 \left (5 C d^2+4 e (B d+2 A e)\right ) b^4-3 \left (4 a (3 C d+2 B e) e^4+c d \left (5 C d^2+7 B e d+16 A e^2\right ) e^2\right ) b^3+3 \left (14 a^2 C e^5+2 a c \left (2 C d^2+e (5 B d-12 A e)\right ) e^3+c^2 d^2 \left (7 C d^2+3 e (B d+5 A e)\right ) e\right ) b^2+c \left (3 a^2 (19 C d+29 B e) e^4-3 a c d \left (19 C d^2-15 B e d-59 A e^2\right ) e^2-c^2 d^3 \left (8 C d^2+4 B e d+5 A e^2\right )\right ) b-6 a c e \left (21 a^2 C e^4-a c \left (13 C d^2-31 B e d+7 A e^2\right ) e^2-c^2 d^2 \left (2 C d^2+B e d-25 A e^2\right )\right )-c \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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 ) x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (c d^2-b e d+a e^2\right )}}{35 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^2-d 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 ) c-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 (7 e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}}{3 e \left (c d^2-b e d+a e^2\right )}-\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}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {\int \frac {2 e^3 \left (5 C d^2+4 e (B d+2 A e)\right ) b^4-3 \left (4 a (3 C d+2 B e) e^4+c d \left (5 C d^2+7 B e d+16 A e^2\right ) e^2\right ) b^3+3 \left (14 a^2 C e^5+2 a c \left (2 C d^2+e (5 B d-12 A e)\right ) e^3+c^2 d^2 \left (7 C d^2+3 e (B d+5 A e)\right ) e\right ) b^2+c \left (3 a^2 (19 C d+29 B e) e^4-3 a c d \left (19 C d^2-15 B e d-59 A e^2\right ) e^2-c^2 d^3 \left (8 C d^2+4 B e d+5 A e^2\right )\right ) b-6 a c e \left (21 a^2 C e^4-a c \left (13 C d^2-31 B e d+7 A e^2\right ) e^2-c^2 d^2 \left (2 C d^2+B e d-25 A e^2\right )\right )-c \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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 ) x}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (c d^2-b e d+a e^2\right )}}{35 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^2-d 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 ) c-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 (7 e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}}{3 e \left (c d^2-b e d+a e^2\right )}-\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}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {-\frac {2 \sqrt {c x^2+b x+a} \left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right )}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {2 \int -\frac {c \left (d e^3 \left (5 C d^2+4 e (B d+2 A e)\right ) b^4+e^2 \left (30 c C d^4-3 c e (4 B d+9 A e) d^2-a e^2 \left (13 C d^2+8 e (B d-A e)\right )\right ) b^3+3 \left (a^2 (C d-4 B e) e^5-2 a c d \left (26 C d^2-B e d+11 A e^2\right ) e^3-c^2 d^3 \left (9 C d^2+4 B e d-10 A e^2\right ) e\right ) b^2+\left (21 a^3 C e^6+3 a^2 c \left (94 C d^2+24 B e d-9 A e^2\right ) e^4+3 a c^2 d^2 \left (15 C d^2+36 B e d+46 A e^2\right ) e^2+c^3 d^4 \left (8 C d^2+4 B e d+5 A e^2\right )\right ) b-2 a c e \left (3 a^2 (38 C d-5 B e) e^4-6 a c d \left (2 C d^2-19 B e d+8 A e^2\right ) e^2+c^2 d^3 \left (2 C d^2+B e d+80 A e^2\right )\right )+\left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{c d^2-b e d+a e^2}}{3 \left (c d^2-b e d+a e^2\right )}}{35 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^2-d 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 ) c-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 (7 e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}}{3 e \left (c d^2-b e d+a e^2\right )}-\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}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {\frac {c \int \frac {d e^3 \left (5 C d^2+4 e (B d+2 A e)\right ) b^4+e^2 \left (30 c C d^4-3 c e (4 B d+9 A e) d^2-a e^2 \left (13 C d^2+8 e (B d-A e)\right )\right ) b^3+3 \left (a^2 (C d-4 B e) e^5-2 a c d \left (26 C d^2-e (B d-11 A e)\right ) e^3-c^2 d^3 \left (9 C d^2+2 e (2 B d-5 A e)\right ) e\right ) b^2+\left (21 a^3 C e^6+3 a^2 c \left (94 C d^2+3 e (8 B d-3 A e)\right ) e^4+3 a c^2 d^2 \left (15 C d^2+36 B e d+46 A e^2\right ) e^2+c^3 \left (8 C d^6+e (4 B d+5 A e) d^4\right )\right ) b-2 a c e \left (3 a^2 (38 C d-5 B e) e^4-6 a c d \left (2 C d^2-e (19 B d-8 A e)\right ) e^2+c^2 \left (2 C d^5+e (B d+80 A e) d^3\right )\right )+\left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{c d^2-b e d+a e^2}-\frac {2 \left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}}{3 \left (c d^2-b e d+a e^2\right )}}{35 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^2-d 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 ) c-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 (7 e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}}{3 e \left (c d^2-b e d+a e^2\right )}-\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}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {\frac {c \left (\frac {\left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (c d^2-b e d+a e^2\right ) \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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 ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}\right )}{c d^2-b e d+a e^2}-\frac {2 \left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}}{3 \left (c d^2-b e d+a e^2\right )}}{35 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^2-d 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 ) c-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 (7 e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}}{3 e \left (c d^2-b e d+a e^2\right )}-\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}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \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 (c d^2-b e d+a e^2\right ) \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{c d^2-b e d+a e^2}-\frac {2 \left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}}{3 \left (c d^2-b e d+a e^2\right )}}{35 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^2-d 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 ) c-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 (7 e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}}{3 e \left (c d^2-b e d+a e^2\right )}-\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}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \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 (c d^2-b e d+a e^2\right ) \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}} \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} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{c d^2-b e d+a e^2}-\frac {2 \left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}}{3 \left (c d^2-b e d+a e^2\right )}}{35 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^2-d 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 ) c-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 (7 e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}}{3 e \left (c d^2-b e d+a e^2\right )}-\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}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) \sqrt {d+e 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} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \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 (c d^2-b e d+a e^2\right ) \left (2 \left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^3+3 d 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 ) c^2+3 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 ) c-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}} \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} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{c d^2-b e d+a e^2}-\frac {2 \left (2 \left (8 C d^6+e (4 B d+5 A e) d^4\right ) c^4-d^2 e \left (56 b C d^3+5 b e (5 B d+4 A e) d-6 a e \left (11 C d^2+8 B e d-34 A e^2\right )\right ) c^3-6 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^2-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 ) c+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 )\right ) \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}}{3 \left (c d^2-b e d+a e^2\right )}}{35 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (8 C d^5+e (4 B d+5 A e) d^3\right ) c^2-d 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 ) c-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 (7 e \left (\frac {2 c C d^2}{e}+B c d-3 b C d-A c e+3 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (2 c C d^3+c e (B d-4 A e) d+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}}{3 e \left (c d^2-b e d+a e^2\right )}-\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}}\) |
(-2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(9*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(9/2)) + ((-2*(c^2*(8*C*d^5 + d^3*e*(4*B*d + 5*A*e)) - e ^2*(3*a^2*e^2*(3*C*d - 5*B*e) - a*b*e*(2*C*d^2 - 17*B*d*e - 10*A*e^2) - b^ 2*d*(5*C*d^2 + 4*B*d*e + 8*A*e^2)) - c*d*e*(3*b*d*(5*C*d^2 + 2*B*d*e + 5*A *e^2) - a*e*(7*C*d^2 + 11*B*d*e + 13*A*e^2)) + e*(7*e*(B*c*d - 3*b*C*d + ( 2*c*C*d^2)/e - A*c*e + 3*a*C*e)*(c*d^2 - e*(b*d - a*e)) + (2*c*d - b*e)*(2 *c*C*d^3 + c*d*e*(B*d - 4*A*e) + 3*a*e^2*(2*C*d - B*e) - b*e*(4*C*d^2 - e* (B*d + 2*A*e))))*x)*Sqrt[a + b*x + c*x^2])/(35*e^2*(c*d^2 - b*d*e + a*e^2) *(d + e*x)^(7/2)) + ((2*(2*c^3*(8*C*d^5 + d^3*e*(4*B*d + 5*A*e)) + 3*c^2*d *e*(2*a*e*(9*C*d^2 + 7*B*d*e - 9*A*e^2) - b*d*(16*C*d^2 + 7*B*d*e + 5*A*e^ 2)) + 3*c*e^2*(2*a^2*e^2*(17*C*d - 5*B*e) - a*b*e*(41*C*d^2 + 5*B*d*e - 9* A*e^2) + b^2*d*(15*C*d^2 + 3*B*d*e + 7*A*e^2)) - b*e^3*(21*a^2*C*e^2 - 6*a *b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e + 8*A*e^2)))*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2)) - ((-2*(2*c^4*(8*C*d^6 + d^4*e*(4*B*d + 5*A*e)) - c^3*d^2*e*(56*b*C*d^3 + 5*b*d*e*(5*B*d + 4*A*e ) - 6*a*e*(11*C*d^2 + 8*B*d*e - 34*A*e^2)) + 2*b^2*e^4*(21*a^2*C*e^2 - 6*a *b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e + 8*A*e^2)) - 6*c^2*e^2*(a*b *d*e*(30*C*d^2 - 5*B*d*e - 34*A*e^2) - a^2*e^2*(30*C*d^2 - 36*B*d*e + 7*A* e^2) - b^2*d^2*(11*C*d^2 + 3*B*d*e + 11*A*e^2)) - c*e^3*(126*a^3*C*e^3 - 3 *a^2*b*e^2*(12*C*d + 29*B*e) - 6*a*b^2*e*(5*C*d^2 + 7*B*d*e - 12*A*e^2)...
3.3.65.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Time = 4.81 (sec) , antiderivative size = 3498, normalized size of antiderivative = 1.84
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(3498\) |
| default | \(\text {Expression too large to display}\) | \(153623\) |
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/9*(A*e ^2-B*d*e+C*d^2)/e^8*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e )^5-2/63*(A*b*e^3-2*A*c*d*e^2+9*B*a*e^3-10*B*b*d*e^2+11*B*c*d^2*e-18*C*a*d *e^2+19*C*b*d^2*e-20*C*c*d^3)/(a*e^2-b*d*e+c*d^2)/e^7*(c*e*x^3+b*e*x^2+c*d *x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^4-2/315*(14*A*a*c*e^4-6*A*b^2*e^4+10*A *b*c*d*e^3-10*A*c^2*d^2*e^2+9*B*a*b*e^4-32*B*a*c*d*e^3-3*B*b^2*d*e^3+17*B* b*c*d^2*e^2-8*B*c^2*d^3*e+63*C*a^2*e^4-144*C*a*b*d*e^3+176*C*a*c*d^2*e^2+7 5*C*b^2*d^2*e^2-170*C*b*c*d^3*e+89*C*c^2*d^4)/e^6/(a*e^2-b*d*e+c*d^2)^2*(c *e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^3+2/315*(27*A*a*b*c* e^5-54*A*a*c^2*d*e^4-8*A*b^3*e^5+21*A*b^2*c*d*e^4-15*A*b*c^2*d^2*e^3+10*A* c^3*d^3*e^2-30*B*a^2*c*e^5+12*B*a*b^2*e^5-15*B*a*b*c*d*e^4+42*B*a*c^2*d^2* e^3-4*B*b^3*d*e^4+9*B*b^2*c*d^2*e^3-21*B*b*c^2*d^3*e^2+8*B*c^3*d^4*e-21*C* a^2*b*e^5+102*C*a^2*c*d*e^4+18*C*a*b^2*d*e^4-123*C*a*b*c*d^2*e^3+54*C*a*c^ 2*d^3*e^2-5*C*b^3*d^2*e^3+45*C*b^2*c*d^3*e^2-48*C*b*c^2*d^4*e+16*C*c^3*d^5 )/e^5/(a*e^2-b*d*e+c*d^2)^3*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2 )/(x+d/e)^2+2/315*(c*e*x^2+b*e*x+a*e)/e^4/(a*e^2-b*d*e+c*d^2)^4*(42*A*a^2* c^2*e^6-72*A*a*b^2*c*e^6+204*A*a*b*c^2*d*e^5-204*A*a*c^3*d^2*e^4+16*A*b^4* e^6-56*A*b^3*c*d*e^5+66*A*b^2*c^2*d^2*e^4-20*A*b*c^3*d^3*e^3+10*A*c^4*d^4* e^2+87*B*a^2*b*c*e^6-216*B*a^2*c^2*d*e^5-24*B*a*b^3*e^6+42*B*a*b^2*c*d*e^5 +30*B*a*b*c^2*d^2*e^4+48*B*a*c^3*d^3*e^3+8*B*b^4*d*e^5-25*B*b^3*c*d^2*e...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.28 (sec) , antiderivative size = 7780, normalized size of antiderivative = 4.09 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\int \frac {\left (A + B x + C x^{2}\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \]
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]