Integrand size = 34, antiderivative size = 1342 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx =\text {Too large to display} \] Output:
2/105*(c^2*(48*C*d^4+2*d^2*e*(3*A*e+4*B*d))+c*e*(2*a*e*(51*C*d^2+e*(-5*A*e +12*B*d))-b*(104*C*d^3+3*d*e*(2*A*e+5*B*d)))+e^2*(70*a^2*C*e^2-7*a*b*e*(B* e+18*C*d)+b^2*(60*C*d^2+e*(4*A*e+3*B*d))))*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2- b*d*e+c*d^2)^2/(e*x+d)^(3/2)+2/105*(c^3*(48*C*d^5+2*d^3*e*(3*A*e+4*B*d))-b *e^3*(35*a^2*C*e^2-14*a*b*e*(B*e+3*C*d)+b^2*(8*A*e^2+6*B*d*e+15*C*d^2))+c^ 2*d*e*(2*a*e*(69*C*d^2+e*(-29*A*e+15*B*d))-b*(128*C*d^3+d*e*(9*A*e+19*B*d) ))+c*e^2*(14*a^2*e^2*(-3*B*e+11*C*d)-a*b*e*(237*C*d^2+e*(-29*A*e+B*d))+b^2 *(103*C*d^3+d*e*(19*A*e+9*B*d))))*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c*d ^2)^3/(e*x+d)^(1/2)-2/35*(24*c*C*d^3/e+c*d*(3*A*e+4*B*d)+7*a*e*(B*e+3*C*d) -b*(25*C*d^2+e*(4*A*e+3*B*d))+5*e*(B*c*d-7*C*b*d+6*c*C*d^2/e-A*c*e+7*C*a*e )*x)*(c*x^2+b*x+a)^(1/2)/e^2/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(5/2)-2/7*(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)^(7/2)-1/10 5*2^(1/2)*(-4*a*c+b^2)^(1/2)*(2*c^3*(24*C*d^5+d^3*e*(3*A*e+4*B*d))-b*e^3*( 35*a^2*C*e^2-14*a*b*e*(B*e+3*C*d)+b^2*(8*A*e^2+6*B*d*e+15*C*d^2))+c^2*d*e* (2*a*e*(69*C*d^2+e*(-29*A*e+15*B*d))-b*(128*C*d^3+d*e*(9*A*e+19*B*d)))+c*e ^2*(14*a^2*e^2*(-3*B*e+11*C*d)-a*b*e*(237*C*d^2+e*(-29*A*e+B*d))+b^2*(103* C*d^3+d*e*(19*A*e+9*B*d))))*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^ (1/2)*EllipticE(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4 *a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/e^4/(a*e^2-b*d* e+c*d^2)^3/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)/(c*x^2+b*...
Result contains complex when optimal does not.
Time = 38.13 (sec) , antiderivative size = 19853, normalized size of antiderivative = 14.79 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(9/2),x]
Output:
Result too large to show
Time = 5.78 (sec) , antiderivative size = 1413, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2181, 27, 1229, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {2 \int -\frac {\left (3 b C d^2-b e (3 B d+4 A e)+7 e (A c d-a C d+a B e)+e \left (\frac {6 c C d^2}{e}+B c d-7 b C d-A c e+7 a C e\right ) x\right ) \sqrt {c x^2+b x+a}}{2 e (d+e x)^{7/2}}dx}{7 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{7 e (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (3 b C d^2-b e (3 B d+4 A e)+7 e (A c d-a C d+a B e)+e \left (\frac {6 c C d^2}{e}+B c d-7 b C d-A c e+7 a C e\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^{7/2}}dx}{7 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{7 e (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {e^2 \left (15 C d^2+6 B e d+8 A e^2\right ) b^3-\left (14 a (3 C d+B e) e^3+c d \left (43 C d^2+3 e (2 B d+5 A e)\right ) e\right ) b^2+\left (35 a^2 C e^4+a c \left (111 C d^2-e (6 B d+29 A e)\right ) e^2+c^2 \left (24 C d^4+e (4 B d+3 A e) d^2\right )\right ) b-6 a c e \left (6 c C d^3+c e (B d-8 A e) d+7 a e^2 (2 C d-B e)\right )+c \left (\left (48 C d^4+2 e (4 B d+3 A e) d^2\right ) c^2+e \left (2 a e \left (51 C d^2+e (12 B d-5 A e)\right )-b \left (104 C d^3+3 e (5 B d+2 A e) d\right )\right ) c+e^2 \left (\left (60 C d^2+e (3 B d+4 A e)\right ) b^2-7 a e (18 C d+B e) b+70 a^2 C e^2\right )\right ) x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{15 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (-e^2 \left (7 a^2 e^2 (C d-3 B e)+a b e \left (12 A e^2+23 B d e+12 C d^2\right )+b^2 (-d) \left (8 A e^2+6 B d e+15 C d^2\right )\right )-c d e \left (b d \left (15 A e^2+6 B d e+43 C d^2\right )-a e \left (19 A e^2+9 B d e+33 C d^2\right )\right )+e x \left (5 e \left (c d^2-e (b d-a e)\right ) \left (7 a C e-A c e-7 b C d+B c d+\frac {6 c C d^2}{e}\right )+(2 c d-b e) \left (7 a e^2 (2 C d-B e)-b e \left (10 C d^2-e (4 A e+3 B d)\right )+c d e (B d-8 A e)+6 c C d^3\right )\right )+c^2 \left (d^3 e (3 A e+4 B d)+24 C d^5\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}}{7 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{7 e (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {e^2 \left (15 C d^2+6 B e d+8 A e^2\right ) b^3-\left (14 a (3 C d+B e) e^3+c d \left (43 C d^2+3 e (2 B d+5 A e)\right ) e\right ) b^2+\left (35 a^2 C e^4+a c \left (111 C d^2-e (6 B d+29 A e)\right ) e^2+c^2 \left (24 C d^4+e (4 B d+3 A e) d^2\right )\right ) b-6 a c e \left (6 c C d^3+c e (B d-8 A e) d+7 a e^2 (2 C d-B e)\right )+c \left (\left (48 C d^4+2 e (4 B d+3 A e) d^2\right ) c^2+e \left (2 a e \left (51 C d^2+e (12 B d-5 A e)\right )-b \left (104 C d^3+3 e (5 B d+2 A e) d\right )\right ) c+e^2 \left (\left (60 C d^2+e (3 B d+4 A e)\right ) b^2-7 a e (18 C d+B e) b+70 a^2 C e^2\right )\right ) x}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{15 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (-e^2 \left (7 a^2 e^2 (C d-3 B e)+a b e \left (12 A e^2+23 B d e+12 C d^2\right )+b^2 (-d) \left (8 A e^2+6 B d e+15 C d^2\right )\right )-c d e \left (b d \left (15 A e^2+6 B d e+43 C d^2\right )-a e \left (19 A e^2+9 B d e+33 C d^2\right )\right )+e x \left (5 e \left (c d^2-e (b d-a e)\right ) \left (7 a C e-A c e-7 b C d+B c d+\frac {6 c C d^2}{e}\right )+(2 c d-b e) \left (7 a e^2 (2 C d-B e)-b e \left (10 C d^2-e (4 A e+3 B d)\right )+c d e (B d-8 A e)+6 c C d^3\right )\right )+c^2 \left (d^3 e (3 A e+4 B d)+24 C d^5\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}}{7 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{7 e (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {\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+15 B e d-29 A e^2\right )-b d \left (128 C d^2+19 B e d+9 A e^2\right )\right ) c^2+e^2 \left (d \left (103 C d^2+9 B e d+19 A e^2\right ) b^2-a e \left (237 C d^2+B e d-29 A e^2\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}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {2 \int \frac {c \left (d e^2 \left (45 C d^2-e (3 B d+4 A e)\right ) b^3-\left (4 a \left (36 C d^2-B e d+A e^2\right ) e^3+c d^2 \left (61 C d^2+9 B e d-9 A e^2\right ) e\right ) b^2+\left (7 a^2 (23 C d+B e) e^4+5 a c d \left (19 C d^2+9 B e d+5 A e^2\right ) e^2+c^2 d^3 \left (24 C d^2+4 B e d+3 A e^2\right )\right ) b-2 a e \left (35 a^2 C e^4+a c \left (9 C d^2+33 B e d-5 A e^2\right ) e^2+c^2 d^2 \left (6 C d^2+B e d+27 A e^2\right )\right )+\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 ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{c d^2-b e d+a e^2}}{15 e^2 \left (c d^2-b e d+a e^2\right )}-\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 (5 e \left (\frac {6 c C d^2}{e}+B c d-7 b C d-A c e+7 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (6 c C d^3+c e (B d-8 A e) d+7 a e^2 (2 C d-B e)-b e \left (10 C d^2-e (3 B d+4 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{15 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}}{7 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}}{7 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\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+15 B e d-29 A e^2\right )-b d \left (128 C d^2+19 B e d+9 A e^2\right )\right ) c^2+e^2 \left (d \left (103 C d^2+9 B e d+19 A e^2\right ) b^2-a e \left (237 C d^2+B e d-29 A e^2\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}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {c \int \frac {d e^2 \left (45 C d^2-e (3 B d+4 A e)\right ) b^3-\left (4 a \left (36 C d^2-B e d+A e^2\right ) e^3+c d^2 \left (61 C d^2+9 B e d-9 A e^2\right ) e\right ) b^2+\left (7 a^2 (23 C d+B e) e^4+5 a c d \left (19 C d^2+9 B e d+5 A e^2\right ) e^2+c^2 d^3 \left (24 C d^2+4 B e d+3 A e^2\right )\right ) b-2 a e \left (35 a^2 C e^4+a c \left (9 C d^2+33 B e d-5 A e^2\right ) e^2+c^2 d^2 \left (6 C d^2+B e d+27 A e^2\right )\right )+\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 ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{c d^2-b e d+a e^2}}{15 e^2 \left (c d^2-b e d+a e^2\right )}-\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 (5 e \left (\frac {6 c C d^2}{e}+B c d-7 b C d-A c e+7 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (6 c C d^3+c e (B d-8 A e) d+7 a e^2 (2 C d-B e)-b e \left (10 C d^2-e (3 B d+4 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{15 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}}{7 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}}{7 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {\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+15 B e d-29 A e^2\right )-b d \left (128 C d^2+19 B e d+9 A e^2\right )\right ) c^2+e^2 \left (d \left (103 C d^2+9 B e d+19 A e^2\right ) b^2-a e \left (237 C d^2+B e d-29 A e^2\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}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {c \left (\frac {\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 ) \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 (\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+12 B e d-5 A e^2\right )-b d \left (104 C d^2+15 B e d+6 A e^2\right )\right ) c+e^2 \left (\left (60 C d^2+3 B e d+4 A e^2\right ) b^2-7 a e (18 C d+B e) b+70 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}}{15 e^2 \left (c d^2-b e d+a e^2\right )}-\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 (5 e \left (\frac {6 c C d^2}{e}+B c d-7 b C d-A c e+7 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (6 c C d^3+c e (B d-8 A e) d+7 a e^2 (2 C d-B e)-b e \left (10 C d^2-e (3 B d+4 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{15 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}}{7 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}}{7 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {\frac {\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+15 B e d-29 A e^2\right )-b d \left (128 C d^2+19 B e d+9 A e^2\right )\right ) c^2+e^2 \left (d \left (103 C d^2+9 B e d+19 A e^2\right ) b^2-a e \left (237 C d^2+B e d-29 A e^2\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}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \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 {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 (\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+12 B e d-5 A e^2\right )-b d \left (104 C d^2+15 B e d+6 A e^2\right )\right ) c+e^2 \left (\left (60 C d^2+3 B e d+4 A e^2\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}} \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}}{15 e^2 \left (c d^2-b e d+a e^2\right )}-\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 (5 e \left (\frac {6 c C d^2}{e}+B c d-7 b C d-A c e+7 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (6 c C d^3+c e (B d-8 A e) d+7 a e^2 (2 C d-B e)-b e \left (10 C d^2-e (3 B d+4 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{15 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}}{7 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}}{7 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {\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+15 B e d-29 A e^2\right )-b d \left (128 C d^2+19 B e d+9 A e^2\right )\right ) c^2+e^2 \left (d \left (103 C d^2+9 B e d+19 A e^2\right ) b^2-a e \left (237 C d^2+B e d-29 A e^2\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}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \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 {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 (\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+12 B e d-5 A e^2\right )-b d \left (104 C d^2+15 B e d+6 A e^2\right )\right ) c+e^2 \left (\left (60 C d^2+3 B e d+4 A e^2\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}} \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}}{15 e^2 \left (c d^2-b e d+a e^2\right )}-\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 (5 e \left (\frac {6 c C d^2}{e}+B c d-7 b C d-A c e+7 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (6 c C d^3+c e (B d-8 A e) d+7 a e^2 (2 C d-B e)-b e \left (10 C d^2-e (3 B d+4 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{15 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}}{7 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}}{7 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\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+15 B e d-29 A e^2\right )-b d \left (128 C d^2+19 B e d+9 A e^2\right )\right ) c^2+e^2 \left (d \left (103 C d^2+9 B e d+19 A e^2\right ) b^2-a e \left (237 C d^2+B e d-29 A e^2\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}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \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 {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 (\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+12 B e d-5 A e^2\right )-b d \left (104 C d^2+15 B e d+6 A e^2\right )\right ) c+e^2 \left (\left (60 C d^2+3 B e d+4 A e^2\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}} \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}}{15 e^2 \left (c d^2-b e d+a e^2\right )}-\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 (5 e \left (\frac {6 c C d^2}{e}+B c d-7 b C d-A c e+7 a C e\right ) \left (c d^2-e (b d-a e)\right )+(2 c d-b e) \left (6 c C d^3+c e (B d-8 A e) d+7 a e^2 (2 C d-B e)-b e \left (10 C d^2-e (3 B d+4 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{15 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}}{7 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}}{7 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}\) |
Input:
Int[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(9/2),x]
Output:
(-2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(7*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/2)) + ((-2*(c^2*(24*C*d^5 + d^3*e*(4*B*d + 3*A*e)) - e^2*(7*a^2*e^2*(C*d - 3*B*e) - b^2*d*(15*C*d^2 + 6*B*d*e + 8*A*e^2) + a*b* e*(12*C*d^2 + 23*B*d*e + 12*A*e^2)) - c*d*e*(b*d*(43*C*d^2 + 6*B*d*e + 15* A*e^2) - a*e*(33*C*d^2 + 9*B*d*e + 19*A*e^2)) + e*(5*e*(B*c*d - 7*b*C*d + (6*c*C*d^2)/e - A*c*e + 7*a*C*e)*(c*d^2 - e*(b*d - a*e)) + (2*c*d - b*e)*( 6*c*C*d^3 + c*d*e*(B*d - 8*A*e) + 7*a*e^2*(2*C*d - B*e) - b*e*(10*C*d^2 - e*(3*B*d + 4*A*e))))*x)*Sqrt[a + b*x + c*x^2])/(15*e^2*(c*d^2 - b*d*e + a* e^2)*(d + e*x)^(5/2)) + ((2*(c^3*(48*C*d^5 + 2*d^3*e*(4*B*d + 3*A*e)) - b* e^3*(35*a^2*C*e^2 - 14*a*b*e*(3*C*d + B*e) + b^2*(15*C*d^2 + 6*B*d*e + 8*A *e^2)) + c^2*d*e*(2*a*e*(69*C*d^2 + 15*B*d*e - 29*A*e^2) - b*d*(128*C*d^2 + 19*B*d*e + 9*A*e^2)) + c*e^2*(14*a^2*e^2*(11*C*d - 3*B*e) - a*b*e*(237*C *d^2 + B*d*e - 29*A*e^2) + b^2*d*(103*C*d^2 + 9*B*d*e + 19*A*e^2)))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) - (c*((Sqrt[2]*Sq rt[b^2 - 4*a*c]*(c^3*(48*C*d^5 + 2*d^3*e*(4*B*d + 3*A*e)) - b*e^3*(35*a^2* C*e^2 - 14*a*b*e*(3*C*d + B*e) + b^2*(15*C*d^2 + 6*B*d*e + 8*A*e^2)) + c^2 *d*e*(2*a*e*(69*C*d^2 + e*(15*B*d - 29*A*e)) - b*(128*C*d^3 + d*e*(19*B*d + 9*A*e))) + c*e^2*(14*a^2*e^2*(11*C*d - 3*B*e) - a*b*e*(237*C*d^2 + e*(B* d - 29*A*e)) + b^2*(103*C*d^3 + d*e*(9*B*d + 19*A*e))))*Sqrt[d + e*x]*Sqrt [-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqr...
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 = 5.12 (sec) , antiderivative size = 2484, normalized size of antiderivative = 1.85
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2484\) |
| default | \(\text {Expression too large to display}\) | \(88790\) |
Input:
int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(9/2),x,method=_RETURNVERBOS E)
Output:
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/7*(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 )^4-2/35*(A*b*e^3-2*A*c*d*e^2+7*B*a*e^3-8*B*b*d*e^2+9*B*c*d^2*e-14*C*a*d*e ^2+15*C*b*d^2*e-16*C*c*d^3)/e^6/(a*e^2-b*d*e+c*d^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)^3-2/105*(10*A*a*c*e^4-4*A*b^2*e^4+6*A*b* c*d*e^3-6*A*c^2*d^2*e^2+7*B*a*b*e^4-24*B*a*c*d*e^3-3*B*b^2*d*e^3+15*B*b*c* d^2*e^2-8*B*c^2*d^3*e+35*C*a^2*e^4-84*C*a*b*d*e^3+108*C*a*c*d^2*e^2+45*C*b ^2*d^2*e^2-106*C*b*c*d^3*e+57*C*c^2*d^4)/e^5/(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)^2+2/105*(c*e*x^2+b*e*x+a* e)/e^4/(a*e^2-b*d*e+c*d^2)^3*(29*A*a*b*c*e^5-58*A*a*c^2*d*e^4-8*A*b^3*e^5+ 19*A*b^2*c*d*e^4-9*A*b*c^2*d^2*e^3+6*A*c^3*d^3*e^2-42*B*a^2*c*e^5+14*B*a*b ^2*e^5-B*a*b*c*d*e^4+30*B*a*c^2*d^2*e^3-6*B*b^3*d*e^4+9*B*b^2*c*d^2*e^3-19 *B*b*c^2*d^3*e^2+8*B*c^3*d^4*e-35*C*a^2*b*e^5+154*C*a^2*c*d*e^4+42*C*a*b^2 *d*e^4-237*C*a*b*c*d^2*e^3+138*C*a*c^2*d^3*e^2-15*C*b^3*d^2*e^3+103*C*b^2* c*d^3*e^2-128*C*b*c^2*d^4*e+48*C*c^3*d^5)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1 /2)+2*(C*c/e^4-1/105*c*(10*A*a*c*e^4-4*A*b^2*e^4+6*A*b*c*d*e^3-6*A*c^2*d^2 *e^2+7*B*a*b*e^4-24*B*a*c*d*e^3-3*B*b^2*d*e^3+15*B*b*c*d^2*e^2-8*B*c^2*d^3 *e+35*C*a^2*e^4-84*C*a*b*d*e^3+108*C*a*c*d^2*e^2+45*C*b^2*d^2*e^2-106*C*b* c*d^3*e+57*C*c^2*d^4)/e^4/(a*e^2-b*d*e+c*d^2)^2+1/105/e^4*(b*e-c*d)*(29*A* a*b*c*e^5-58*A*a*c^2*d*e^4-8*A*b^3*e^5+19*A*b^2*c*d*e^4-9*A*b*c^2*d^2*e...
Leaf count of result is larger than twice the leaf count of optimal. 4543 vs. \(2 (1278) = 2556\).
Time = 0.78 (sec) , antiderivative size = 4543, normalized size of antiderivative = 3.39 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(9/2),x, algorithm="fr icas")
Output:
Too large to include
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int \frac {\left (A + B x + C x^{2}\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \] Input:
integrate((c*x**2+b*x+a)**(1/2)*(C*x**2+B*x+A)/(e*x+d)**(9/2),x)
Output:
Integral((A + B*x + C*x**2)*sqrt(a + b*x + c*x**2)/(d + e*x)**(9/2), x)
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(9/2),x, algorithm="ma xima")
Output:
integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(9/2), x)
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(9/2),x, algorithm="gi ac")
Output:
integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(9/2), x)
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{9/2}} \,d x \] Input:
int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(9/2),x)
Output:
int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(9/2), x)
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int \frac {\sqrt {c \,x^{2}+b x +a}\, \left (C \,x^{2}+B x +A \right )}{\left (e x +d \right )^{\frac {9}{2}}}d x \] Input:
int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(9/2),x)
Output:
int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(9/2),x)