\(\int \frac {\sqrt {a+b x+c x^2} (A+B x+C x^2)}{(d+e x)^{11/2}} \, dx\) [265]

Optimal result

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=\frac {2 \left (2 c^3 d^3 \left (8 C d^2+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 d^4 \left (8 C d^2+e (4 B d+5 A e)\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 )+c^3 d^2 e \left (6 a e \left (11 C d^2+8 B d e-34 A e^2\right )-b d \left (56 C d^2+5 e (5 B d+4 A e)\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 d^3 \left (8 C d^2+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 \left (3 c^2 d^2 \left (6 C d^2+e (3 B d-5 A e)\right )+c e \left (a e \left (47 C d^2+B d e-7 A e^2\right )-3 b d \left (15 C d^2+2 B d e-5 A e^2\right )\right )+e^2 \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 d^4 \left (8 C d^2+e (4 B d+5 A e)\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 )+c^3 d^2 e \left (6 a e \left (11 C d^2+8 B d e-34 A e^2\right )-b d \left (56 C d^2+5 e (5 B d+4 A e)\right )\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} 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 d^3 \left (8 C d^2+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}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} 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}} \]

[Out]

-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*(-10*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*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*(-3
4*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))))*EllipticE(1/2*((b+2*c*x+(-4
*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^
(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/e^4/(a*e^2-b*d*e+c*d^2)^
4/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)+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+1
7*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)))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2
^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-c*(c*x^2
+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/e^4/(a*e^2-b*d*e+c*d^2)^3/(e*x+
d)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 3.79 (sec) , antiderivative size = 1904, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1664, 824, 848, 857, 732, 435, 430} \[ \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 (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 (\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^2 \left (\frac {3 \left (6 C d^4+e (3 B d-5 A e) d^2\right ) c^2}{e}+\left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+e (2 B d-5 A e) d\right )\right ) c+e \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 \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}}{315 e^3 \left (c d^2-b e d+a e^2\right )^4 \sqrt {d+e x}}+\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}}{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 \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 )}{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 \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 )}{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}} \]

[In]

Int[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(11/2),x]

[Out]

(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])/(315*e^3*(c*d^2 - b*d*e + a*e^2)^3*(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) + b^3*d*(
20*C*d^2 + 25*B*d*e + 56*A*e^2)))*Sqrt[a + b*x + c*x^2])/(315*e^3*(c*d^2 - b*d*e + a*e^2)^4*Sqrt[d + e*x]) - (
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^2*((3*c^2*(6*C*d^4 + d^2*e*(3*B*d - 5*A*e)))/e + c*(a*e*(47*C*d^2 + e*(B*d - 7*A*e)) - 3*
b*(15*C*d^3 + d*e*(2*B*d - 5*A*e))) + e*(21*a^2*C*e^2 - 3*a*b*e*(16*C*d - B*e) + b^2*(25*C*d^2 - e*(B*d + 2*A*
e))))*x)*Sqrt[a + b*x + c*x^2])/(105*e^3*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2)) - (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)) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(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) + b^3*d*(20*C*d^2 + 25*B*d*e + 56*A*e^2)))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))
]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/
(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(315*e^4*(c*d^2 - b*d*e + a*e^2)^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt
[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(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[(c*(d + e*x))/(2*c*d - (b + Sqr
t[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c]
+ 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(315*e^4*
(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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
]

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(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] - Dist[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] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3,
0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(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] + Dist[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] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1664

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[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] + Dist[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)*Q + 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]

Rubi steps \begin{align*} \text {integral}& = -\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} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 38.31 (sec) , antiderivative size = 29140, normalized size of antiderivative = 15.30 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(11/2),x]

[Out]

Result too large to show

Maple [A] (verified)

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	\(\text {Expression too large to display}\)	\(153623\)

[In]

int((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(11/2),x,method=_RETURNVERBOSE)

[Out]

((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+1
7*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+75*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+2
04*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+1
0*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^4+18*B*b^2*c^2*d^3*e^3-25*B*b*c^3*d^4*e^2+8*B*c^4*d^5*e-126*C*
a^3*c*e^6+42*C*a^2*b^2*e^6+36*C*a^2*b*c*d*e^5+180*C*a^2*c^2*d^2*e^4-36*C*a*b^3*d*e^5+30*C*a*b^2*c*d^2*e^4-180*
C*a*b*c^2*d^3*e^3+66*C*a*c^3*d^4*e^2+10*C*b^4*d^2*e^4-20*C*b^3*c*d^3*e^3+66*C*b^2*c^2*d^4*e^2-56*C*b*c^3*d^5*e
+16*C*c^4*d^6)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2*(1/315*c*(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^4/(a*e^2-b*d*e+c*d^2)^3+1/315/e^4*(b*e-c*d)*(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^4+18*B*b^2*c^2*d^3*e^3-25*B*b*c^3*d^4*e^2+8*B*c^4*d^5*e-126*C*a^3*c*e^6+42*C*a^2*b^2
*e^6+36*C*a^2*b*c*d*e^5+180*C*a^2*c^2*d^2*e^4-36*C*a*b^3*d*e^5+30*C*a*b^2*c*d^2*e^4-180*C*a*b*c^2*d^3*e^3+66*C
*a*c^3*d^4*e^2+10*C*b^4*d^2*e^4-20*C*b^3*c*d^3*e^3+66*C*b^2*c^2*d^4*e^2-56*C*b*c^3*d^5*e+16*C*c^4*d^6)/(a*e^2-
b*d*e+c*d^2)^4-1/315*b/e^3/(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^4+18*B*b^2*c^2*d^3*e^3-25*B*b*c^3*d^4*e^2+8*B*c^4*d^5*e-126*C*a^3*c*e^6+42*C*a^2*b^2*e^
6+36*C*a^2*b*c*d*e^5+180*C*a^2*c^2*d^2*e^4-36*C*a*b^3*d*e^5+30*C*a*b^2*c*d^2*e^4-180*C*a*b*c^2*d^3*e^3+66*C*a*
c^3*d^4*e^2+10*C*b^4*d^2*e^4-20*C*b^3*c*d^3*e^3+66*C*b^2*c^2*d^4*e^2-56*C*b*c^3*d^5*e+16*C*c^4*d^6))*(d/e-1/2*
(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))
)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2
))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2)
)/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))-2/315*c/e^3*(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^4+18*B*b^2*c^2*d^3*e^3-25*B*b*
c^3*d^4*e^2+8*B*c^4*d^5*e-126*C*a^3*c*e^6+42*C*a^2*b^2*e^6+36*C*a^2*b*c*d*e^5+180*C*a^2*c^2*d^2*e^4-36*C*a*b^3
*d*e^5+30*C*a*b^2*c*d^2*e^4-180*C*a*b*c^2*d^3*e^3+66*C*a*c^3*d^4*e^2+10*C*b^4*d^2*e^4-20*C*b^3*c*d^3*e^3+66*C*
b^2*c^2*d^4*e^2-56*C*b*c^3*d^5*e+16*C*c^4*d^6)/(a*e^2-b*d*e+c*d^2)^4*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/
e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(
1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*
d*x^2+a*e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)
^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+
(-4*a*c+b^2)^(1/2))*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1
/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))))

Fricas [C] (verification not implemented)

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 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\text {Too large to display} \]

[In]

integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\int \frac {\left (A + B x + C x^{2}\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \]

[In]

integrate((C*x**2+B*x+A)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**(11/2),x)

[Out]

Integral((A + B*x + C*x**2)*sqrt(a + b*x + c*x**2)/(d + e*x)**(11/2), x)

Maxima [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(11/2), x)

Giac [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(11/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx=\int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]

[In]

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(11/2),x)

[Out]

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(11/2), x)