3.3.64 \(\int \frac {\sqrt {a+b x+c x^2} (A+B x+C x^2)}{(d+e x)^{9/2}} \, dx\) [264]

3.3.64.1 Optimal result

Integrand size = 34, antiderivative size = 1363 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\frac {2 \left (2 c^3 d^3 \left (24 C d^2+e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b d \left (128 C d^2+e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 d \left (103 C d^2+e (9 B d+19 A e)\right )\right )\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {2 \left (c^2 d^3 \left (24 C d^2+e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 d^2 \left (6 C d^2+e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 c^3 d^3 \left (24 C d^2+e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b d \left (128 C d^2+e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 d \left (103 C d^2+e (9 B d+19 A e)\right )\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\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 )}{105 e^4 \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (2 c^2 d^2 \left (24 C d^2+e (4 B d+3 A e)\right )+c e \left (2 a e \left (51 C d^2+e (12 B d-5 A e)\right )-b d \left (104 C d^2+3 e (5 B d+2 A e)\right )\right )+e^2 \left (70 a^2 C e^2-7 a b e (18 C d+B e)+b^2 \left (60 C d^2+e (3 B d+4 A e)\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 )}{105 e^4 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

output

-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)-2/105*(c^2*d^3*(24*C*d^2+e*(3*A*e+4*B*d))-e^2*(7*a^2*e^2*(-3*B*e+C 
*d)-b^2*d*(8*A*e^2+6*B*d*e+15*C*d^2)+a*b*e*(12*A*e^2+23*B*d*e+12*C*d^2))-c 
*d*e*(b*d*(15*A*e^2+6*B*d*e+43*C*d^2)-a*e*(19*A*e^2+9*B*d*e+33*C*d^2))+e*( 
7*c^2*d^2*(6*C*d^2+e*(-3*A*e+B*d))+e^2*(35*a^2*C*e^2-7*a*b*e*(-B*e+12*C*d) 
+b^2*(-4*A*e^2-3*B*d*e+45*C*d^2))+c*e*(a*e*(-5*A*e^2-9*B*d*e+93*C*d^2)-b*( 
-21*A*d*e^2+91*C*d^3)))*x)*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c*d^2)^2/( 
e*x+d)^(5/2)+2/105*(2*c^3*d^3*(24*C*d^2+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*d*(128*C*d^2+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*d*(103*C*d^2+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)-1/105*(2*c^3*d^3*(24*C*d^2+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*d*(128*C*d^2+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*d*(103*C*d^2+e*(19*A*e+9 
*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)^3/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c...
 

3.3.64.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 36.64 (sec) , antiderivative size = 19853, normalized size of antiderivative = 14.57 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\text {Result too large to show} \]

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
 

3.3.64.3 Rubi [A] (verified)

Time = 3.27 (sec) , antiderivative size = 1413, normalized size of antiderivative = 1.04, 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...
 

3.3.64.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

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])
 

rule 327

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]
 

rule 1172

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]
 

rule 1229

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]
 

rule 1237

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])
 

rule 1269

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]
 

rule 2181

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]
 

3.3.64.4 Maple [A] (verified)

Time = 4.21 (sec) , antiderivative size = 2484, normalized size of antiderivative = 1.82

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)*(c*x^2+b*x+a)^(1/2)/(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)/(a*e^2-b*d*e+c*d^2)/e^6*(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...
 

3.3.64.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.78 (sec) , antiderivative size = 4543, normalized size of antiderivative = 3.33 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \]

input

integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(9/2),x, algorithm="fr 
icas")
 

output

2/315*((48*C*c^4*d^10 - 8*(19*C*b*c^3 - B*c^4)*d^9*e + (158*C*b^2*c^2 + 6* 
A*c^4 + (174*C*a - 23*B*b)*c^3)*d^8*e^2 - (47*C*b^3*c - 12*(3*B*a - A*b)*c 
^3 + (384*C*a*b - 17*B*b^2)*c^2)*d^7*e^3 - (15*C*b^4 - 104*A*a*c^3 - (208* 
C*a^2 - 106*B*a*b - 17*A*b^2)*c^2 - 3*(79*C*a*b^2 + 4*B*b^3)*c)*d^6*e^4 + 
(42*C*a*b^3 - 6*B*b^4 + 52*(3*B*a^2 - 2*A*a*b)*c^2 - (364*C*a^2*b - B*a*b^ 
2 - 23*A*b^3)*c)*d^5*e^5 - (35*C*a^2*b^2 - 14*B*a*b^3 + 8*A*b^4 + 30*A*a^2 
*c^2 - (210*C*a^3 - 63*B*a^2*b + 41*A*a*b^2)*c)*d^4*e^6 + (48*C*c^4*d^6*e^ 
4 - 8*(19*C*b*c^3 - B*c^4)*d^5*e^5 + (158*C*b^2*c^2 + 6*A*c^4 + (174*C*a - 
 23*B*b)*c^3)*d^4*e^6 - (47*C*b^3*c - 12*(3*B*a - A*b)*c^3 + (384*C*a*b - 
17*B*b^2)*c^2)*d^3*e^7 - (15*C*b^4 - 104*A*a*c^3 - (208*C*a^2 - 106*B*a*b 
- 17*A*b^2)*c^2 - 3*(79*C*a*b^2 + 4*B*b^3)*c)*d^2*e^8 + (42*C*a*b^3 - 6*B* 
b^4 + 52*(3*B*a^2 - 2*A*a*b)*c^2 - (364*C*a^2*b - B*a*b^2 - 23*A*b^3)*c)*d 
*e^9 - (35*C*a^2*b^2 - 14*B*a*b^3 + 8*A*b^4 + 30*A*a^2*c^2 - (210*C*a^3 - 
63*B*a^2*b + 41*A*a*b^2)*c)*e^10)*x^4 + 4*(48*C*c^4*d^7*e^3 - 8*(19*C*b*c^ 
3 - B*c^4)*d^6*e^4 + (158*C*b^2*c^2 + 6*A*c^4 + (174*C*a - 23*B*b)*c^3)*d^ 
5*e^5 - (47*C*b^3*c - 12*(3*B*a - A*b)*c^3 + (384*C*a*b - 17*B*b^2)*c^2)*d 
^4*e^6 - (15*C*b^4 - 104*A*a*c^3 - (208*C*a^2 - 106*B*a*b - 17*A*b^2)*c^2 
- 3*(79*C*a*b^2 + 4*B*b^3)*c)*d^3*e^7 + (42*C*a*b^3 - 6*B*b^4 + 52*(3*B*a^ 
2 - 2*A*a*b)*c^2 - (364*C*a^2*b - B*a*b^2 - 23*A*b^3)*c)*d^2*e^8 - (35*C*a 
^2*b^2 - 14*B*a*b^3 + 8*A*b^4 + 30*A*a^2*c^2 - (210*C*a^3 - 63*B*a^2*b ...
 

3.3.64.6 Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int \frac {\left (A + B x + C x^{2}\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]

input

integrate((C*x**2+B*x+A)*(c*x**2+b*x+a)**(1/2)/(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)
 

3.3.64.7 Maxima [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]

input

integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(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)
 

3.3.64.8 Giac [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]

input

integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(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)
 

3.3.64.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]

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)