\(\int \frac {\sqrt {a+b x+c x^2} (A+B x+C x^2)}{(d+e x)^{7/2}} \, dx\) [99]

Optimal result

Integrand size = 34, antiderivative size = 976 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (23 b C d^2-\frac {24 c C d^3}{e}+c d (4 B d+A e)-b e (3 B d+2 A e)-5 a e (5 C d-B e)+3 e \left (B c d+5 b C d-\frac {6 c C d^2}{e}-A c e-5 a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 A e^2\right )\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+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 )}{15 e^4 \left (c d^2-b d e+a e^2\right )^2 \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 (15 b C e^2 (b d-a e)+c^2 \left (48 C d^3-2 d e (4 B d+A e)\right )-c e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B e)\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 {1+\frac {b+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 )}{15 c e^4 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/15*(c^2*(48*C*d^4-2*d^2*e*(A*e+4*B*d))+e^2*(30*a^2*C*e^2-5*a*b*e*(-B*e+ 
14*C*d)+b^2*(-2*A*e^2-3*B*d*e+38*C*d^2))-c*e*(b*d*(-2*A*e^2-13*B*d*e+88*C* 
d^2)-2*a*e*(3*A*e^2-8*B*d*e+43*C*d^2)))*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d 
*e+c*d^2)^2/(e*x+d)^(1/2)-2/15*(23*b*C*d^2-24*c*C*d^3/e+c*d*(A*e+4*B*d)-b* 
e*(2*A*e+3*B*d)-5*a*e*(-B*e+5*C*d)+3*e*(B*c*d+5*C*b*d-6*c*C*d^2/e-A*c*e-5* 
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)^(3/2)-2/5*(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)^(5/2) 
+1/15*2^(1/2)*(-4*a*c+b^2)^(1/2)*(c^2*(48*C*d^4-2*d^2*e*(A*e+4*B*d))+e^2*( 
30*a^2*C*e^2-5*a*b*e*(-B*e+14*C*d)+b^2*(-2*A*e^2-3*B*d*e+38*C*d^2))-c*e*(b 
*d*(-2*A*e^2-13*B*d*e+88*C*d^2)-2*a*e*(3*A*e^2-8*B*d*e+43*C*d^2)))*(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)^2/(c*(e*x+d)/(2*c*d-(b+(-4 
*a*c+b^2)^(1/2))*e))^(1/2)/(c*x^2+b*x+a)^(1/2)-2/15*2^(1/2)*(-4*a*c+b^2)^( 
1/2)*(15*b*C*e^2*(-a*e+b*d)+c^2*(48*C*d^3-2*d*e*(A*e+4*B*d))-c*e*(64*b*C*d 
^2-b*e*(A*e+9*B*d)-10*a*e*(-B*e+5*C*d)))*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2) 
^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF(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))/c/e^4/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2) 
/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 37.08 (sec) , antiderivative size = 12997, normalized size of antiderivative = 13.32 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(7/2),x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 3.29 (sec) , antiderivative size = 1013, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2181, 27, 1229, 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.

\(\displaystyle \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 2181

\(\displaystyle -\frac {2 \int -\frac {\left (3 b C d^2-b e (3 B d+2 A e)+5 e (A c d-a C d+a B e)-e \left (-\frac {6 c C d^2}{e}+B c d+5 b C d-A c e-5 a C e\right ) x\right ) \sqrt {c x^2+b x+a}}{2 e (d+e x)^{5/2}}dx}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{5 e (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (3 b C d^2-b e (3 B d+2 A e)+5 e (A c d-a C d+a B e)-e \left (-\frac {6 c C d^2}{e}+B c d+5 b C d-A c e-5 a C e\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^{5/2}}dx}{5 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{5 e (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {-\frac {2 \int -\frac {15 C d^2 e^2 b^3-d e \left (41 c C d^2+30 a C e^2-c e (6 B d-A e)\right ) b^2+\left (15 a^2 C e^4+a c \left (59 C d^2-e (14 B d+A e)\right ) e^2+c^2 \left (24 C d^4-d^2 e (4 B d+A e)\right )\right ) b-2 a c e \left (6 c C d^3-c e (B d+4 A e) d+5 a e^2 (2 C d-B e)\right )+c \left (\left (48 C d^4-2 d^2 e (4 B d+A e)\right ) c^2-e \left (b d \left (88 C d^2-13 B e d-2 A e^2\right )-2 a e \left (43 C d^2-8 B e d+3 A e^2\right )\right ) c+e^2 \left (\left (38 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (14 C d-B e) b+30 a^2 C e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e^2 \left (5 a^2 e^2 (B e+C d)-a b e \left (2 A e^2+3 B d e+22 C d^2\right )+15 b^2 C d^3\right )-c d e \left (b d \left (A e^2-6 B d e+41 C d^2\right )-a e \left (7 A e^2-7 B d e+37 C d^2\right )\right )-e x \left (3 e \left (c d^2-e (b d-a e)\right ) \left (-5 a C e-A c e+5 b C d+B c d-\frac {6 c C d^2}{e}\right )-(2 c d-b e) \left (5 a e^2 (2 C d-B e)-b e \left (8 C d^2-e (2 A e+3 B d)\right )-c d e (4 A e+B d)+6 c C d^3\right )\right )+c^2 \left (24 C d^5-d^3 e (A e+4 B d)\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{5 e (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {15 C d^2 e^2 b^3-d e \left (41 c C d^2+30 a C e^2-c e (6 B d-A e)\right ) b^2+\left (15 a^2 C e^4+a c \left (59 C d^2-e (14 B d+A e)\right ) e^2+c^2 \left (24 C d^4-d^2 e (4 B d+A e)\right )\right ) b-2 a c e \left (6 c C d^3-c e (B d+4 A e) d+5 a e^2 (2 C d-B e)\right )+c \left (\left (48 C d^4-2 d^2 e (4 B d+A e)\right ) c^2-e \left (b d \left (88 C d^2-13 B e d-2 A e^2\right )-2 a e \left (43 C d^2-8 B e d+3 A e^2\right )\right ) c+e^2 \left (\left (38 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (14 C d-B e) b+30 a^2 C e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e^2 \left (5 a^2 e^2 (B e+C d)-a b e \left (2 A e^2+3 B d e+22 C d^2\right )+15 b^2 C d^3\right )-c d e \left (b d \left (A e^2-6 B d e+41 C d^2\right )-a e \left (7 A e^2-7 B d e+37 C d^2\right )\right )-e x \left (3 e \left (c d^2-e (b d-a e)\right ) \left (-5 a C e-A c e+5 b C d+B c d-\frac {6 c C d^2}{e}\right )-(2 c d-b e) \left (5 a e^2 (2 C d-B e)-b e \left (8 C d^2-e (2 A e+3 B d)\right )-c d e (4 A e+B d)+6 c C d^3\right )\right )+c^2 \left (24 C d^5-d^3 e (A e+4 B d)\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{5 e (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {c \left (e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (-2 A e^2-3 B d e+38 C d^2\right )\right )-c e \left (b d \left (-2 A e^2-13 B d e+88 C d^2\right )-2 a e \left (3 A e^2-8 B d e+43 C d^2\right )\right )+c^2 \left (48 C d^4-2 d^2 e (A e+4 B d)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-c e \left (-10 a e (5 C d-B e)-b e (A e+9 B d)+64 b C d^2\right )+15 b C e^2 (b d-a e)+c^2 \left (48 C d^3-2 d e (A e+4 B d)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e^2 \left (5 a^2 e^2 (B e+C d)-a b e \left (2 A e^2+3 B d e+22 C d^2\right )+15 b^2 C d^3\right )-c d e \left (b d \left (A e^2-6 B d e+41 C d^2\right )-a e \left (7 A e^2-7 B d e+37 C d^2\right )\right )-e x \left (3 e \left (c d^2-e (b d-a e)\right ) \left (-5 a C e-A c e+5 b C d+B c d-\frac {6 c C d^2}{e}\right )-(2 c d-b e) \left (5 a e^2 (2 C d-B e)-b e \left (8 C d^2-e (2 A e+3 B d)\right )-c d e (4 A e+B d)+6 c C d^3\right )\right )+c^2 \left (24 C d^5-d^3 e (A e+4 B d)\right )\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{5 e (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\left (48 C d^4-2 d^2 e (4 B d+A e)\right ) c^2-e \left (b d \left (88 C d^2-13 B e d-2 A e^2\right )-2 a e \left (43 C d^2-8 B e d+3 A e^2\right )\right ) c+e^2 \left (\left (38 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (14 C d-B e) b+30 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}}}{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^3-2 d e (4 B d+A e)\right ) c^2-e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B e)\right ) c+15 b C e^2 (b d-a e)\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}}}{3 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (24 C d^5-d^3 e (4 B d+A e)\right ) c^2-d e \left (b d \left (41 C d^2-6 B e d+A e^2\right )-a e \left (37 C d^2-7 B e d+7 A e^2\right )\right ) c+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B e d+2 A e^2\right )\right )-e \left (3 e \left (-\frac {6 c C d^2}{e}+B c d+5 b C d-A c e-5 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+4 A e) d+5 a e^2 (2 C d-B e)-b e \left (8 C d^2-e (3 B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{3 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}}{5 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}}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\left (48 C d^4-2 d^2 e (4 B d+A e)\right ) c^2-e \left (b d \left (88 C d^2-13 B e d-2 A e^2\right )-2 a e \left (43 C d^2-8 B e d+3 A e^2\right )\right ) c+e^2 \left (\left (38 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (14 C d-B e) b+30 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}}}{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^3-2 d e (4 B d+A e)\right ) c^2-e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B e)\right ) c+15 b C e^2 (b d-a e)\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}}}{3 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (24 C d^5-d^3 e (4 B d+A e)\right ) c^2-d e \left (b d \left (41 C d^2-6 B e d+A e^2\right )-a e \left (37 C d^2-7 B e d+7 A e^2\right )\right ) c+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B e d+2 A e^2\right )\right )-e \left (3 e \left (-\frac {6 c C d^2}{e}+B c d+5 b C d-A c e-5 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+4 A e) d+5 a e^2 (2 C d-B e)-b e \left (8 C d^2-e (3 B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{3 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}}{5 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}}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\left (48 C d^4-2 d^2 e (4 B d+A e)\right ) c^2-e \left (b d \left (88 C d^2-13 B e d-2 A e^2\right )-2 a e \left (43 C d^2-8 B e d+3 A e^2\right )\right ) c+e^2 \left (\left (38 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (14 C d-B e) b+30 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 )}{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^3-2 d e (4 B d+A e)\right ) c^2-e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B e)\right ) c+15 b C e^2 (b d-a e)\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}}}{3 e^2 \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (\left (24 C d^5-d^3 e (4 B d+A e)\right ) c^2-d e \left (b d \left (41 C d^2-6 B e d+A e^2\right )-a e \left (37 C d^2-7 B e d+7 A e^2\right )\right ) c+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B e d+2 A e^2\right )\right )-e \left (3 e \left (-\frac {6 c C d^2}{e}+B c d+5 b C d-A c e-5 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+4 A e) d+5 a e^2 (2 C d-B e)-b e \left (8 C d^2-e (3 B d+2 A e)\right )\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{3 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}}{5 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}}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}\)

Input:

Int[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(7/2),x]
 

Output:

(-2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(5*e*(c*d^2 - b*d*e + 
 a*e^2)*(d + e*x)^(5/2)) + ((-2*(c^2*(24*C*d^5 - d^3*e*(4*B*d + A*e)) + e^ 
2*(15*b^2*C*d^3 + 5*a^2*e^2*(C*d + B*e) - a*b*e*(22*C*d^2 + 3*B*d*e + 2*A* 
e^2)) - c*d*e*(b*d*(41*C*d^2 - 6*B*d*e + A*e^2) - a*e*(37*C*d^2 - 7*B*d*e 
+ 7*A*e^2)) - e*(3*e*(B*c*d + 5*b*C*d - (6*c*C*d^2)/e - A*c*e - 5*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 + 4*A*e) + 
5*a*e^2*(2*C*d - B*e) - b*e*(8*C*d^2 - e*(3*B*d + 2*A*e))))*x)*Sqrt[a + b* 
x + c*x^2])/(3*e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2)) + ((Sqrt[2]*Sq 
rt[b^2 - 4*a*c]*(c^2*(48*C*d^4 - 2*d^2*e*(4*B*d + A*e)) + e^2*(30*a^2*C*e^ 
2 - 5*a*b*e*(14*C*d - B*e) + b^2*(38*C*d^2 - 3*B*d*e - 2*A*e^2)) - c*e*(b* 
d*(88*C*d^2 - 13*B*d*e - 2*A*e^2) - 2*a*e*(43*C*d^2 - 8*B*d*e + 3*A*e^2))) 
*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcS 
in[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*S 
qrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(e*Sqrt[(c*(d + 
e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqr 
t[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(15*b*C*e^2*(b*d - a*e) + c 
^2*(48*C*d^3 - 2*d*e*(4*B*d + A*e)) - c*e*(64*b*C*d^2 - b*e*(9*B*d + A*e) 
- 10*a*e*(5*C*d - B*e)))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[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...
 

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 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]
 

Maple [A] (verified)

Time = 4.16 (sec) , antiderivative size = 1766, normalized size of antiderivative = 1.81

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1766\)
default	\(\text {Expression too large to display}\)	\(48427\)

Input:

int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(7/2),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)*(-2/5*(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 
)^3-2/15*(A*b*e^3-2*A*c*d*e^2+5*B*a*e^3-6*B*b*d*e^2+7*B*c*d^2*e-10*C*a*d*e 
^2+11*C*b*d^2*e-12*C*c*d^3)/e^5/(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)^2-2/15*(c*e*x^2+b*e*x+a*e)/e^4/(a*e^2-b* 
d*e+c*d^2)^2*(6*A*a*c*e^4-2*A*b^2*e^4+2*A*b*c*d*e^3-2*A*c^2*d^2*e^2+5*B*a* 
b*e^4-16*B*a*c*d*e^3-3*B*b^2*d*e^3+13*B*b*c*d^2*e^2-8*B*c^2*d^3*e+15*C*a^2 
*e^4-40*C*a*b*d*e^3+56*C*a*c*d^2*e^2+23*C*b^2*d^2*e^2-58*C*b*c*d^3*e+33*C* 
c^2*d^4)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2*((B*c*e+C*b*e-3*C*c*d)/e^4- 
1/15*c*(A*b*e^3-2*A*c*d*e^2+5*B*a*e^3-6*B*b*d*e^2+7*B*c*d^2*e-10*C*a*d*e^2 
+11*C*b*d^2*e-12*C*c*d^3)/e^4/(a*e^2-b*d*e+c*d^2)-1/15/e^4*(b*e-c*d)*(6*A* 
a*c*e^4-2*A*b^2*e^4+2*A*b*c*d*e^3-2*A*c^2*d^2*e^2+5*B*a*b*e^4-16*B*a*c*d*e 
^3-3*B*b^2*d*e^3+13*B*b*c*d^2*e^2-8*B*c^2*d^3*e+15*C*a^2*e^4-40*C*a*b*d*e^ 
3+56*C*a*c*d^2*e^2+23*C*b^2*d^2*e^2-58*C*b*c*d^3*e+33*C*c^2*d^4)/(a*e^2-b* 
d*e+c*d^2)^2+1/15*b/e^3/(a*e^2-b*d*e+c*d^2)^2*(6*A*a*c*e^4-2*A*b^2*e^4+2*A 
*b*c*d*e^3-2*A*c^2*d^2*e^2+5*B*a*b*e^4-16*B*a*c*d*e^3-3*B*b^2*d*e^3+13*B*b 
*c*d^2*e^2-8*B*c^2*d^3*e+15*C*a^2*e^4-40*C*a*b*d*e^3+56*C*a*c*d^2*e^2+23*C 
*b^2*d^2*e^2-58*C*b*c*d^3*e+33*C*c^2*d^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+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2430 vs. \(2 (921) = 1842\).

Time = 0.30 (sec) , antiderivative size = 2430, normalized size of antiderivative = 2.49 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(7/2),x, algorithm="fr 
icas")
 

Output:

-2/45*((48*C*c^3*d^8 - 8*(14*C*b*c^2 + B*c^3)*d^7*e + (73*C*b^2*c - 2*A*c^ 
3 + (122*C*a + 17*B*b)*c^2)*d^6*e^2 - (7*C*b^3 + (22*B*a - 3*A*b)*c^2 + (1 
61*C*a*b + 8*B*b^2)*c)*d^5*e^3 + (20*C*a*b^2 - 3*B*b^3 - 18*A*a*c^2 + (90* 
C*a^2 + 31*B*a*b + 3*A*b^2)*c)*d^4*e^4 - (15*C*a^2*b - 5*B*a*b^2 + 2*A*b^3 
 + 3*(10*B*a^2 - 3*A*a*b)*c)*d^3*e^5 + (48*C*c^3*d^5*e^3 - 8*(14*C*b*c^2 + 
 B*c^3)*d^4*e^4 + (73*C*b^2*c - 2*A*c^3 + (122*C*a + 17*B*b)*c^2)*d^3*e^5 
- (7*C*b^3 + (22*B*a - 3*A*b)*c^2 + (161*C*a*b + 8*B*b^2)*c)*d^2*e^6 + (20 
*C*a*b^2 - 3*B*b^3 - 18*A*a*c^2 + (90*C*a^2 + 31*B*a*b + 3*A*b^2)*c)*d*e^7 
 - (15*C*a^2*b - 5*B*a*b^2 + 2*A*b^3 + 3*(10*B*a^2 - 3*A*a*b)*c)*e^8)*x^3 
+ 3*(48*C*c^3*d^6*e^2 - 8*(14*C*b*c^2 + B*c^3)*d^5*e^3 + (73*C*b^2*c - 2*A 
*c^3 + (122*C*a + 17*B*b)*c^2)*d^4*e^4 - (7*C*b^3 + (22*B*a - 3*A*b)*c^2 + 
 (161*C*a*b + 8*B*b^2)*c)*d^3*e^5 + (20*C*a*b^2 - 3*B*b^3 - 18*A*a*c^2 + ( 
90*C*a^2 + 31*B*a*b + 3*A*b^2)*c)*d^2*e^6 - (15*C*a^2*b - 5*B*a*b^2 + 2*A* 
b^3 + 3*(10*B*a^2 - 3*A*a*b)*c)*d*e^7)*x^2 + 3*(48*C*c^3*d^7*e - 8*(14*C*b 
*c^2 + B*c^3)*d^6*e^2 + (73*C*b^2*c - 2*A*c^3 + (122*C*a + 17*B*b)*c^2)*d^ 
5*e^3 - (7*C*b^3 + (22*B*a - 3*A*b)*c^2 + (161*C*a*b + 8*B*b^2)*c)*d^4*e^4 
 + (20*C*a*b^2 - 3*B*b^3 - 18*A*a*c^2 + (90*C*a^2 + 31*B*a*b + 3*A*b^2)*c) 
*d^3*e^5 - (15*C*a^2*b - 5*B*a*b^2 + 2*A*b^3 + 3*(10*B*a^2 - 3*A*a*b)*c)*d 
^2*e^6)*x)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3 
*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a...
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx=\int \frac {\left (A + B x + C x^{2}\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(1/2)*(C*x**2+B*x+A)/(e*x+d)**(7/2),x)
 

Output:

Integral((A + B*x + C*x**2)*sqrt(a + b*x + c*x**2)/(d + e*x)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(7/2),x, algorithm="ma 
xima")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(7/2),x, algorithm="gi 
ac")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx=\int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{7/2}} \,d x \] Input:

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(7/2),x)
 

Output:

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {c \,x^{2}+b x +a}\, \left (C \,x^{2}+B x +A \right )}{\left (e x +d \right )^{\frac {7}{2}}}d x \] Input:

int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(7/2),x)
 

Output:

int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(7/2),x)