\(\int \frac {(f+g x)^2 (A+B x+C x^2)}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx\) [7]

Optimal result

Integrand size = 41, antiderivative size = 995 \[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \left (6 c C d^2-C e (b d-a e)-5 c e (B d-A e)\right ) (e f-d g)^2 \sqrt {a+b x+c x^2}}{5 c e^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 g (4 b C e g-c (4 C e f-6 C d g+5 B e g)) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2 e^3}+\frac {2 C (f+g x)^2 \sqrt {a+b x+c x^2}}{5 c e \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^2 C e^3 (b d-a e) g^2+c e^2 g \left (9 a^2 C e^2 g+2 a b e (10 C e f-13 C d g+5 B e g)-b^2 d (20 C e f-9 C d g+10 B e g)\right )-c^3 \left (2 C d^2 \left (15 e^2 f^2-40 d e f g+24 d^2 g^2\right )+5 e \left (3 A e \left (e^2 f^2-2 d e f g+2 d^2 g^2\right )-B d \left (3 e^2 f^2-12 d e f g+8 d^2 g^2\right )\right )\right )+c^2 e \left (b d \left (15 e g (2 B e f-B d g+A e g)+C \left (15 e^2 f^2-30 d e f g+16 d^2 g^2\right )\right )-a e \left (5 e g (6 B e f-5 B d g+3 A e g)+C \left (15 e^2 f^2-50 d e f g+24 d^2 g^2\right )\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 c^3 e^4 \left (c d^2-b d e+a e^2\right ) \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 (4 b C e^2 (b d-a e) g^2+c e g (a e (10 C e f-12 C d g+5 B e g)-b d (10 C e f-8 C d g+5 B e g))+c^2 \left (2 C d \left (15 e^2 f^2-40 d e f g+24 d^2 g^2\right )-5 e \left (6 A e g (e f-d g)+B \left (3 e^2 f^2-12 d e f g+8 d^2 g^2\right )\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 {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^3 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/5*(6*c*C*d^2-C*e*(-a*e+b*d)-5*c*e*(-A*e+B*d))*(-d*g+e*f)^2*(c*x^2+b*x+a 
)^(1/2)/c/e^3/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)-2/15*g*(4*b*C*e*g-c*(5*B*e 
*g-6*C*d*g+4*C*e*f))*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2/e^3+2/5*C*(g*x+ 
f)^2*(c*x^2+b*x+a)^(1/2)/c/e/(e*x+d)^(1/2)-1/15*2^(1/2)*(-4*a*c+b^2)^(1/2) 
*(8*b^2*C*e^3*(-a*e+b*d)*g^2+c*e^2*g*(9*a^2*C*e^2*g+2*a*b*e*(5*B*e*g-13*C* 
d*g+10*C*e*f)-b^2*d*(10*B*e*g-9*C*d*g+20*C*e*f))-c^3*(2*C*d^2*(24*d^2*g^2- 
40*d*e*f*g+15*e^2*f^2)+5*e*(3*A*e*(2*d^2*g^2-2*d*e*f*g+e^2*f^2)-B*d*(8*d^2 
*g^2-12*d*e*f*g+3*e^2*f^2)))+c^2*e*(b*d*(15*e*g*(A*e*g-B*d*g+2*B*e*f)+C*(1 
6*d^2*g^2-30*d*e*f*g+15*e^2*f^2))-a*e*(5*e*g*(3*A*e*g-5*B*d*g+6*B*e*f)+C*( 
24*d^2*g^2-50*d*e*f*g+15*e^2*f^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))/c^3/e^ 
4/(a*e^2-b*d*e+c*d^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)*(4*b*C*e^2*(-a*e+b*d)*g 
^2+c*e*g*(a*e*(5*B*e*g-12*C*d*g+10*C*e*f)-b*d*(5*B*e*g-8*C*d*g+10*C*e*f))+ 
c^2*(2*C*d*(24*d^2*g^2-40*d*e*f*g+15*e^2*f^2)-5*e*(6*A*e*g*(-d*g+e*f)+B*(8 
*d^2*g^2-12*d*e*f*g+3*e^2*f^2))))*(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^3/e^4/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 38.38 (sec) , antiderivative size = 23948, normalized size of antiderivative = 24.07 \[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[((f + g*x)^2*(A + B*x + C*x^2))/((d + e*x)^(3/2)*Sqrt[a + b*x + 
c*x^2]),x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 6.94 (sec) , antiderivative size = 1052, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.268, Rules used = {2181, 27, 25, 2184, 27, 2184, 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.

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

\(\Big \downarrow \) 2181

\(\displaystyle -\frac {2 \int \frac {C \left (-\frac {c d^2}{e}+b d-a e\right ) g^2 x^3-\frac {\left (c d^2-b e d+a e^2\right ) g (2 C e f-C d g+B e g) x^2}{e^2}-\frac {\left (2 c C d^2 (e f-d g)^2-e (b d-a e) \left (C (e f-d g)^2+e g (2 B e f-B d g+A e g)\right )-c e \left (B d \left (e^2 f^2-4 d e g f+2 d^2 g^2\right )-A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )\right )\right ) x}{e^3}+\frac {-A \left (c d f^2+a g (2 e f-d g)\right ) e^3+a (C d-B e) (e f-d g)^2 e-b d \left (C d (e f-d g)^2-e \left (B (e f-d g)^2+A e g (2 e f-d g)\right )\right )}{e^3}}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g)^2 \left (C d^2-e (B d-A e)\right )}{e^3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -\frac {-C \left (-\frac {c d^2}{e}+b d-a e\right ) g^2 x^3+\frac {\left (c d^2-b e d+a e^2\right ) g (2 C e f-C d g+B e g) x^2}{e^2}-\frac {\left (e (b d-a e) \left (C (e f-d g)^2+e g (2 B e f-B d g+A e g)\right )-c \left (2 C d^2 (e f-d g)^2-e \left (B d \left (e^2 f^2-4 d e g f+2 d^2 g^2\right )-A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )\right )\right )\right ) x}{e^3}+A c d f^2-\frac {a (C d-B e) (e f-d g)^2}{e^2}+a A g (2 e f-d g)+\frac {b d \left (C d (e f-d g)^2-e \left (B (e f-d g)^2+A e g (2 e f-d g)\right )\right )}{e^3}}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g)^2 \left (C d^2-e (B d-A e)\right )}{e^3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-C \left (-\frac {c d^2}{e}+b d-a e\right ) g^2 x^3+\frac {\left (c d^2-b e d+a e^2\right ) g (2 C e f-C d g+B e g) x^2}{e^2}+\frac {\left (2 c C d^2 (e f-d g)^2-e (b d-a e) \left (C (e f-d g)^2+e g (2 B e f-B d g+A e g)\right )-c e \left (B d \left (e^2 f^2-4 d e g f+2 d^2 g^2\right )-A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )\right )\right ) x}{e^3}+A c d f^2-\frac {a (C d-B e) (e f-d g)^2}{e^2}+a A g (2 e f-d g)+\frac {b d \left (C d (e f-d g)^2-e \left (B (e f-d g)^2+A e g (2 e f-d g)\right )\right )}{e^3}}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g)^2 \left (C d^2-e (B d-A e)\right )}{e^3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {\frac {2 \int \frac {b^2 C e g^2 d^3+b \left (2 a C d e^2 g^2+c C d \left (5 e^2 f^2-10 d e g f+4 d^2 g^2\right )-5 c e \left (B (e f-d g)^2+A e g (2 e f-d g)\right )\right ) d+e \left (c d^2-b e d+a e^2\right ) g (10 c C e f-12 c C d g+5 B c e g-4 b C e g) x^2+5 A c e^3 \left (c d f^2+a g (2 e f-d g)\right )-a e \left (3 a C d e^2 g^2-5 B c e (e f-d g)^2+c C d \left (5 e^2 f^2-10 d e g f+8 d^2 g^2\right )\right )+\left (\left (2 C d^2 \left (5 e^2 f^2-10 d e g f+4 d^2 g^2\right )-5 e \left (B d \left (e^2 f^2-4 d e g f+2 d^2 g^2\right )-A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )\right )\right ) c^2+\left (5 a e^3 (C f (e f-2 d g)+g (2 B e f-B d g+A e g))-b d e \left (5 e g (2 B e f-B d g+A e g)+C \left (5 e^2 f^2-10 d e g f+8 d^2 g^2\right )\right )\right ) c+C e^2 \left (5 b^2 d^2-2 a b e d-3 a^2 e^2\right ) g^2\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{5 c e^3}+\frac {2 C g^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{5 c e^3}}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g)^2 \left (C d^2-e (B d-A e)\right )}{e^3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {b^2 C e g^2 d^3+b \left (2 a C d e^2 g^2+c C d \left (5 e^2 f^2-10 d e g f+4 d^2 g^2\right )-5 c e \left (B (e f-d g)^2+A e g (2 e f-d g)\right )\right ) d+e \left (c d^2-b e d+a e^2\right ) g (10 c C e f-12 c C d g+5 B c e g-4 b C e g) x^2+5 A c e^3 \left (c d f^2+a g (2 e f-d g)\right )-a e \left (3 a C d e^2 g^2-5 B c e (e f-d g)^2+c C d \left (5 e^2 f^2-10 d e g f+8 d^2 g^2\right )\right )+\left (\left (2 C d^2 \left (5 e^2 f^2-10 d e g f+4 d^2 g^2\right )-5 e \left (B d \left (e^2 f^2-4 d e g f+2 d^2 g^2\right )-A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )\right )\right ) c^2+\left (5 a e^3 (C f (e f-2 d g)+g (2 B e f-B d g+A e g))-b d e \left (5 e g (2 B e f-B d g+A e g)+C \left (5 e^2 f^2-10 d e g f+8 d^2 g^2\right )\right )\right ) c+C e^2 \left (5 b^2 d^2-2 a b e d-3 a^2 e^2\right ) g^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{5 c e^3}+\frac {2 C g^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{5 c e^3}}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g)^2 \left (C d^2-e (B d-A e)\right )}{e^3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {\frac {2 C \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2} \sqrt {c x^2+b x+a} g^2}{5 c e^3}+\frac {\frac {2 \left (c d^2-b e d+a e^2\right ) g \sqrt {d+e x} \sqrt {c x^2+b x+a} (5 B c e g-4 b C e g+2 c C (5 e f-6 d g))}{3 c}+\frac {2 \int -\frac {e^2 \left (4 C d^2 e^2 g^2 b^3-5 c d^2 e g (2 C e f-C d g+B e g) b^2-\left (4 a^2 C g^2 e^4+10 a c C d^2 g^2 e^2+c^2 d \left (C d \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right )-5 e \left (3 A e g (2 e f-d g)+B \left (3 e^2 f^2-6 d e g f+4 d^2 g^2\right )\right )\right )\right ) b-c e \left (15 A c e^2 \left (c d f^2+a g (2 e f-d g)\right )-a \left (a g (10 C e f-3 C d g+5 B e g) e^2-5 B c \left (3 e^2 f^2-6 d e g f+2 d^2 g^2\right ) e+c C d \left (15 e^2 f^2-20 d e g f+12 d^2 g^2\right )\right )\right )+\left (-\left (\left (2 C \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right ) d^2+5 e \left (3 A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )-B d \left (3 e^2 f^2-12 d e g f+8 d^2 g^2\right )\right )\right ) c^3\right )+e \left (b d \left (15 e g (2 B e f-B d g+A e g)+C \left (15 e^2 f^2-30 d e g f+16 d^2 g^2\right )\right )-a e \left (5 e g (6 B e f-5 B d g+3 A e g)+C \left (15 e^2 f^2-50 d e g f+24 d^2 g^2\right )\right )\right ) c^2+e^2 g \left (-d (20 C e f-9 C d g+10 B e g) b^2+2 a e (10 C e f-13 C d g+5 B e g) b+9 a^2 C e^2 g\right ) c+8 b^2 C e^3 (b d-a e) g^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c e^2}}{5 c e^3}}{c d^2-b e d+a e^2}-\frac {2 \left (C d^2-e (B d-A e)\right ) (e f-d g)^2 \sqrt {c x^2+b x+a}}{e^3 \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 C \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2} \sqrt {c x^2+b x+a} g^2}{5 c e^3}+\frac {\frac {2 \left (c d^2-b e d+a e^2\right ) g (5 B c e g-4 b C e g+2 c C (5 e f-6 d g)) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {\int \frac {4 C d^2 e^2 g^2 b^3-5 c d^2 e g (2 C e f-C d g+B e g) b^2-\left (4 a^2 C g^2 e^4+10 a c C d^2 g^2 e^2+c^2 d \left (C d \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right )-5 e \left (3 A e g (2 e f-d g)+B \left (3 e^2 f^2-6 d e g f+4 d^2 g^2\right )\right )\right )\right ) b-c e \left (15 A c e^2 \left (c d f^2+a g (2 e f-d g)\right )-a \left (a g (10 C e f-3 C d g+5 B e g) e^2-5 B c \left (3 e^2 f^2-6 d e g f+2 d^2 g^2\right ) e+c C d \left (15 e^2 f^2-20 d e g f+12 d^2 g^2\right )\right )\right )+\left (-\left (\left (2 C \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right ) d^2+5 e \left (3 A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )-B d \left (3 e^2 f^2-12 d e g f+8 d^2 g^2\right )\right )\right ) c^3\right )+e \left (b d \left (15 e g (2 B e f-B d g+A e g)+C \left (15 e^2 f^2-30 d e g f+16 d^2 g^2\right )\right )-a e \left (5 e g (6 B e f-5 B d g+3 A e g)+C \left (15 e^2 f^2-50 d e g f+24 d^2 g^2\right )\right )\right ) c^2+e^2 g \left (-d (20 C e f-9 C d g+10 B e g) b^2+2 a e (10 C e f-13 C d g+5 B e g) b+9 a^2 C e^2 g\right ) c+8 b^2 C e^3 (b d-a e) g^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}}{5 c e^3}}{c d^2-b e d+a e^2}-\frac {2 \left (C d^2-e (B d-A e)\right ) (e f-d g)^2 \sqrt {c x^2+b x+a}}{e^3 \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {2 g \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) (-4 b C e g+5 B c e g+2 c C (5 e f-6 d g))}{3 c}-\frac {\frac {\left (c e^2 g \left (9 a^2 C e^2 g+2 a b e (5 B e g-13 C d g+10 C e f)+b^2 (-d) (10 B e g-9 C d g+20 C e f)\right )+c^2 e \left (b d \left (15 e g (A e g-B d g+2 B e f)+C \left (16 d^2 g^2-30 d e f g+15 e^2 f^2\right )\right )-a e \left (5 e g (3 A e g-5 B d g+6 B e f)+C \left (24 d^2 g^2-50 d e f g+15 e^2 f^2\right )\right )\right )+8 b^2 C e^3 g^2 (b d-a e)-\left (c^3 \left (5 e \left (3 A e \left (2 d^2 g^2-2 d e f g+e^2 f^2\right )-B d \left (8 d^2 g^2-12 d e f g+3 e^2 f^2\right )\right )+2 C d^2 \left (24 d^2 g^2-40 d e f g+15 e^2 f^2\right )\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}+\frac {\left (a e^2-b d e+c d^2\right ) \left (c e g (a e (5 B e g-12 C d g+10 C e f)-b d (5 B e g-8 C d g+10 C e f))+4 b C e^2 g^2 (b d-a e)+c^2 \left (2 C d \left (24 d^2 g^2-40 d e f g+15 e^2 f^2\right )-5 e \left (6 A e g (e f-d g)+B \left (8 d^2 g^2-12 d e f g+3 e^2 f^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{3 c}}{5 c e^3}+\frac {2 C g^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{5 c e^3}}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g)^2 \left (C d^2-e (B d-A e)\right )}{e^3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {\frac {2 C \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2} \sqrt {c x^2+b x+a} g^2}{5 c e^3}+\frac {\frac {2 \left (c d^2-b e d+a e^2\right ) g (5 B c e g-4 b C e g+2 c C (5 e f-6 d g)) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (\left (2 C d \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right )-5 e \left (6 A e g (e f-d g)+B \left (3 e^2 f^2-12 d e g f+8 d^2 g^2\right )\right )\right ) c^2+e g (a e (10 C e f-12 C d g+5 B e g)-b d (10 C e f-8 C d g+5 B e g)) c+4 b C e^2 (b d-a e) g^2\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}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (-\left (\left (2 C \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right ) d^2+5 e \left (3 A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )-B d \left (3 e^2 f^2-12 d e g f+8 d^2 g^2\right )\right )\right ) c^3\right )+e \left (b d \left (15 e g (2 B e f-B d g+A e g)+C \left (15 e^2 f^2-30 d e g f+16 d^2 g^2\right )\right )-a e \left (5 e g (6 B e f-5 B d g+3 A e g)+C \left (15 e^2 f^2-50 d e g f+24 d^2 g^2\right )\right )\right ) c^2+e^2 g \left (-d (20 C e f-9 C d g+10 B e g) b^2+2 a e (10 C e f-13 C d g+5 B e g) b+9 a^2 C e^2 g\right ) c+8 b^2 C e^3 (b d-a e) g^2\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}}}{3 c}}{5 c e^3}}{c d^2-b e d+a e^2}-\frac {2 \left (C d^2-e (B d-A e)\right ) (e f-d g)^2 \sqrt {c x^2+b x+a}}{e^3 \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {2 C \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2} \sqrt {c x^2+b x+a} g^2}{5 c e^3}+\frac {\frac {2 \left (c d^2-b e d+a e^2\right ) g (5 B c e g-4 b C e g+2 c C (5 e f-6 d g)) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (\left (2 C d \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right )-5 e \left (6 A e g (e f-d g)+B \left (3 e^2 f^2-12 d e g f+8 d^2 g^2\right )\right )\right ) c^2+e g (a e (10 C e f-12 C d g+5 B e g)-b d (10 C e f-8 C d g+5 B e g)) c+4 b C e^2 (b d-a e) g^2\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}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (-\left (\left (2 C \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right ) d^2+5 e \left (3 A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )-B d \left (3 e^2 f^2-12 d e g f+8 d^2 g^2\right )\right )\right ) c^3\right )+e \left (b d \left (15 e g (2 B e f-B d g+A e g)+C \left (15 e^2 f^2-30 d e g f+16 d^2 g^2\right )\right )-a e \left (5 e g (6 B e f-5 B d g+3 A e g)+C \left (15 e^2 f^2-50 d e g f+24 d^2 g^2\right )\right )\right ) c^2+e^2 g \left (-d (20 C e f-9 C d g+10 B e g) b^2+2 a e (10 C e f-13 C d g+5 B e g) b+9 a^2 C e^2 g\right ) c+8 b^2 C e^3 (b d-a e) g^2\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}}}{3 c}}{5 c e^3}}{c d^2-b e d+a e^2}-\frac {2 \left (C d^2-e (B d-A e)\right ) (e f-d g)^2 \sqrt {c x^2+b x+a}}{e^3 \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2 C \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2} \sqrt {c x^2+b x+a} g^2}{5 c e^3}+\frac {\frac {2 \left (c d^2-b e d+a e^2\right ) g (5 B c e g-4 b C e g+2 c C (5 e f-6 d g)) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (-\left (\left (2 C \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right ) d^2+5 e \left (3 A e \left (e^2 f^2-2 d e g f+2 d^2 g^2\right )-B d \left (3 e^2 f^2-12 d e g f+8 d^2 g^2\right )\right )\right ) c^3\right )+e \left (b d \left (15 e g (2 B e f-B d g+A e g)+C \left (15 e^2 f^2-30 d e g f+16 d^2 g^2\right )\right )-a e \left (5 e g (6 B e f-5 B d g+3 A e g)+C \left (15 e^2 f^2-50 d e g f+24 d^2 g^2\right )\right )\right ) c^2+e^2 g \left (-d (20 C e f-9 C d g+10 B e g) b^2+2 a e (10 C e f-13 C d g+5 B e g) b+9 a^2 C e^2 g\right ) c+8 b^2 C e^3 (b d-a e) g^2\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 (2 C d \left (15 e^2 f^2-40 d e g f+24 d^2 g^2\right )-5 e \left (6 A e g (e f-d g)+B \left (3 e^2 f^2-12 d e g f+8 d^2 g^2\right )\right )\right ) c^2+e g (a e (10 C e f-12 C d g+5 B e g)-b d (10 C e f-8 C d g+5 B e g)) c+4 b C e^2 (b d-a e) g^2\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 c}}{5 c e^3}}{c d^2-b e d+a e^2}-\frac {2 \left (C d^2-e (B d-A e)\right ) (e f-d g)^2 \sqrt {c x^2+b x+a}}{e^3 \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}\)

Input:

Int[((f + g*x)^2*(A + B*x + C*x^2))/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2] 
),x]
 

Output:

(-2*(C*d^2 - e*(B*d - A*e))*(e*f - d*g)^2*Sqrt[a + b*x + c*x^2])/(e^3*(c*d 
^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + ((2*C*(c*d^2 - b*d*e + a*e^2)*g^2*(d 
+ e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(5*c*e^3) + ((2*(c*d^2 - b*d*e + a*e^2 
)*g*(5*B*c*e*g - 4*b*C*e*g + 2*c*C*(5*e*f - 6*d*g))*Sqrt[d + e*x]*Sqrt[a + 
 b*x + c*x^2])/(3*c) - ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8*b^2*C*e^3*(b*d - a*e 
)*g^2 + c*e^2*g*(9*a^2*C*e^2*g + 2*a*b*e*(10*C*e*f - 13*C*d*g + 5*B*e*g) - 
 b^2*d*(20*C*e*f - 9*C*d*g + 10*B*e*g)) - c^3*(2*C*d^2*(15*e^2*f^2 - 40*d* 
e*f*g + 24*d^2*g^2) + 5*e*(3*A*e*(e^2*f^2 - 2*d*e*f*g + 2*d^2*g^2) - B*d*( 
3*e^2*f^2 - 12*d*e*f*g + 8*d^2*g^2))) + c^2*e*(b*d*(15*e*g*(2*B*e*f - B*d* 
g + A*e*g) + C*(15*e^2*f^2 - 30*d*e*f*g + 16*d^2*g^2)) - a*e*(5*e*g*(6*B*e 
*f - 5*B*d*g + 3*A*e*g) + C*(15*e^2*f^2 - 50*d*e*f*g + 24*d^2*g^2))))*Sqrt 
[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sq 
rt[(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)])/(c*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*Sqrt[2 
]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(4*b*C*e^2*(b*d - a*e)*g^2 + c 
*e*g*(a*e*(10*C*e*f - 12*C*d*g + 5*B*e*g) - b*d*(10*C*e*f - 8*C*d*g + 5*B* 
e*g)) + c^2*(2*C*d*(15*e^2*f^2 - 40*d*e*f*g + 24*d^2*g^2) - 5*e*(6*A*e*g*( 
e*f - d*g) + B*(3*e^2*f^2 - 12*d*e*f*g + 8*d^2*g^2))))*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 ...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 11.27 (sec) , antiderivative size = 1643, normalized size of antiderivative = 1.65

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1643\)
risch	\(\text {Expression too large to display}\)	\(2058\)
default	\(\text {Expression too large to display}\)	\(31952\)

Input:

int((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x,method=_RE 
TURNVERBOSE)
 

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*(c*e*x 
^2+b*e*x+a*e)/(a*e^2-b*d*e+c*d^2)/e^4*(A*d^2*e^2*g^2-2*A*d*e^3*f*g+A*e^4*f 
^2-B*d^3*e*g^2+2*B*d^2*e^2*f*g-B*d*e^3*f^2+C*d^4*g^2-2*C*d^3*e*f*g+C*d^2*e 
^2*f^2)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2/5*C*g^2/c/e^2*x*(c*e*x^3+b*e 
*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(1/e^2*g*(B*e*g-C*d*g+2*C*e*f)-2/5 
/c/e^2*(2*b*e+2*c*d)*C*g^2)/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^ 
(1/2)+2*(-(A*d*e^2*g^2-2*A*e^3*f*g-B*d^2*e*g^2+2*B*d*e^2*f*g-B*e^3*f^2+C*d 
^3*g^2-2*C*d^2*e*f*g+C*d*e^2*f^2)/e^4-1/e^4*(b*e-c*d)*(A*d^2*e^2*g^2-2*A*d 
*e^3*f*g+A*e^4*f^2-B*d^3*e*g^2+2*B*d^2*e^2*f*g-B*d*e^3*f^2+C*d^4*g^2-2*C*d 
^3*e*f*g+C*d^2*e^2*f^2)/(a*e^2-b*d*e+c*d^2)+b/e^3/(a*e^2-b*d*e+c*d^2)*(A*d 
^2*e^2*g^2-2*A*d*e^3*f*g+A*e^4*f^2-B*d^3*e*g^2+2*B*d^2*e^2*f*g-B*d*e^3*f^2 
+C*d^4*g^2-2*C*d^3*e*f*g+C*d^2*e^2*f^2)-2/5*a/c*d/e^2*C*g^2-2/3*(1/e^2*g*( 
B*e*g-C*d*g+2*C*e*f)-2/5/c/e^2*(2*b*e+2*c*d)*C*g^2)/c/e*(1/2*a*e+1/2*b*d)) 
*(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)*Ellip 
ticF(((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*(1/e^3*( 
A*e^2*g^2-B*d*e*g^2+2*B*e^2*f*g+C*d^2*g^2-2*C*d*e*f*g+C*e^2*f^2)+1/e^3*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2367 vs. \(2 (937) = 1874\).

Time = 0.17 (sec) , antiderivative size = 2367, normalized size of antiderivative = 2.38 \[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, alg 
orithm="fricas")
 

Output:

-2/45*((15*(2*C*c^4*d^4*e^2 - (2*C*b*c^3 + B*c^4)*d^3*e^3 - (C*b^2*c^2 + 2 
*A*c^4 - 2*(2*C*a + B*b)*c^3)*d^2*e^4 + (C*a*b*c^2 - (3*B*a - A*b)*c^3)*d* 
e^5)*f^2 - 10*(8*C*c^4*d^5*e - (7*C*b*c^3 + 6*B*c^4)*d^4*e^2 - (2*C*b^2*c^ 
2 - 3*A*c^4 - (11*C*a + 6*B*b)*c^3)*d^3*e^3 - (2*C*b^3*c + 6*(2*B*a + A*b) 
*c^3 - (7*C*a*b + 3*B*b^2)*c^2)*d^2*e^4 + (2*C*a*b^2*c + 9*A*a*c^3 - 3*(C* 
a^2 + B*a*b)*c^2)*d*e^5)*f*g + (48*C*c^4*d^6 - 40*(C*b*c^3 + B*c^4)*d^5*e 
- 5*(2*C*b^2*c^2 - 6*A*c^4 - (12*C*a + 7*B*b)*c^3)*d^4*e^2 - 5*(C*b^3*c + 
(11*B*a + 6*A*b)*c^3 - 2*(2*C*a*b + B*b^2)*c^2)*d^3*e^3 - (8*C*b^4 - 60*A* 
a*c^3 + (18*C*a^2 + 35*B*a*b + 15*A*b^2)*c^2 - 2*(17*C*a*b^2 + 5*B*b^3)*c) 
*d^2*e^4 + (8*C*a*b^3 + 15*(B*a^2 + A*a*b)*c^2 - (21*C*a^2*b + 10*B*a*b^2) 
*c)*d*e^5)*g^2 + (15*(2*C*c^4*d^3*e^3 - (2*C*b*c^3 + B*c^4)*d^2*e^4 - (C*b 
^2*c^2 + 2*A*c^4 - 2*(2*C*a + B*b)*c^3)*d*e^5 + (C*a*b*c^2 - (3*B*a - A*b) 
*c^3)*e^6)*f^2 - 10*(8*C*c^4*d^4*e^2 - (7*C*b*c^3 + 6*B*c^4)*d^3*e^3 - (2* 
C*b^2*c^2 - 3*A*c^4 - (11*C*a + 6*B*b)*c^3)*d^2*e^4 - (2*C*b^3*c + 6*(2*B* 
a + A*b)*c^3 - (7*C*a*b + 3*B*b^2)*c^2)*d*e^5 + (2*C*a*b^2*c + 9*A*a*c^3 - 
 3*(C*a^2 + B*a*b)*c^2)*e^6)*f*g + (48*C*c^4*d^5*e - 40*(C*b*c^3 + B*c^4)* 
d^4*e^2 - 5*(2*C*b^2*c^2 - 6*A*c^4 - (12*C*a + 7*B*b)*c^3)*d^3*e^3 - 5*(C* 
b^3*c + (11*B*a + 6*A*b)*c^3 - 2*(2*C*a*b + B*b^2)*c^2)*d^2*e^4 - (8*C*b^4 
 - 60*A*a*c^3 + (18*C*a^2 + 35*B*a*b + 15*A*b^2)*c^2 - 2*(17*C*a*b^2 + 5*B 
*b^3)*c)*d*e^5 + (8*C*a*b^3 + 15*(B*a^2 + A*a*b)*c^2 - (21*C*a^2*b + 10...
 

Sympy [F]

\[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (f + g x\right )^{2} \left (A + B x + C x^{2}\right )}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((g*x+f)**2*(C*x**2+B*x+A)/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(1/2),x 
)
 

Output:

Integral((f + g*x)**2*(A + B*x + C*x**2)/((d + e*x)**(3/2)*sqrt(a + b*x + 
c*x**2)), x)
 

Maxima [F]

\[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (g x + f\right )}^{2}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, alg 
orithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*(g*x + f)^2/(sqrt(c*x^2 + b*x + a)*(e*x + d)^( 
3/2)), x)
 

Giac [F]

\[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (g x + f\right )}^{2}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, alg 
orithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*(g*x + f)^2/(sqrt(c*x^2 + b*x + a)*(e*x + d)^( 
3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (C\,x^2+B\,x+A\right )}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int(((f + g*x)^2*(A + B*x + C*x^2))/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/ 
2)),x)
 

Output:

int(((f + g*x)^2*(A + B*x + C*x^2))/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/ 
2)), x)
 

Reduce [F]

\[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (g x +f \right )^{2} \left (C \,x^{2}+B x +A \right )}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}d x \] Input:

int((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

int((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)