3.100 \(\int \sqrt{a+b x+c x^2} \left (d+e x+f x^2\right )^2 \, dx\)

Optimal. Leaf size=436 \[ -\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{1024 c^{11/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{512 c^5}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (-8 c^2 \left (32 a e f+25 b \left (2 d f+e^2\right )\right )+28 b c f (7 a f+10 b e)-105 b^3 f^2+640 c^3 d e\right )}{960 c^4}+\frac{x \left (a+b x+c x^2\right )^{3/2} \left (-4 c f (5 a f+14 b e)+21 b^2 f^2+40 c^2 \left (2 d f+e^2\right )\right )}{160 c^3}+\frac{f x^2 \left (a+b x+c x^2\right )^{3/2} (8 c e-3 b f)}{20 c^2}+\frac{f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c} \]

[Out]

((128*c^4*d^2 + 21*b^4*f^2 - 56*b^2*c*f*(b*e + a*f) - 32*c^3*(4*b*d*e + a*(e^2 +
2*d*f)) + 8*c^2*(12*a*b*e*f + 2*a^2*f^2 + 5*b^2*(e^2 + 2*d*f)))*(b + 2*c*x)*Sqr
t[a + b*x + c*x^2])/(512*c^5) + ((640*c^3*d*e - 105*b^3*f^2 + 28*b*c*f*(10*b*e +
7*a*f) - 8*c^2*(32*a*e*f + 25*b*(e^2 + 2*d*f)))*(a + b*x + c*x^2)^(3/2))/(960*c
^4) + ((21*b^2*f^2 - 4*c*f*(14*b*e + 5*a*f) + 40*c^2*(e^2 + 2*d*f))*x*(a + b*x +
c*x^2)^(3/2))/(160*c^3) + (f*(8*c*e - 3*b*f)*x^2*(a + b*x + c*x^2)^(3/2))/(20*c
^2) + (f^2*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) - ((b^2 - 4*a*c)*(128*c^4*d^2 + 21
*b^4*f^2 - 56*b^2*c*f*(b*e + a*f) - 32*c^3*(4*b*d*e + a*(e^2 + 2*d*f)) + 8*c^2*(
12*a*b*e*f + 2*a^2*f^2 + 5*b^2*(e^2 + 2*d*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sq
rt[a + b*x + c*x^2])])/(1024*c^(11/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.497, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{1024 c^{11/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{512 c^5}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (-8 c^2 \left (32 a e f+25 b \left (2 d f+e^2\right )\right )+28 b c f (7 a f+10 b e)-105 b^3 f^2+640 c^3 d e\right )}{960 c^4}+\frac{x \left (a+b x+c x^2\right )^{3/2} \left (-4 c f (5 a f+14 b e)+21 b^2 f^2+40 c^2 \left (2 d f+e^2\right )\right )}{160 c^3}+\frac{f x^2 \left (a+b x+c x^2\right )^{3/2} (8 c e-3 b f)}{20 c^2}+\frac{f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)^2,x]

[Out]

((128*c^4*d^2 + 21*b^4*f^2 - 56*b^2*c*f*(b*e + a*f) - 32*c^3*(4*b*d*e + a*(e^2 +
2*d*f)) + 8*c^2*(12*a*b*e*f + 2*a^2*f^2 + 5*b^2*(e^2 + 2*d*f)))*(b + 2*c*x)*Sqr
t[a + b*x + c*x^2])/(512*c^5) + ((640*c^3*d*e - 105*b^3*f^2 + 28*b*c*f*(10*b*e +
7*a*f) - 8*c^2*(32*a*e*f + 25*b*(e^2 + 2*d*f)))*(a + b*x + c*x^2)^(3/2))/(960*c
^4) + ((21*b^2*f^2 - 4*c*f*(14*b*e + 5*a*f) + 40*c^2*(e^2 + 2*d*f))*x*(a + b*x +
c*x^2)^(3/2))/(160*c^3) + (f*(8*c*e - 3*b*f)*x^2*(a + b*x + c*x^2)^(3/2))/(20*c
^2) + (f^2*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) - ((b^2 - 4*a*c)*(128*c^4*d^2 + 21
*b^4*f^2 - 56*b^2*c*f*(b*e + a*f) - 32*c^3*(4*b*d*e + a*(e^2 + 2*d*f)) + 8*c^2*(
12*a*b*e*f + 2*a^2*f^2 + 5*b^2*(e^2 + 2*d*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sq
rt[a + b*x + c*x^2])])/(1024*c^(11/2))

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(1/2)*(f*x**2+e*x+d)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 1.19413, size = 456, normalized size = 1.05 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b c^2 \left (113 a^2 f^2-2 a c \left (f \left (130 d+17 f x^2\right )+65 e^2+58 e f x\right )+4 c^2 \left (30 d^2+10 d x (2 e+f x)+x^2 \left (5 e^2+6 e f x+2 f^2 x^2\right )\right )\right )-32 c^3 \left (a^2 f (64 e+15 f x)-2 a c \left (80 d e+30 d f x+15 e^2 x+16 e f x^2+5 f^2 x^3\right )-4 c^2 x \left (30 d^2+10 d x (4 e+3 f x)+x^2 \left (15 e^2+24 e f x+10 f^2 x^2\right )\right )\right )+8 b^3 c \left (c \left (3 f \left (50 d+7 f x^2\right )+75 e^2+70 e f x\right )-210 a f^2\right )-16 b^2 c^2 \left (c \left (120 d e+50 d f x+25 e^2 x+28 e f x^2+9 f^2 x^3\right )-2 a f (115 e+28 f x)\right )+315 b^5 f^2-210 b^4 c f (4 e+f x)\right )-15 \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{15360 c^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)^2,x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(315*b^5*f^2 - 210*b^4*c*f*(4*e + f*x) - 16*b^2
*c^2*(-2*a*f*(115*e + 28*f*x) + c*(120*d*e + 25*e^2*x + 50*d*f*x + 28*e*f*x^2 +
9*f^2*x^3)) + 8*b^3*c*(-210*a*f^2 + c*(75*e^2 + 70*e*f*x + 3*f*(50*d + 7*f*x^2))
) + 16*b*c^2*(113*a^2*f^2 - 2*a*c*(65*e^2 + 58*e*f*x + f*(130*d + 17*f*x^2)) + 4
*c^2*(30*d^2 + 10*d*x*(2*e + f*x) + x^2*(5*e^2 + 6*e*f*x + 2*f^2*x^2))) - 32*c^3
*(a^2*f*(64*e + 15*f*x) - 2*a*c*(80*d*e + 15*e^2*x + 30*d*f*x + 16*e*f*x^2 + 5*f
^2*x^3) - 4*c^2*x*(30*d^2 + 10*d*x*(4*e + 3*f*x) + x^2*(15*e^2 + 24*e*f*x + 10*f
^2*x^2)))) - 15*(b^2 - 4*a*c)*(128*c^4*d^2 + 21*b^4*f^2 - 56*b^2*c*f*(b*e + a*f)
- 32*c^3*(4*b*d*e + a*(e^2 + 2*d*f)) + 8*c^2*(12*a*b*e*f + 2*a^2*f^2 + 5*b^2*(e
^2 + 2*d*f)))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(15360*c^(11/2))

_______________________________________________________________________________________

Maple [B]  time = 0.027, size = 1429, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)^2,x)

[Out]

7/24*e*f*b^2/c^3*(c*x^2+b*x+a)^(3/2)+2/5*e*f*x^2*(c*x^2+b*x+a)^(3/2)/c+3/8*e*f*b
/c^2*a*(c*x^2+b*x+a)^(1/2)*x+1/6*f^2*x^3*(c*x^2+b*x+a)^(3/2)/c+1/4*d^2/c*(c*x^2+
b*x+a)^(1/2)*b+1/2*d^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/8
*d^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2+2/3*d*e*(c*x^2+b*x+
a)^(3/2)/c+1/4*x*(c*x^2+b*x+a)^(3/2)/c*e^2-5/24*b/c^2*(c*x^2+b*x+a)^(3/2)*e^2+5/
64*b^3/c^3*(c*x^2+b*x+a)^(1/2)*e^2-5/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*e^2-1/8/c^(3/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*e^2-7/64*f^2*b^3/c^4*(c*x^2+b*x+a)^(3/2)+21/512*f^2*b^5/c^5*(c*x^2+b*x+a)^(1/2)
-21/1024*f^2*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/16*f^2/c
^(5/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/16*e*f*b^3/c^(7/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/16*e*f*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)
+3/8*e*f*b/c^(5/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/8/c^2*a*(c*
x^2+b*x+a)^(1/2)*b*d*f-1/2*d*e*b/c*(c*x^2+b*x+a)^(1/2)*x-1/2*d*e*b/c^(3/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-7/20*e*f*b/c^2*x*(c*x^2+b*x+a)^(3/2)-7/
32*e*f*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x-7/32*f^2*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)*x+5/
16*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*d*f+3/8*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*a*d*f-1/4/c*a*(c*x^2+b*x+a)^(1/2)*x*d*f-7/64*e*f*b^4/c^4*(c*x^2
+b*x+a)^(1/2)+7/128*e*f*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-
4/15*e*f/c^2*a*(c*x^2+b*x+a)^(3/2)+1/2*x*(c*x^2+b*x+a)^(3/2)/c*d*f-5/12*b/c^2*(c
*x^2+b*x+a)^(3/2)*d*f+5/32*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*e^2+5/32*b^3/c^3*(c*x^2
+b*x+a)^(1/2)*d*f+3/16*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a
*e^2-5/64*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f-1/8/c*a*(c
*x^2+b*x+a)^(1/2)*x*e^2-1/16/c^2*a*(c*x^2+b*x+a)^(1/2)*b*e^2-1/4/c^(3/2)*a^2*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f-1/4*d*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)
+1/8*d*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-7/64*f^2*b^3/c^
4*a*(c*x^2+b*x+a)^(1/2)-15/64*f^2*b^2/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))+49/240*f^2*b/c^3*a*(c*x^2+b*x+a)^(3/2)-1/8*f^2/c^2*a*x*(c*x^2+b*x+
a)^(3/2)+1/16*f^2/c^2*a^2*(c*x^2+b*x+a)^(1/2)*x+1/32*f^2/c^3*a^2*(c*x^2+b*x+a)^(
1/2)*b-3/20*f^2*b/c^2*x^2*(c*x^2+b*x+a)^(3/2)+21/160*f^2*b^2/c^3*x*(c*x^2+b*x+a)
^(3/2)+21/256*f^2*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x+35/256*f^2*b^4/c^(9/2)*ln((1/2*b
+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/2*d^2*(c*x^2+b*x+a)^(1/2)*x

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(f*x^2 + e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.564142, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(f*x^2 + e*x + d)^2,x, algorithm="fricas")

[Out]

[1/30720*(4*(1280*c^5*f^2*x^5 + 1920*b*c^4*d^2 + 128*(24*c^5*e*f + b*c^4*f^2)*x^
4 + 16*(120*c^5*e^2 - (9*b^2*c^3 - 20*a*c^4)*f^2 + 24*(10*c^5*d + b*c^4*e)*f)*x^
3 - 640*(3*b^2*c^3 - 8*a*c^4)*d*e + 40*(15*b^3*c^2 - 52*a*b*c^3)*e^2 + (315*b^5
- 1680*a*b^3*c + 1808*a^2*b*c^2)*f^2 + 8*(640*c^5*d*e + 40*b*c^4*e^2 + (21*b^3*c
^2 - 68*a*b*c^3)*f^2 + 8*(10*b*c^4*d - (7*b^2*c^3 - 16*a*c^4)*e)*f)*x^2 + 8*(10*
(15*b^3*c^2 - 52*a*b*c^3)*d - (105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*e)*f + 2
*(1920*c^5*d^2 + 640*b*c^4*d*e - 40*(5*b^2*c^3 - 12*a*c^4)*e^2 - (105*b^4*c - 44
8*a*b^2*c^2 + 240*a^2*c^3)*f^2 - 8*(10*(5*b^2*c^3 - 12*a*c^4)*d - (35*b^3*c^2 -
116*a*b*c^3)*e)*f)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 15*(128*(b^2*c^4 - 4*a*c^5
)*d^2 - 128*(b^3*c^3 - 4*a*b*c^4)*d*e + 8*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4
)*e^2 + (21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*f^2 + 8*(2*(5*b^4*
c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d - (7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*e)*
f)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4
*a*c)*sqrt(c)))/c^(11/2), 1/15360*(2*(1280*c^5*f^2*x^5 + 1920*b*c^4*d^2 + 128*(2
4*c^5*e*f + b*c^4*f^2)*x^4 + 16*(120*c^5*e^2 - (9*b^2*c^3 - 20*a*c^4)*f^2 + 24*(
10*c^5*d + b*c^4*e)*f)*x^3 - 640*(3*b^2*c^3 - 8*a*c^4)*d*e + 40*(15*b^3*c^2 - 52
*a*b*c^3)*e^2 + (315*b^5 - 1680*a*b^3*c + 1808*a^2*b*c^2)*f^2 + 8*(640*c^5*d*e +
40*b*c^4*e^2 + (21*b^3*c^2 - 68*a*b*c^3)*f^2 + 8*(10*b*c^4*d - (7*b^2*c^3 - 16*
a*c^4)*e)*f)*x^2 + 8*(10*(15*b^3*c^2 - 52*a*b*c^3)*d - (105*b^4*c - 460*a*b^2*c^
2 + 256*a^2*c^3)*e)*f + 2*(1920*c^5*d^2 + 640*b*c^4*d*e - 40*(5*b^2*c^3 - 12*a*c
^4)*e^2 - (105*b^4*c - 448*a*b^2*c^2 + 240*a^2*c^3)*f^2 - 8*(10*(5*b^2*c^3 - 12*
a*c^4)*d - (35*b^3*c^2 - 116*a*b*c^3)*e)*f)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) -
15*(128*(b^2*c^4 - 4*a*c^5)*d^2 - 128*(b^3*c^3 - 4*a*b*c^4)*d*e + 8*(5*b^4*c^2 -
24*a*b^2*c^3 + 16*a^2*c^4)*e^2 + (21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a
^3*c^3)*f^2 + 8*(2*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d - (7*b^5*c - 40*a*b
^3*c^2 + 48*a^2*b*c^3)*e)*f)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x +
a)*c)))/(sqrt(-c)*c^5)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b x + c x^{2}} \left (d + e x + f x^{2}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(1/2)*(f*x**2+e*x+d)**2,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)*(d + e*x + f*x**2)**2, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.280034, size = 861, normalized size = 1.97 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, f^{2} x + \frac{b c^{4} f^{2} + 24 \, c^{5} f e}{c^{5}}\right )} x + \frac{240 \, c^{5} d f - 9 \, b^{2} c^{3} f^{2} + 20 \, a c^{4} f^{2} + 24 \, b c^{4} f e + 120 \, c^{5} e^{2}}{c^{5}}\right )} x + \frac{80 \, b c^{4} d f + 21 \, b^{3} c^{2} f^{2} - 68 \, a b c^{3} f^{2} + 640 \, c^{5} d e - 56 \, b^{2} c^{3} f e + 128 \, a c^{4} f e + 40 \, b c^{4} e^{2}}{c^{5}}\right )} x + \frac{1920 \, c^{5} d^{2} - 400 \, b^{2} c^{3} d f + 960 \, a c^{4} d f - 105 \, b^{4} c f^{2} + 448 \, a b^{2} c^{2} f^{2} - 240 \, a^{2} c^{3} f^{2} + 640 \, b c^{4} d e + 280 \, b^{3} c^{2} f e - 928 \, a b c^{3} f e - 200 \, b^{2} c^{3} e^{2} + 480 \, a c^{4} e^{2}}{c^{5}}\right )} x + \frac{1920 \, b c^{4} d^{2} + 1200 \, b^{3} c^{2} d f - 4160 \, a b c^{3} d f + 315 \, b^{5} f^{2} - 1680 \, a b^{3} c f^{2} + 1808 \, a^{2} b c^{2} f^{2} - 1920 \, b^{2} c^{3} d e + 5120 \, a c^{4} d e - 840 \, b^{4} c f e + 3680 \, a b^{2} c^{2} f e - 2048 \, a^{2} c^{3} f e + 600 \, b^{3} c^{2} e^{2} - 2080 \, a b c^{3} e^{2}}{c^{5}}\right )} + \frac{{\left (128 \, b^{2} c^{4} d^{2} - 512 \, a c^{5} d^{2} + 80 \, b^{4} c^{2} d f - 384 \, a b^{2} c^{3} d f + 256 \, a^{2} c^{4} d f + 21 \, b^{6} f^{2} - 140 \, a b^{4} c f^{2} + 240 \, a^{2} b^{2} c^{2} f^{2} - 64 \, a^{3} c^{3} f^{2} - 128 \, b^{3} c^{3} d e + 512 \, a b c^{4} d e - 56 \, b^{5} c f e + 320 \, a b^{3} c^{2} f e - 384 \, a^{2} b c^{3} f e + 40 \, b^{4} c^{2} e^{2} - 192 \, a b^{2} c^{3} e^{2} + 128 \, a^{2} c^{4} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(f*x^2 + e*x + d)^2,x, algorithm="giac")

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*f^2*x + (b*c^4*f^2 + 24*c^5*f*e)/c^
5)*x + (240*c^5*d*f - 9*b^2*c^3*f^2 + 20*a*c^4*f^2 + 24*b*c^4*f*e + 120*c^5*e^2)
/c^5)*x + (80*b*c^4*d*f + 21*b^3*c^2*f^2 - 68*a*b*c^3*f^2 + 640*c^5*d*e - 56*b^2
*c^3*f*e + 128*a*c^4*f*e + 40*b*c^4*e^2)/c^5)*x + (1920*c^5*d^2 - 400*b^2*c^3*d*
f + 960*a*c^4*d*f - 105*b^4*c*f^2 + 448*a*b^2*c^2*f^2 - 240*a^2*c^3*f^2 + 640*b*
c^4*d*e + 280*b^3*c^2*f*e - 928*a*b*c^3*f*e - 200*b^2*c^3*e^2 + 480*a*c^4*e^2)/c
^5)*x + (1920*b*c^4*d^2 + 1200*b^3*c^2*d*f - 4160*a*b*c^3*d*f + 315*b^5*f^2 - 16
80*a*b^3*c*f^2 + 1808*a^2*b*c^2*f^2 - 1920*b^2*c^3*d*e + 5120*a*c^4*d*e - 840*b^
4*c*f*e + 3680*a*b^2*c^2*f*e - 2048*a^2*c^3*f*e + 600*b^3*c^2*e^2 - 2080*a*b*c^3
*e^2)/c^5) + 1/1024*(128*b^2*c^4*d^2 - 512*a*c^5*d^2 + 80*b^4*c^2*d*f - 384*a*b^
2*c^3*d*f + 256*a^2*c^4*d*f + 21*b^6*f^2 - 140*a*b^4*c*f^2 + 240*a^2*b^2*c^2*f^2
- 64*a^3*c^3*f^2 - 128*b^3*c^3*d*e + 512*a*b*c^4*d*e - 56*b^5*c*f*e + 320*a*b^3
*c^2*f*e - 384*a^2*b*c^3*f*e + 40*b^4*c^2*e^2 - 192*a*b^2*c^3*e^2 + 128*a^2*c^4*
e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)