3.2173 \(\int (d+e x)^2 (f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\)

Optimal. Leaf size=339 \[ \frac{(2 c d-b e)^4 (-7 b e g+4 c d g+10 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{9/2} e^2}+\frac{(b+2 c x) (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+4 c d g+10 c e f)}{128 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+4 c d g+10 c e f)}{48 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+4 c d g+10 c e f)}{40 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2} \]

[Out]

((2*c*d - b*e)^2*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2])/(128*c^4*e) - ((2*c*d - b*e)*(10*c*e*f + 4*c*d*g - 7*b*e*
g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(48*c^3*e^2) - ((10*c*e*f + 4*c*
d*g - 7*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(40*c^2*e^
2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(5*c*e^2) + ((2
*c*d - b*e)^4*(10*c*e*f + 4*c*d*g - 7*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*S
qrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(256*c^(9/2)*e^2)

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Rubi [A]  time = 0.844142, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.159 \[ \frac{(2 c d-b e)^4 (-7 b e g+4 c d g+10 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{9/2} e^2}+\frac{(b+2 c x) (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+4 c d g+10 c e f)}{128 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+4 c d g+10 c e f)}{48 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+4 c d g+10 c e f)}{40 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*d - b*e)^2*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2])/(128*c^4*e) - ((2*c*d - b*e)*(10*c*e*f + 4*c*d*g - 7*b*e*
g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(48*c^3*e^2) - ((10*c*e*f + 4*c*
d*g - 7*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(40*c^2*e^
2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(5*c*e^2) + ((2
*c*d - b*e)^4*(10*c*e*f + 4*c*d*g - 7*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*S
qrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(256*c^(9/2)*e^2)

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Rubi in Sympy [A]  time = 104.476, size = 326, normalized size = 0.96 \[ - \frac{g \left (d + e x\right )^{2} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{5 c e^{2}} + \frac{\left (d + e x\right ) \left (7 b e g - 4 c d g - 10 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{40 c^{2} e^{2}} - \frac{\left (b e - 2 c d\right ) \left (7 b e g - 4 c d g - 10 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{48 c^{3} e^{2}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right )^{2} \left (7 b e g - 4 c d g - 10 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{128 c^{4} e} - \frac{\left (b e - 2 c d\right )^{4} \left (7 b e g - 4 c d g - 10 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{256 c^{\frac{9}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-g*(d + e*x)**2*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(5*c*e**2) + (
d + e*x)*(7*b*e*g - 4*c*d*g - 10*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d
))**(3/2)/(40*c**2*e**2) - (b*e - 2*c*d)*(7*b*e*g - 4*c*d*g - 10*c*e*f)*(-b*e**2
*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(48*c**3*e**2) - (b + 2*c*x)*(b*e - 2*
c*d)**2*(7*b*e*g - 4*c*d*g - 10*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e +
c*d))/(128*c**4*e) - (b*e - 2*c*d)**4*(7*b*e*g - 4*c*d*g - 10*c*e*f)*atan(-e*(-b
- 2*c*x)/(2*sqrt(c)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))))/(256*c**(9
/2)*e**2)

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Mathematica [C]  time = 1.00682, size = 376, normalized size = 1.11 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\frac{210 b^4 e^2 g}{c^4}+\frac{20 b^3 e (76 d g+15 e f)}{c^3}+\frac{16 x^2 \left (-7 b^2 e^2 g+2 b c e (18 d g+5 e f)+32 c^2 d (2 d g+5 e f)\right )}{c^2}-\frac{8 b^2 d (499 d g+250 e f)}{c^2}+\frac{4 x \left (35 b^3 e^3 g-2 b^2 c e^2 (108 d g+25 e f)+4 b c^2 d e (109 d g+70 e f)-120 c^3 d^2 (2 d g-3 e f)\right )}{c^3 e}+\frac{15 i (b e-2 c d)^4 (-7 b e g+4 c d g+10 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{c^{9/2} e^2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}}+\frac{16 b d^2 (274 d g+285 e f)}{c e}+\frac{96 e x^3 (b e g+10 c (2 d g+e f))}{c}-\frac{256 d^3 (7 d g+10 e f)}{e^2}+768 e^2 g x^4\right )}{3840} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((-210*b^4*e^2*g)/c^4 - (256*d^3*(10*e*f
+ 7*d*g))/e^2 + (20*b^3*e*(15*e*f + 76*d*g))/c^3 + (16*b*d^2*(285*e*f + 274*d*g
))/(c*e) - (8*b^2*d*(250*e*f + 499*d*g))/c^2 + (4*(35*b^3*e^3*g - 120*c^3*d^2*(-
3*e*f + 2*d*g) - 2*b^2*c*e^2*(25*e*f + 108*d*g) + 4*b*c^2*d*e*(70*e*f + 109*d*g)
)*x)/(c^3*e) + (16*(-7*b^2*e^2*g + 32*c^2*d*(5*e*f + 2*d*g) + 2*b*c*e*(5*e*f + 1
8*d*g))*x^2)/c^2 + (96*e*(b*e*g + 10*c*(e*f + 2*d*g))*x^3)/c + 768*e^2*g*x^4 + (
(15*I)*(-2*c*d + b*e)^4*(10*c*e*f + 4*c*d*g - 7*b*e*g)*Log[((-I)*e*(b + 2*c*x))/
Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)]])/(c^(9/2)*e^2*Sqrt[d + e*x
]*Sqrt[-(b*e) + c*(d - e*x)])))/3840

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Maple [B]  time = 0.016, size = 1618, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/48*g*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+5/24*b/c^2*(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(3/2)*f+5/8*d^2*f*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-1/
4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c*f+5/8*d^4*f*c/(c*e^2)^(1/2)*arctan(
(c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))+7/40*g*b/c^2*x
*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-7/128*e^2*g*b^4/c^4*(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(1/2)-7/15*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2*d^2*g+1/4*b
^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*e*g+5/128*b^4/c^3*e^4/(c*e^2)^(1
/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*f-1
1/16/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*d^2*g+5/32*b^2/c^2*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*e^2*f+1/8/c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/
2)*b*d^3*g-1/2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e*d*g+1/4*c/e/(c*e^2)^
(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d
^5*g+11/20*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/e*d*g-7/64*e^2*g*b^3/c^3
*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-7/256*e^4*g*b^5/c^4/(c*e^2)^(1/2)*arct
an((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))-5/4*d^3*f*e
/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2))*b-5/16/c^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d*f-5/8/c*e*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*d*f+1/4/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(1/2)*x*d^3*g-15/16/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(1/2))*b*d^4*g-11/32/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*b^2*d^2*g-1/5*g*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c-2/3*(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e*d*f+5/64*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(1/2)*e^2*f+5/16*d^2*f/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b+1/2*b^2/c^
2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*e*g+15/16*d^2*f/c*e^2/(c*e^2)^(1/2)
*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^2+15
/64*b^4/c^3*e^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(1/2))*d*g-25/32*b^3/c^2*e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*
(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^2*g-5/16*b^3/c^2*e^3/(c*e^
2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)
)*d*f+5/4*b^2/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(1/2))*d^3*e*g

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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*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^2*(g*x + f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.0413, 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*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^2*(g*x + f),x, algorithm="fricas")

[Out]

[1/7680*(4*(384*c^4*e^4*g*x^4 + 48*(10*c^4*e^4*f + (20*c^4*d*e^3 + b*c^3*e^4)*g)
*x^3 + 8*(10*(16*c^4*d*e^3 + b*c^3*e^4)*f + (64*c^4*d^2*e^2 + 36*b*c^3*d*e^3 - 7
*b^2*c^2*e^4)*g)*x^2 - 10*(128*c^4*d^3*e - 228*b*c^3*d^2*e^2 + 100*b^2*c^2*d*e^3
- 15*b^3*c*e^4)*f - (896*c^4*d^4 - 2192*b*c^3*d^3*e + 1996*b^2*c^2*d^2*e^2 - 76
0*b^3*c*d*e^3 + 105*b^4*e^4)*g + 2*(10*(36*c^4*d^2*e^2 + 28*b*c^3*d*e^3 - 5*b^2*
c^2*e^4)*f - (240*c^4*d^3*e - 436*b*c^3*d^2*e^2 + 216*b^2*c^2*d*e^3 - 35*b^3*c*e
^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) - 15*(10*(16*c^5*d
^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8*b^3*c^2*d*e^4 + b^4*c*e^5)*f +
(64*c^5*d^5 - 240*b*c^4*d^4*e + 320*b^2*c^3*d^3*e^2 - 200*b^3*c^2*d^2*e^3 + 60*b
^4*c*d*e^4 - 7*b^5*e^5)*g)*log(-4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*
c^2*e*x + b*c*e) + (8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^
2)*sqrt(-c)))/(sqrt(-c)*c^4*e^2), 1/3840*(2*(384*c^4*e^4*g*x^4 + 48*(10*c^4*e^4*
f + (20*c^4*d*e^3 + b*c^3*e^4)*g)*x^3 + 8*(10*(16*c^4*d*e^3 + b*c^3*e^4)*f + (64
*c^4*d^2*e^2 + 36*b*c^3*d*e^3 - 7*b^2*c^2*e^4)*g)*x^2 - 10*(128*c^4*d^3*e - 228*
b*c^3*d^2*e^2 + 100*b^2*c^2*d*e^3 - 15*b^3*c*e^4)*f - (896*c^4*d^4 - 2192*b*c^3*
d^3*e + 1996*b^2*c^2*d^2*e^2 - 760*b^3*c*d*e^3 + 105*b^4*e^4)*g + 2*(10*(36*c^4*
d^2*e^2 + 28*b*c^3*d*e^3 - 5*b^2*c^2*e^4)*f - (240*c^4*d^3*e - 436*b*c^3*d^2*e^2
+ 216*b^2*c^2*d*e^3 - 35*b^3*c*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b
*d*e)*sqrt(c) + 15*(10*(16*c^5*d^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8
*b^3*c^2*d*e^4 + b^4*c*e^5)*f + (64*c^5*d^5 - 240*b*c^4*d^4*e + 320*b^2*c^3*d^3*
e^2 - 200*b^3*c^2*d^2*e^3 + 60*b^4*c*d*e^4 - 7*b^5*e^5)*g)*arctan(1/2*(2*c*e*x +
b*e)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c))))/(c^(9/2)*e^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.33214, size = 710, normalized size = 2.09 \[ \frac{1}{1920} \, \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, g x e^{2} + \frac{{\left (20 \, c^{4} d g e^{7} + 10 \, c^{4} f e^{8} + b c^{3} g e^{8}\right )} e^{\left (-6\right )}}{c^{4}}\right )} x + \frac{{\left (64 \, c^{4} d^{2} g e^{6} + 160 \, c^{4} d f e^{7} + 36 \, b c^{3} d g e^{7} + 10 \, b c^{3} f e^{8} - 7 \, b^{2} c^{2} g e^{8}\right )} e^{\left (-6\right )}}{c^{4}}\right )} x - \frac{{\left (240 \, c^{4} d^{3} g e^{5} - 360 \, c^{4} d^{2} f e^{6} - 436 \, b c^{3} d^{2} g e^{6} - 280 \, b c^{3} d f e^{7} + 216 \, b^{2} c^{2} d g e^{7} + 50 \, b^{2} c^{2} f e^{8} - 35 \, b^{3} c g e^{8}\right )} e^{\left (-6\right )}}{c^{4}}\right )} x - \frac{{\left (896 \, c^{4} d^{4} g e^{4} + 1280 \, c^{4} d^{3} f e^{5} - 2192 \, b c^{3} d^{3} g e^{5} - 2280 \, b c^{3} d^{2} f e^{6} + 1996 \, b^{2} c^{2} d^{2} g e^{6} + 1000 \, b^{2} c^{2} d f e^{7} - 760 \, b^{3} c d g e^{7} - 150 \, b^{3} c f e^{8} + 105 \, b^{4} g e^{8}\right )} e^{\left (-6\right )}}{c^{4}}\right )} + \frac{{\left (64 \, c^{5} d^{5} g + 160 \, c^{5} d^{4} f e - 240 \, b c^{4} d^{4} g e - 320 \, b c^{4} d^{3} f e^{2} + 320 \, b^{2} c^{3} d^{3} g e^{2} + 240 \, b^{2} c^{3} d^{2} f e^{3} - 200 \, b^{3} c^{2} d^{2} g e^{3} - 80 \, b^{3} c^{2} d f e^{4} + 60 \, b^{4} c d g e^{4} + 10 \, b^{4} c f e^{5} - 7 \, b^{5} g e^{5}\right )} \sqrt{-c e^{2}} e^{\left (-3\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{-c e^{2}} x - \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt{-c e^{2}} b \right |}\right )}{256 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(6*(8*g*x*e^2 + (20*c^4*
d*g*e^7 + 10*c^4*f*e^8 + b*c^3*g*e^8)*e^(-6)/c^4)*x + (64*c^4*d^2*g*e^6 + 160*c^
4*d*f*e^7 + 36*b*c^3*d*g*e^7 + 10*b*c^3*f*e^8 - 7*b^2*c^2*g*e^8)*e^(-6)/c^4)*x -
(240*c^4*d^3*g*e^5 - 360*c^4*d^2*f*e^6 - 436*b*c^3*d^2*g*e^6 - 280*b*c^3*d*f*e^
7 + 216*b^2*c^2*d*g*e^7 + 50*b^2*c^2*f*e^8 - 35*b^3*c*g*e^8)*e^(-6)/c^4)*x - (89
6*c^4*d^4*g*e^4 + 1280*c^4*d^3*f*e^5 - 2192*b*c^3*d^3*g*e^5 - 2280*b*c^3*d^2*f*e
^6 + 1996*b^2*c^2*d^2*g*e^6 + 1000*b^2*c^2*d*f*e^7 - 760*b^3*c*d*g*e^7 - 150*b^3
*c*f*e^8 + 105*b^4*g*e^8)*e^(-6)/c^4) + 1/256*(64*c^5*d^5*g + 160*c^5*d^4*f*e -
240*b*c^4*d^4*g*e - 320*b*c^4*d^3*f*e^2 + 320*b^2*c^3*d^3*g*e^2 + 240*b^2*c^3*d^
2*f*e^3 - 200*b^3*c^2*d^2*g*e^3 - 80*b^3*c^2*d*f*e^4 + 60*b^4*c*d*g*e^4 + 10*b^4
*c*f*e^5 - 7*b^5*g*e^5)*sqrt(-c*e^2)*e^(-3)*ln(abs(-2*(sqrt(-c*e^2)*x - sqrt(-c*
x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*c - sqrt(-c*e^2)*b))/c^5