3.2217 \(\int \frac{(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=287 \[ -\frac{3 (2 c d-b e) (-5 b e g+6 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{7/2} e^2}+\frac{3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{4 c^3 e^2}+\frac{(d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{2 c^2 e^2 (2 c d-b e)}+\frac{2 (d+e x)^3 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^3)/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2]) + (3*(4*c*e*f + 6*c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2])/(4*c^3*e^2) + ((4*c*e*f + 6*c*d*g - 5*b*e*g)*(d + e*x)*Sqr
t[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(2*c^2*e^2*(2*c*d - b*e)) - (3*(2*c*d -
b*e)*(4*c*e*f + 6*c*d*g - 5*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2])])/(8*c^(7/2)*e^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.981733, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ -\frac{3 (2 c d-b e) (-5 b e g+6 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{7/2} e^2}+\frac{3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{4 c^3 e^2}+\frac{(d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{2 c^2 e^2 (2 c d-b e)}+\frac{2 (d+e x)^3 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^3*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^3)/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2]) + (3*(4*c*e*f + 6*c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2])/(4*c^3*e^2) + ((4*c*e*f + 6*c*d*g - 5*b*e*g)*(d + e*x)*Sqr
t[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(2*c^2*e^2*(2*c*d - b*e)) - (3*(2*c*d -
b*e)*(4*c*e*f + 6*c*d*g - 5*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2])])/(8*c^(7/2)*e^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 101.477, size = 279, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{3} \left (b e g - c d g - c e f\right )}{c e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{\left (d + e x\right ) \left (5 b e g - 6 c d g - 4 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{2 c^{2} e^{2} \left (b e - 2 c d\right )} - \frac{3 \left (5 b e g - 6 c d g - 4 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{4 c^{3} e^{2}} - \frac{3 \left (b e - 2 c d\right ) \left (5 b e g - 6 c d g - 4 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{8 c^{\frac{7}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

2*(d + e*x)**3*(b*e*g - c*d*g - c*e*f)/(c*e**2*(b*e - 2*c*d)*sqrt(-b*e**2*x - c*
e**2*x**2 + d*(-b*e + c*d))) + (d + e*x)*(5*b*e*g - 6*c*d*g - 4*c*e*f)*sqrt(-b*e
**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(2*c**2*e**2*(b*e - 2*c*d)) - 3*(5*b*e*g -
6*c*d*g - 4*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(4*c**3*e**2)
- 3*(b*e - 2*c*d)*(5*b*e*g - 6*c*d*g - 4*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))))/(8*c**(7/2)*e**2)

_______________________________________________________________________________________

Mathematica [C]  time = 0.812836, size = 229, normalized size = 0.8 \[ \frac{2 \sqrt{c} (d+e x)^2 (c (d-e x)-b e) \left (15 b^2 e^2 g+b c e (-43 d g-12 e f+5 e g x)+2 c^2 \left (14 d^2 g+5 d e (2 f-g x)-e^2 x (2 f+g x)\right )\right )-3 i (d+e x)^{3/2} (2 c d-b e) (c (d-e x)-b e)^{3/2} (-5 b e g+6 c d g+4 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{8 c^{7/2} e^2 ((d+e x) (c (d-e x)-b e))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^3*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[c]*(d + e*x)^2*(-(b*e) + c*(d - e*x))*(15*b^2*e^2*g + b*c*e*(-12*e*f - 4
3*d*g + 5*e*g*x) + 2*c^2*(14*d^2*g + 5*d*e*(2*f - g*x) - e^2*x*(2*f + g*x))) - (
3*I)*(2*c*d - b*e)*(4*c*e*f + 6*c*d*g - 5*b*e*g)*(d + e*x)^(3/2)*(-(b*e) + c*(d
- e*x))^(3/2)*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-(b*e) + c
*(d - e*x)]])/(8*c^(7/2)*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.031, size = 2032, normalized size = 7.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-43/2/c*e^3*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(1/2)*x*d^2*g-7/c*e^4*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*f+11/2*b^2/c^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*d*g+3/4*b^2/c^3*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-7/2/c^2/(-c*e^2
*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d*f+7/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
1/2)*d^3*g+5/c/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*f-1/2*e*g*x^3/c/(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-15/16*e*g*b^3/c^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(1/2)+14*b/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(1/2)*x*d^3*e^2*g+7/c*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*e^2*g+5/c*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e
^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*e^3*f-15/8*e^5*g*b^4/c^3/(-b^2*e^
4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+3/2*b^3/c^
2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^
5*x*f+11/2*b^4/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(1/2)*e^4*d*g+10*b/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(1/2)*x*d^2*e^3*f-43/4/c^2*e^3*b^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2
*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*g-7/2/c^2*e^4*b^3/(-b^2*e^4
+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f+3*x/c/(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-x^2/c*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2)*f-6*b/c^2*x/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*g-3/2*b/c^2*x*e/(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-9/2/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^2*g+9/2*x/c/e/(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*g-3/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*f+2*d^3*f*(-2*c*e^2*x-b*e^2)/(-4*c
*e^2*(-b*d*e+c*d^2)-b^2*e^4)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-3*x^2/c/(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*g+11*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^
2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^4*x*d*g+3/4*b^4/c^3/(-b^2*e^4+4*
b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^5*f+6*b/c^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*g+3/2*b/c^2*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))*f-43/4/c^2/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*
b*d^2*g-15/8*e*g*b^2/c^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))+5/4*e*g*b/c^2*x^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(1/2)+15/8*e*g*b^2/c^3*x/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-15/16*e^5*g*b
^5/c^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/
2)

_______________________________________________________________________________________

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((e*x + d)^3*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 1.1832, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (2 \, c^{2} e^{2} g x^{2} - 4 \,{\left (5 \, c^{2} d e - 3 \, b c e^{2}\right )} f -{\left (28 \, c^{2} d^{2} - 43 \, b c d e + 15 \, b^{2} e^{2}\right )} g +{\left (4 \, c^{2} e^{2} f + 5 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{-c} - 3 \,{\left (4 \,{\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f +{\left (12 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 21 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g -{\left (4 \,{\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} f +{\left (12 \, c^{3} d^{2} e - 16 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3}\right )} g\right )} x\right )} \log \left (-4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{16 \,{\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )} \sqrt{-c}}, \frac{2 \,{\left (2 \, c^{2} e^{2} g x^{2} - 4 \,{\left (5 \, c^{2} d e - 3 \, b c e^{2}\right )} f -{\left (28 \, c^{2} d^{2} - 43 \, b c d e + 15 \, b^{2} e^{2}\right )} g +{\left (4 \, c^{2} e^{2} f + 5 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c} + 3 \,{\left (4 \,{\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f +{\left (12 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 21 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g -{\left (4 \,{\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} f +{\left (12 \, c^{3} d^{2} e - 16 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3}\right )} g\right )} x\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{8 \,{\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )} \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(4*(2*c^2*e^2*g*x^2 - 4*(5*c^2*d*e - 3*b*c*e^2)*f - (28*c^2*d^2 - 43*b*c*d
*e + 15*b^2*e^2)*g + (4*c^2*e^2*f + 5*(2*c^2*d*e - b*c*e^2)*g)*x)*sqrt(-c*e^2*x^
2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) - 3*(4*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*
c*e^3)*f + (12*c^3*d^3 - 28*b*c^2*d^2*e + 21*b^2*c*d*e^2 - 5*b^3*e^3)*g - (4*(2*
c^3*d*e^2 - b*c^2*e^3)*f + (12*c^3*d^2*e - 16*b*c^2*d*e^2 + 5*b^2*c*e^3)*g)*x)*l
og(-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)))/((c^4*e^3*x -
c^4*d*e^2 + b*c^3*e^3)*sqrt(-c)), 1/8*(2*(2*c^2*e^2*g*x^2 - 4*(5*c^2*d*e - 3*b*c
*e^2)*f - (28*c^2*d^2 - 43*b*c*d*e + 15*b^2*e^2)*g + (4*c^2*e^2*f + 5*(2*c^2*d*e
- b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c) + 3*(4*(2*c
^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*f + (12*c^3*d^3 - 28*b*c^2*d^2*e + 21*b^2*
c*d*e^2 - 5*b^3*e^3)*g - (4*(2*c^3*d*e^2 - b*c^2*e^3)*f + (12*c^3*d^2*e - 16*b*c
^2*d*e^2 + 5*b^2*c*e^3)*g)*x)*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^4*e^3*x - c^4*d*e^2 + b*c^3*e^3)*sqrt(c))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Integral((d + e*x)**3*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.326967, size = 841, normalized size = 2.93 \[ \frac{\sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left ({\left ({\left (\frac{2 \,{\left (4 \, c^{4} d^{2} g e^{5} - 4 \, b c^{3} d g e^{6} + b^{2} c^{2} g e^{7}\right )} x}{4 \, c^{5} d^{2} e^{4} - 4 \, b c^{4} d e^{5} + b^{2} c^{3} e^{6}} + \frac{48 \, c^{4} d^{3} g e^{4} + 16 \, c^{4} d^{2} f e^{5} - 68 \, b c^{3} d^{2} g e^{5} - 16 \, b c^{3} d f e^{6} + 32 \, b^{2} c^{2} d g e^{6} + 4 \, b^{2} c^{2} f e^{7} - 5 \, b^{3} c g e^{7}}{4 \, c^{5} d^{2} e^{4} - 4 \, b c^{4} d e^{5} + b^{2} c^{3} e^{6}}\right )} x - \frac{72 \, c^{4} d^{4} g e^{3} + 64 \, c^{4} d^{3} f e^{4} - 224 \, b c^{3} d^{3} g e^{4} - 112 \, b c^{3} d^{2} f e^{5} + 230 \, b^{2} c^{2} d^{2} g e^{5} + 64 \, b^{2} c^{2} d f e^{6} - 98 \, b^{3} c d g e^{6} - 12 \, b^{3} c f e^{7} + 15 \, b^{4} g e^{7}}{4 \, c^{5} d^{2} e^{4} - 4 \, b c^{4} d e^{5} + b^{2} c^{3} e^{6}}\right )} x - \frac{112 \, c^{4} d^{5} g e^{2} + 80 \, c^{4} d^{4} f e^{3} - 284 \, b c^{3} d^{4} g e^{3} - 128 \, b c^{3} d^{3} f e^{4} + 260 \, b^{2} c^{2} d^{3} g e^{4} + 68 \, b^{2} c^{2} d^{2} f e^{5} - 103 \, b^{3} c d^{2} g e^{5} - 12 \, b^{3} c d f e^{6} + 15 \, b^{4} d g e^{6}}{4 \, c^{5} d^{2} e^{4} - 4 \, b c^{4} d e^{5} + b^{2} c^{3} e^{6}}\right )}}{4 \,{\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}} - \frac{3 \,{\left (12 \, c^{2} d^{2} g + 8 \, c^{2} d f e - 16 \, b c d g e - 4 \, b c f e^{2} + 5 \, b^{2} g e^{2}\right )} \sqrt{-c e^{2}} e^{\left (-3\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{-c e^{2}} x - \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt{-c e^{2}} b \right |}\right )}{8 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2),x, algorithm="giac")

[Out]

1/4*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(((2*(4*c^4*d^2*g*e^5 - 4*b*c^3*d
*g*e^6 + b^2*c^2*g*e^7)*x/(4*c^5*d^2*e^4 - 4*b*c^4*d*e^5 + b^2*c^3*e^6) + (48*c^
4*d^3*g*e^4 + 16*c^4*d^2*f*e^5 - 68*b*c^3*d^2*g*e^5 - 16*b*c^3*d*f*e^6 + 32*b^2*
c^2*d*g*e^6 + 4*b^2*c^2*f*e^7 - 5*b^3*c*g*e^7)/(4*c^5*d^2*e^4 - 4*b*c^4*d*e^5 +
b^2*c^3*e^6))*x - (72*c^4*d^4*g*e^3 + 64*c^4*d^3*f*e^4 - 224*b*c^3*d^3*g*e^4 - 1
12*b*c^3*d^2*f*e^5 + 230*b^2*c^2*d^2*g*e^5 + 64*b^2*c^2*d*f*e^6 - 98*b^3*c*d*g*e
^6 - 12*b^3*c*f*e^7 + 15*b^4*g*e^7)/(4*c^5*d^2*e^4 - 4*b*c^4*d*e^5 + b^2*c^3*e^6
))*x - (112*c^4*d^5*g*e^2 + 80*c^4*d^4*f*e^3 - 284*b*c^3*d^4*g*e^3 - 128*b*c^3*d
^3*f*e^4 + 260*b^2*c^2*d^3*g*e^4 + 68*b^2*c^2*d^2*f*e^5 - 103*b^3*c*d^2*g*e^5 -
12*b^3*c*d*f*e^6 + 15*b^4*d*g*e^6)/(4*c^5*d^2*e^4 - 4*b*c^4*d*e^5 + b^2*c^3*e^6)
)/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e) - 3/8*(12*c^2*d^2*g + 8*c^2*d*f*e - 16*b
*c*d*g*e - 4*b*c*f*e^2 + 5*b^2*g*e^2)*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^4