3.2185 \(\int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=297 \[ \frac{(2 c d-b e)^5 (-7 b e g+2 c d g+12 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{9/2} e^2}+\frac{(b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+2 c d g+12 c e f)}{512 c^4 e}+\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+2 c d g+12 c e f)}{192 c^3 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2} \]

[Out]

((2*c*d - b*e)^3*(12*c*e*f + 2*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])/(512*c^4*e) + ((2*c*d - b*e)*(12*c*e*f + 2*c*d*g - 7*b*e*
g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(192*c^3*e) + ((7*b*
e*g - 12*c*(e*f + d*g) - 10*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2)
)/(60*c^2*e^2) + ((2*c*d - b*e)^5*(12*c*e*f + 2*c*d*g - 7*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])])/(1024*c^(9/2)*e^
2)

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Rubi [A]  time = 0.88849, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{(2 c d-b e)^5 (-7 b e g+2 c d g+12 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{9/2} e^2}+\frac{(b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+2 c d g+12 c e f)}{512 c^4 e}+\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+2 c d g+12 c e f)}{192 c^3 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*d - b*e)^3*(12*c*e*f + 2*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])/(512*c^4*e) + ((2*c*d - b*e)*(12*c*e*f + 2*c*d*g - 7*b*e*
g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(192*c^3*e) + ((7*b*
e*g - 12*c*(e*f + d*g) - 10*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2)
)/(60*c^2*e^2) + ((2*c*d - b*e)^5*(12*c*e*f + 2*c*d*g - 7*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])])/(1024*c^(9/2)*e^
2)

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Rubi in Sympy [A]  time = 88.5189, size = 325, normalized size = 1.09 \[ - \frac{g \left (d + e x\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{6 c e^{2}} + \frac{\left (\frac{7 b e g}{2} - c d g - 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{30 c^{2} e^{2}} + \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (7 b e g - 2 c d g - 12 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{192 c^{3} e} + \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right )^{3} \left (7 b e g - 2 c d g - 12 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{512 c^{4} e} + \frac{\left (b e - 2 c d\right )^{5} \left (7 b e g - 2 c d g - 12 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{1024 c^{\frac{9}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-g*(d + e*x)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(6*c*e**2) + (7*b
*e*g/2 - c*d*g - 6*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(30*
c**2*e**2) + (b + 2*c*x)*(b*e - 2*c*d)*(7*b*e*g - 2*c*d*g - 12*c*e*f)*(-b*e**2*x
- c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(192*c**3*e) + (b + 2*c*x)*(b*e - 2*c*d)
**3*(7*b*e*g - 2*c*d*g - 12*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)
)/(512*c**4*e) + (b*e - 2*c*d)**5*(7*b*e*g - 2*c*d*g - 12*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))))/(1024*c**(9/2)
*e**2)

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Mathematica [C]  time = 3.63195, size = 475, normalized size = 1.6 \[ \frac{((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{\sqrt{c} \left (-210 b^5 e^5 g+20 b^4 c e^4 (94 d g+18 e f+7 e g x)-16 b^3 c^2 e^3 \left (407 d^2 g+3 d e (65 f+23 g x)+e^2 x (15 f+7 g x)\right )+96 b^2 c^3 e^2 \left (111 d^3 g+d^2 e (107 f+33 g x)+d e^2 x (19 f+8 g x)+e^3 x^2 (2 f+g x)\right )+32 b c^4 e \left (-273 d^4 g-6 d^3 e (57 f+17 g x)+6 d^2 e^2 x (43 f+29 g x)+4 d e^3 x^2 (93 f+68 g x)+4 e^4 x^3 (33 f+26 g x)\right )+64 c^5 \left (48 d^5 g+3 d^4 e (16 f+5 g x)-6 d^3 e^2 x (25 f+16 g x)-2 d^2 e^3 x^2 (48 f+35 g x)+12 d e^4 x^3 (5 f+4 g x)+8 e^5 x^4 (6 f+5 g x)\right )\right )}{(d+e x) (b e-c d+c e x)}+\frac{15 i (2 c d-b e)^5 (2 c (d g+6 e f)-7 b e g) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{(d+e x)^{3/2} (c (d-e x)-b e)^{3/2}}\right )}{15360 c^{9/2} e^2} \]

Antiderivative was successfully verified.

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

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((Sqrt[c]*(-210*b^5*e^5*g + 20*b^4*c*e
^4*(18*e*f + 94*d*g + 7*e*g*x) - 16*b^3*c^2*e^3*(407*d^2*g + e^2*x*(15*f + 7*g*x
) + 3*d*e*(65*f + 23*g*x)) + 96*b^2*c^3*e^2*(111*d^3*g + e^3*x^2*(2*f + g*x) + d
*e^2*x*(19*f + 8*g*x) + d^2*e*(107*f + 33*g*x)) + 64*c^5*(48*d^5*g + 12*d*e^4*x^
3*(5*f + 4*g*x) + 8*e^5*x^4*(6*f + 5*g*x) + 3*d^4*e*(16*f + 5*g*x) - 6*d^3*e^2*x
*(25*f + 16*g*x) - 2*d^2*e^3*x^2*(48*f + 35*g*x)) + 32*b*c^4*e*(-273*d^4*g - 6*d
^3*e*(57*f + 17*g*x) + 4*e^4*x^3*(33*f + 26*g*x) + 6*d^2*e^2*x*(43*f + 29*g*x) +
4*d*e^3*x^2*(93*f + 68*g*x))))/((d + e*x)*(-(c*d) + b*e + c*e*x)) + ((15*I)*(2*
c*d - b*e)^5*(-7*b*e*g + 2*c*(6*e*f + d*g))*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] + 2
*Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)]])/((d + e*x)^(3/2)*(-(b*e) + c*(d - e*
x))^(3/2))))/(15360*c^(9/2)*e^2)

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Maple [B]  time = 0.019, size = 2117, 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)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-3/8*b*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^5*g-1/6*b/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d*g-1/8*b
/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*e*f+1/16/e*g*c*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(1/2)*x*d^4+1/16/e*g*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^6+7/256*e^3*g*b^4/c^3*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2)*x+7/96*e*g*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
3/2)*x+7/1024*e^5*g*b^6/c^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))+1/48/e*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
3/2)*b*d^2-5/16*b*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3*g+7/60/e*g*b/c^2*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)+1/8*f*d/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(3/2)*b+3/8*f*d^3*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-1/5*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2*d*g-1/12*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(3/2)*d*g+3/8*f*d^5*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))-3/128*b^4/c^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(1/2)*f+7/512*e^3*g*b^5/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+7/192*e*g
*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+1/32/e*g*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(1/2)*b*d^4-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e*f+1/4*f*d*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x+3/16*f*d^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(1/2)*b-3/64*b^3/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*f+9/64*f*d
/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^3-9/32*f*d^2/c*e*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2)*b^2+15/16*f*d^3*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^2-9/16*f*d^2*e*(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b+3/16*b^3/c^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2)*d^2*g-3/256*b^5/c^3*e^5/(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+45/64*b^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))*d^4*g-5/32*b^2/c*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*g-15/16*f*d^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))*b-11/64*b^3/c^2*
e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*g+3/8*b^2/c*e*(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2)*x*d^2*g-5/8*b^3/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))*d^3*g-9/128*b^5/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))*
d*g+75/256*b^4/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^2*g+9/32*f*d/c*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(1/2)*x*b^2+15/128*f*d/c^2*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))*b^4-15/32*f*d^2/c*e^3/(c*e^2)^(1/2)*ar
ctan((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^3-11/12
8*b^4/c^3*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*g-1/6/e*g*x*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(5/2)/c+1/24/e*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^
2-1/16*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e*f

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.34433, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30720*(4*(1280*c^5*e^5*g*x^5 + 128*(12*c^5*e^5*f + (12*c^5*d*e^4 + 13*b*c^4*
e^5)*g)*x^4 + 16*(12*(10*c^5*d*e^4 + 11*b*c^4*e^5)*f - (140*c^5*d^2*e^3 - 272*b*
c^4*d*e^4 - 3*b^2*c^3*e^5)*g)*x^3 - 8*(12*(32*c^5*d^2*e^3 - 62*b*c^4*d*e^4 - b^2
*c^3*e^5)*f + (384*c^5*d^3*e^2 - 348*b*c^4*d^2*e^3 - 48*b^2*c^3*d*e^4 + 7*b^3*c^
2*e^5)*g)*x^2 + 12*(128*c^5*d^4*e - 456*b*c^4*d^3*e^2 + 428*b^2*c^3*d^2*e^3 - 13
0*b^3*c^2*d*e^4 + 15*b^4*c*e^5)*f + (1536*c^5*d^5 - 4368*b*c^4*d^4*e + 5328*b^2*
c^3*d^3*e^2 - 3256*b^3*c^2*d^2*e^3 + 940*b^4*c*d*e^4 - 105*b^5*e^5)*g - 2*(12*(2
00*c^5*d^3*e^2 - 172*b*c^4*d^2*e^3 - 38*b^2*c^3*d*e^4 + 5*b^3*c^2*e^5)*f - (240*
c^5*d^4*e - 816*b*c^4*d^3*e^2 + 792*b^2*c^3*d^2*e^3 - 276*b^3*c^2*d*e^4 + 35*b^4
*c*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) - 15*(12*(32*c
^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c
^2*d*e^5 - b^5*c*e^6)*f + (64*c^6*d^6 - 384*b*c^5*d^5*e + 720*b^2*c^4*d^4*e^2 -
640*b^3*c^3*d^3*e^3 + 300*b^4*c^2*d^2*e^4 - 72*b^5*c*d*e^5 + 7*b^6*e^6)*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),
-1/15360*(2*(1280*c^5*e^5*g*x^5 + 128*(12*c^5*e^5*f + (12*c^5*d*e^4 + 13*b*c^4*
e^5)*g)*x^4 + 16*(12*(10*c^5*d*e^4 + 11*b*c^4*e^5)*f - (140*c^5*d^2*e^3 - 272*b*
c^4*d*e^4 - 3*b^2*c^3*e^5)*g)*x^3 - 8*(12*(32*c^5*d^2*e^3 - 62*b*c^4*d*e^4 - b^2
*c^3*e^5)*f + (384*c^5*d^3*e^2 - 348*b*c^4*d^2*e^3 - 48*b^2*c^3*d*e^4 + 7*b^3*c^
2*e^5)*g)*x^2 + 12*(128*c^5*d^4*e - 456*b*c^4*d^3*e^2 + 428*b^2*c^3*d^2*e^3 - 13
0*b^3*c^2*d*e^4 + 15*b^4*c*e^5)*f + (1536*c^5*d^5 - 4368*b*c^4*d^4*e + 5328*b^2*
c^3*d^3*e^2 - 3256*b^3*c^2*d^2*e^3 + 940*b^4*c*d*e^4 - 105*b^5*e^5)*g - 2*(12*(2
00*c^5*d^3*e^2 - 172*b*c^4*d^2*e^3 - 38*b^2*c^3*d*e^4 + 5*b^3*c^2*e^5)*f - (240*
c^5*d^4*e - 816*b*c^4*d^3*e^2 + 792*b^2*c^3*d^2*e^3 - 276*b^3*c^2*d*e^4 + 35*b^4
*c*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c) - 15*(12*(32*c^
6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^
2*d*e^5 - b^5*c*e^6)*f + (64*c^6*d^6 - 384*b*c^5*d^5*e + 720*b^2*c^4*d^4*e^2 - 6
40*b^3*c^3*d^3*e^3 + 300*b^4*c^2*d^2*e^4 - 72*b^5*c*d*e^5 + 7*b^6*e^6)*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. \[ \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \left (d + e x\right ) \left (f + g x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

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GIAC/XCAS [A]  time = 0.339625, size = 956, normalized size = 3.22 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7680*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(10*c*g*x*e^3 + (
12*c^6*d*g*e^10 + 12*c^6*f*e^11 + 13*b*c^5*g*e^11)*e^(-8)/c^5)*x - (140*c^6*d^2*
g*e^9 - 120*c^6*d*f*e^10 - 272*b*c^5*d*g*e^10 - 132*b*c^5*f*e^11 - 3*b^2*c^4*g*e
^11)*e^(-8)/c^5)*x - (384*c^6*d^3*g*e^8 + 384*c^6*d^2*f*e^9 - 348*b*c^5*d^2*g*e^
9 - 744*b*c^5*d*f*e^10 - 48*b^2*c^4*d*g*e^10 - 12*b^2*c^4*f*e^11 + 7*b^3*c^3*g*e
^11)*e^(-8)/c^5)*x + (240*c^6*d^4*g*e^7 - 2400*c^6*d^3*f*e^8 - 816*b*c^5*d^3*g*e
^8 + 2064*b*c^5*d^2*f*e^9 + 792*b^2*c^4*d^2*g*e^9 + 456*b^2*c^4*d*f*e^10 - 276*b
^3*c^3*d*g*e^10 - 60*b^3*c^3*f*e^11 + 35*b^4*c^2*g*e^11)*e^(-8)/c^5)*x + (1536*c
^6*d^5*g*e^6 + 1536*c^6*d^4*f*e^7 - 4368*b*c^5*d^4*g*e^7 - 5472*b*c^5*d^3*f*e^8
+ 5328*b^2*c^4*d^3*g*e^8 + 5136*b^2*c^4*d^2*f*e^9 - 3256*b^3*c^3*d^2*g*e^9 - 156
0*b^3*c^3*d*f*e^10 + 940*b^4*c^2*d*g*e^10 + 180*b^4*c^2*f*e^11 - 105*b^5*c*g*e^1
1)*e^(-8)/c^5) + 1/1024*(64*c^6*d^6*g + 384*c^6*d^5*f*e - 384*b*c^5*d^5*g*e - 96
0*b*c^5*d^4*f*e^2 + 720*b^2*c^4*d^4*g*e^2 + 960*b^2*c^4*d^3*f*e^3 - 640*b^3*c^3*
d^3*g*e^3 - 480*b^3*c^3*d^2*f*e^4 + 300*b^4*c^2*d^2*g*e^4 + 120*b^4*c^2*d*f*e^5
- 72*b^5*c*d*g*e^5 - 12*b^5*c*f*e^6 + 7*b^6*g*e^6)*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