\(\int (d+e x) (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\) [149]

Optimal result

Integrand size = 42, antiderivative size = 292 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\frac {(2 c d-b e)^3 (12 c e f+2 c d g-7 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^4 e}+\frac {(2 c d-b e) (12 c e f+2 c d g-7 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^3 e}+\frac {(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac {(2 c d-b e)^5 (12 c e f+2 c d g-7 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{512 c^{9/2} e^2} \] Output:

1/512*(-b*e+2*c*d)^3*(-7*b*e*g+2*c*d*g+12*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b 
*e^2*x-c*e^2*x^2)^(1/2)/c^4/e+1/192*(-b*e+2*c*d)*(-7*b*e*g+2*c*d*g+12*c*e* 
f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^3/e+1/60*(7*b*e*g-12 
*c*(d*g+e*f)-10*c*e*g*x)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^2/e^2+1/ 
512*(-b*e+2*c*d)^5*(-7*b*e*g+2*c*d*g+12*c*e*f)*arctan(c^(1/2)*(e*x+d)/(d*( 
-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(9/2)/e^2
 

Mathematica [A] (verified)

Time = 2.22 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.62 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\frac {(2 c d-b e)^5 ((d+e x) (-b e+c (d-e x)))^{3/2} \left (-\frac {\sqrt {c} \left (-105 b^5 e^5 g+10 b^4 c e^4 (18 e f+94 d g+7 e g x)-8 b^3 c^2 e^3 \left (407 d^2 g+e^2 x (15 f+7 g x)+3 d e (65 f+23 g x)\right )+48 b^2 c^3 e^2 \left (111 d^3 g+e^3 x^2 (2 f+g x)+d e^2 x (19 f+8 g x)+d^2 e (107 f+33 g x)\right )+32 c^5 \left (48 d^5 g+12 d e^4 x^3 (5 f+4 g x)+8 e^5 x^4 (6 f+5 g x)+3 d^4 e (16 f+5 g x)-6 d^3 e^2 x (25 f+16 g x)-2 d^2 e^3 x^2 (48 f+35 g x)\right )+16 b c^4 e \left (-273 d^4 g-6 d^3 e (57 f+17 g x)+4 e^4 x^3 (33 f+26 g x)+6 d^2 e^2 x (43 f+29 g x)+4 d e^3 x^2 (93 f+68 g x)\right )\right )}{(2 c d-b e)^5 (d+e x) (-b e+c (d-e x))}+\frac {15 (7 b e g-2 c (6 e f+d g)) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{3/2} (-b e+c (d-e x))^{3/2}}\right )}{7680 c^{9/2} e^2} \] Input:

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]
 

Output:

((2*c*d - b*e)^5*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-((Sqrt[c]*(-10 
5*b^5*e^5*g + 10*b^4*c*e^4*(18*e*f + 94*d*g + 7*e*g*x) - 8*b^3*c^2*e^3*(40 
7*d^2*g + e^2*x*(15*f + 7*g*x) + 3*d*e*(65*f + 23*g*x)) + 48*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)) + 32*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)) + 16*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*(9 
3*f + 68*g*x))))/((2*c*d - b*e)^5*(d + e*x)*(-(b*e) + c*(d - e*x)))) + (15 
*(7*b*e*g - 2*c*(6*e*f + d*g))*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqr 
t[d + e*x])])/((d + e*x)^(3/2)*(-(b*e) + c*(d - e*x))^(3/2))))/(7680*c^(9/ 
2)*e^2)
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {1225, 1087, 1087, 1092, 217}

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.

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {(2 c d-b e) (-7 b e g+2 c d g+12 c e f) \int \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{24 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c d-b e) (-7 b e g+2 c d g+12 c e f) \left (\frac {3 (2 c d-b e)^2 \int \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c d-b e) (-7 b e g+2 c d g+12 c e f) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(2 c d-b e) (-7 b e g+2 c d g+12 c e f) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right ) (-7 b e g+2 c d g+12 c e f)}{24 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2}\)

Input:

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]
 

Output:

((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)*(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))/(8*c) + (3*(2* 
c*d - b*e)^2*(((b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c 
) + ((2*c*d - b*e)^2*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^(3/2)*e)))/(16*c)))/(24*c^2*e)
 

Defintions of rubi rules used

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1067\) vs. \(2(272)=544\).

Time = 2.41 (sec) , antiderivative size = 1068, normalized size of antiderivative = 3.66

method	result	size
default	\(\text {Expression too large to display}\)	\(1068\)

Input:

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,method=_RETUR 
NVERBOSE)
 

Output:

d*f*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)- 
3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^ 
2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2* 
e^4)/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*g+e*f)*(-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2) 
^(5/2)/c/e^2-1/2*b/c*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b* 
d*e+c*d^2)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c* 
e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(- 
b*d*e+c*d^2)-b^2*e^4)/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)))))+e*g*(-1/6*x*(-c*e^2*x^2-b*e^2* 
x-b*d*e+c*d^2)^(5/2)/c/e^2-7/12*b/c*(-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2) 
^(5/2)/c/e^2-1/2*b/c*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b* 
d*e+c*d^2)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c* 
e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(- 
b*d*e+c*d^2)-b^2*e^4)/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)))))+1/6*(-b*d*e+c*d^2)/c/e^2*(-1/8 
*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-3/16*(-4* 
c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2 
*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (272) = 544\).

Time = 0.78 (sec) , antiderivative size = 1473, normalized size of antiderivative = 5.04 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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, algori 
thm="fricas")
 

Output:

[-1/30720*(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)*sqrt(-c)*log(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 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 
 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) + 4*(1280*c^6*e^5*g*x^5 + 128*(12*c^6* 
e^5*f + (12*c^6*d*e^4 + 13*b*c^5*e^5)*g)*x^4 + 16*(12*(10*c^6*d*e^4 + 11*b 
*c^5*e^5)*f - (140*c^6*d^2*e^3 - 272*b*c^5*d*e^4 - 3*b^2*c^4*e^5)*g)*x^3 - 
 8*(12*(32*c^6*d^2*e^3 - 62*b*c^5*d*e^4 - b^2*c^4*e^5)*f + (384*c^6*d^3*e^ 
2 - 348*b*c^5*d^2*e^3 - 48*b^2*c^4*d*e^4 + 7*b^3*c^3*e^5)*g)*x^2 + 12*(128 
*c^6*d^4*e - 456*b*c^5*d^3*e^2 + 428*b^2*c^4*d^2*e^3 - 130*b^3*c^3*d*e^4 + 
 15*b^4*c^2*e^5)*f + (1536*c^6*d^5 - 4368*b*c^5*d^4*e + 5328*b^2*c^4*d^3*e 
^2 - 3256*b^3*c^3*d^2*e^3 + 940*b^4*c^2*d*e^4 - 105*b^5*c*e^5)*g - 2*(12*( 
200*c^6*d^3*e^2 - 172*b*c^5*d^2*e^3 - 38*b^2*c^4*d*e^4 + 5*b^3*c^3*e^5)*f 
- (240*c^6*d^4*e - 816*b*c^5*d^3*e^2 + 792*b^2*c^4*d^2*e^3 - 276*b^3*c^3*d 
*e^4 + 35*b^4*c^2*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/( 
c^5*e^2), -1/15360*(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)*sqrt(c)*arctan(1/2*sqrt(-c...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3228 vs. \(2 (280) = 560\).

Time = 2.09 (sec) , antiderivative size = 3228, normalized size of antiderivative = 11.05 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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)
 

Output:

Piecewise((sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)*(-c*e**3*g*x**5/ 
6 - x**4*(13*b*c*e**5*g/12 + c**2*d*e**4*g + c**2*e**5*f)/(5*c*e**2) - x** 
3*(b**2*e**5*g + 4*b*c*d*e**4*g + 2*b*c*e**5*f - 9*b*(13*b*c*e**5*g/12 + c 
**2*d*e**4*g + c**2*e**5*f)/(10*c) - 2*c**2*d**2*e**3*g + c**2*d*e**4*f + 
c*e**3*g*(-5*b*d*e + 5*c*d**2)/6)/(4*c*e**2) - x**2*(3*b**2*d*e**4*g + b** 
2*e**5*f + 4*b*c*d*e**4*f - 7*b*(b**2*e**5*g + 4*b*c*d*e**4*g + 2*b*c*e**5 
*f - 9*b*(13*b*c*e**5*g/12 + c**2*d*e**4*g + c**2*e**5*f)/(10*c) - 2*c**2* 
d**2*e**3*g + c**2*d*e**4*f + c*e**3*g*(-5*b*d*e + 5*c*d**2)/6)/(8*c) - 2* 
c**2*d**3*e**2*g - 2*c**2*d**2*e**3*f + (-4*b*d*e + 4*c*d**2)*(13*b*c*e**5 
*g/12 + c**2*d*e**4*g + c**2*e**5*f)/(5*c*e**2))/(3*c*e**2) - x*(3*b**2*d* 
*2*e**3*g + 3*b**2*d*e**4*f - 4*b*c*d**3*e**2*g - 5*b*(3*b**2*d*e**4*g + b 
**2*e**5*f + 4*b*c*d*e**4*f - 7*b*(b**2*e**5*g + 4*b*c*d*e**4*g + 2*b*c*e* 
*5*f - 9*b*(13*b*c*e**5*g/12 + c**2*d*e**4*g + c**2*e**5*f)/(10*c) - 2*c** 
2*d**2*e**3*g + c**2*d*e**4*f + c*e**3*g*(-5*b*d*e + 5*c*d**2)/6)/(8*c) - 
2*c**2*d**3*e**2*g - 2*c**2*d**2*e**3*f + (-4*b*d*e + 4*c*d**2)*(13*b*c*e* 
*5*g/12 + c**2*d*e**4*g + c**2*e**5*f)/(5*c*e**2))/(6*c) + c**2*d**4*e*g - 
 2*c**2*d**3*e**2*f + (-3*b*d*e + 3*c*d**2)*(b**2*e**5*g + 4*b*c*d*e**4*g 
+ 2*b*c*e**5*f - 9*b*(13*b*c*e**5*g/12 + c**2*d*e**4*g + c**2*e**5*f)/(10* 
c) - 2*c**2*d**2*e**3*g + c**2*d*e**4*f + c*e**3*g*(-5*b*d*e + 5*c*d**2)/6 
)/(4*c*e**2))/(2*c*e**2) - (b**2*d**3*e**2*g + 3*b**2*d**2*e**3*f - 2*b...
 

Maxima [F(-2)]

Exception generated. \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:

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, algori 
thm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (272) = 544\).

Time = 0.38 (sec) , antiderivative size = 765, normalized size of antiderivative = 2.62 \[ \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=-\frac {1}{7680} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c e^{3} g x + \frac {12 \, c^{6} e^{11} f + 12 \, c^{6} d e^{10} g + 13 \, b c^{5} e^{11} g}{c^{5} e^{8}}\right )} x + \frac {120 \, c^{6} d e^{10} f + 132 \, b c^{5} e^{11} f - 140 \, c^{6} d^{2} e^{9} g + 272 \, b c^{5} d e^{10} g + 3 \, b^{2} c^{4} e^{11} g}{c^{5} e^{8}}\right )} x - \frac {384 \, c^{6} d^{2} e^{9} f - 744 \, b c^{5} d e^{10} f - 12 \, b^{2} c^{4} e^{11} f + 384 \, c^{6} d^{3} e^{8} g - 348 \, b c^{5} d^{2} e^{9} g - 48 \, b^{2} c^{4} d e^{10} g + 7 \, b^{3} c^{3} e^{11} g}{c^{5} e^{8}}\right )} x - \frac {2400 \, c^{6} d^{3} e^{8} f - 2064 \, b c^{5} d^{2} e^{9} f - 456 \, b^{2} c^{4} d e^{10} f + 60 \, b^{3} c^{3} e^{11} f - 240 \, c^{6} d^{4} e^{7} g + 816 \, b c^{5} d^{3} e^{8} g - 792 \, b^{2} c^{4} d^{2} e^{9} g + 276 \, b^{3} c^{3} d e^{10} g - 35 \, b^{4} c^{2} e^{11} g}{c^{5} e^{8}}\right )} x + \frac {1536 \, c^{6} d^{4} e^{7} f - 5472 \, b c^{5} d^{3} e^{8} f + 5136 \, b^{2} c^{4} d^{2} e^{9} f - 1560 \, b^{3} c^{3} d e^{10} f + 180 \, b^{4} c^{2} e^{11} f + 1536 \, c^{6} d^{5} e^{6} g - 4368 \, b c^{5} d^{4} e^{7} g + 5328 \, b^{2} c^{4} d^{3} e^{8} g - 3256 \, b^{3} c^{3} d^{2} e^{9} g + 940 \, b^{4} c^{2} d e^{10} g - 105 \, b^{5} c e^{11} g}{c^{5} e^{8}}\right )} - \frac {{\left (384 \, c^{6} d^{5} e f - 960 \, b c^{5} d^{4} e^{2} f + 960 \, b^{2} c^{4} d^{3} e^{3} f - 480 \, b^{3} c^{3} d^{2} e^{4} f + 120 \, b^{4} c^{2} d e^{5} f - 12 \, b^{5} c e^{6} f + 64 \, c^{6} d^{6} g - 384 \, b c^{5} d^{5} e g + 720 \, b^{2} c^{4} d^{4} e^{2} g - 640 \, b^{3} c^{3} d^{3} e^{3} g + 300 \, b^{4} c^{2} d^{2} e^{4} g - 72 \, b^{5} c d e^{5} g + 7 \, b^{6} e^{6} g\right )} \log \left ({\left | -b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} {\left | e \right |} \right |}\right )}{1024 \, \sqrt {-c} c^{4} e {\left | e \right |}} \] Input:

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, algori 
thm="giac")
 

Output:

-1/7680*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(2*(8*(10*c*e^3*g 
*x + (12*c^6*e^11*f + 12*c^6*d*e^10*g + 13*b*c^5*e^11*g)/(c^5*e^8))*x + (1 
20*c^6*d*e^10*f + 132*b*c^5*e^11*f - 140*c^6*d^2*e^9*g + 272*b*c^5*d*e^10* 
g + 3*b^2*c^4*e^11*g)/(c^5*e^8))*x - (384*c^6*d^2*e^9*f - 744*b*c^5*d*e^10 
*f - 12*b^2*c^4*e^11*f + 384*c^6*d^3*e^8*g - 348*b*c^5*d^2*e^9*g - 48*b^2* 
c^4*d*e^10*g + 7*b^3*c^3*e^11*g)/(c^5*e^8))*x - (2400*c^6*d^3*e^8*f - 2064 
*b*c^5*d^2*e^9*f - 456*b^2*c^4*d*e^10*f + 60*b^3*c^3*e^11*f - 240*c^6*d^4* 
e^7*g + 816*b*c^5*d^3*e^8*g - 792*b^2*c^4*d^2*e^9*g + 276*b^3*c^3*d*e^10*g 
 - 35*b^4*c^2*e^11*g)/(c^5*e^8))*x + (1536*c^6*d^4*e^7*f - 5472*b*c^5*d^3* 
e^8*f + 5136*b^2*c^4*d^2*e^9*f - 1560*b^3*c^3*d*e^10*f + 180*b^4*c^2*e^11* 
f + 1536*c^6*d^5*e^6*g - 4368*b*c^5*d^4*e^7*g + 5328*b^2*c^4*d^3*e^8*g - 3 
256*b^3*c^3*d^2*e^9*g + 940*b^4*c^2*d*e^10*g - 105*b^5*c*e^11*g)/(c^5*e^8) 
) - 1/1024*(384*c^6*d^5*e*f - 960*b*c^5*d^4*e^2*f + 960*b^2*c^4*d^3*e^3*f 
- 480*b^3*c^3*d^2*e^4*f + 120*b^4*c^2*d*e^5*f - 12*b^5*c*e^6*f + 64*c^6*d^ 
6*g - 384*b*c^5*d^5*e*g + 720*b^2*c^4*d^4*e^2*g - 640*b^3*c^3*d^3*e^3*g + 
300*b^4*c^2*d^2*e^4*g - 72*b^5*c*d*e^5*g + 7*b^6*e^6*g)*log(abs(-b*e^2 + 2 
*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*ab 
s(e)))/(sqrt(-c)*c^4*e*abs(e))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.92 (sec) , antiderivative size = 2564, normalized size of antiderivative = 8.78 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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)
 

Output:

(i*(105*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d)) 
*b**7*e**7*g - 1290*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b 
*e + 2*c*d))*b**6*c*d*e**6*g - 180*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e* 
x)*i)/sqrt( - b*e + 2*c*d))*b**6*c*e**7*f + 6660*sqrt(c)*asinh((sqrt( - b* 
e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**5*c**2*d**2*e**5*g + 2160*sqr 
t(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**5*c**2* 
d*e**6*f - 18600*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e 
+ 2*c*d))*b**4*c**3*d**3*e**4*g - 10800*sqrt(c)*asinh((sqrt( - b*e + c*d - 
 c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c**3*d**2*e**5*f + 30000*sqrt(c)*asi 
nh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**4*d**4*e** 
3*g + 28800*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c 
*d))*b**3*c**4*d**3*e**4*f - 27360*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e* 
x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**5*d**5*e**2*g - 43200*sqrt(c)*asinh((s 
qrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**5*d**4*e**3*f + 
 12480*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))* 
b*c**6*d**6*e*g + 34560*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( 
 - b*e + 2*c*d))*b*c**6*d**5*e**2*f - 1920*sqrt(c)*asinh((sqrt( - b*e + c* 
d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**7*d**7*g - 11520*sqrt(c)*asinh((sqr 
t( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**7*d**6*e*f + 105*sqrt( 
d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e...