\(\int (d+e x)^2 (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\) [148]

Optimal result

Integrand size = 44, antiderivative size = 351 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\frac {(2 c d-b e)^4 (14 c e f+4 c d g-9 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{1024 c^5 e}+\frac {(2 c d-b e)^2 (14 c e f+4 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{384 c^4 e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}-\frac {(14 c e f+4 c d g-9 b e g) (24 c d-7 b e+10 c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{840 c^3 e^2}+\frac {(2 c d-b e)^6 (14 c e f+4 c d g-9 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{11/2} e^2} \] Output:

1/1024*(-b*e+2*c*d)^4*(-9*b*e*g+4*c*d*g+14*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)- 
b*e^2*x-c*e^2*x^2)^(1/2)/c^5/e+1/384*(-b*e+2*c*d)^2*(-9*b*e*g+4*c*d*g+14*c 
*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^4/e-1/7*g*(e*x+d) 
^2*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c/e^2-1/840*(-9*b*e*g+4*c*d*g+14 
*c*e*f)*(10*c*e*x-7*b*e+24*c*d)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^3 
/e^2+1/1024*(-b*e+2*c*d)^6*(-9*b*e*g+4*c*d*g+14*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^(11/2)/e^2
 

Mathematica [A] (verified)

Time = 2.92 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.69 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\frac {(-2 c d+b e)^6 ((d+e x) (-b e+c (d-e x)))^{3/2} \left (-\frac {\sqrt {c} \left (945 b^6 e^6 g-210 b^5 c e^5 (7 e f+50 d g+3 e g x)+64 c^6 \left (432 d^6 g+112 d e^5 x^4 (6 f+5 g x)+42 d^5 e (16 f+5 g x)+40 e^6 x^5 (7 f+6 g x)-2 d^2 e^4 x^3 (35 f+24 g x)-28 d^3 e^3 x^2 (48 f+35 g x)-3 d^4 e^2 x (315 f+208 g x)\right )+28 b^4 c^2 e^4 \left (1708 d^2 g+e^2 x (35 f+18 g x)+d e (560 f+226 g x)\right )+48 b^2 c^4 e^2 \left (3037 d^4 g+2 e^4 x^3 (7 f+4 g x)+4 d e^3 x^2 (35 f+18 g x)+14 d^2 e^2 x (52 f+23 g x)+4 d^3 e (763 f+255 g x)\right )-16 b^3 c^3 e^3 \left (7090 d^3 g+e^3 x^2 (49 f+27 g x)+4 d e^2 x (147 f+71 g x)+2 d^2 e (2107 f+786 g x)\right )+32 b c^5 e \left (-3054 d^5 g-123 d^4 e (35 f+11 g x)+12 d^3 e^2 x (91 f+75 g x)+8 e^5 x^4 (91 f+75 g x)+4 d e^4 x^3 (707 f+556 g x)+2 d^2 e^3 x^2 (1911 f+1409 g x)\right )\right )}{(-2 c d+b e)^6 (d+e x) (-b e+c (d-e x))}-\frac {105 (14 c e f+4 c d g-9 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{3/2} (-b e+c (d-e x))^{3/2}}\right )}{107520 c^{11/2} e^2} \] Input:

Integrate[(d + e*x)^2*(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)^6*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-((Sqrt[c]*(94 
5*b^6*e^6*g - 210*b^5*c*e^5*(7*e*f + 50*d*g + 3*e*g*x) + 64*c^6*(432*d^6*g 
 + 112*d*e^5*x^4*(6*f + 5*g*x) + 42*d^5*e*(16*f + 5*g*x) + 40*e^6*x^5*(7*f 
 + 6*g*x) - 2*d^2*e^4*x^3*(35*f + 24*g*x) - 28*d^3*e^3*x^2*(48*f + 35*g*x) 
 - 3*d^4*e^2*x*(315*f + 208*g*x)) + 28*b^4*c^2*e^4*(1708*d^2*g + e^2*x*(35 
*f + 18*g*x) + d*e*(560*f + 226*g*x)) + 48*b^2*c^4*e^2*(3037*d^4*g + 2*e^4 
*x^3*(7*f + 4*g*x) + 4*d*e^3*x^2*(35*f + 18*g*x) + 14*d^2*e^2*x*(52*f + 23 
*g*x) + 4*d^3*e*(763*f + 255*g*x)) - 16*b^3*c^3*e^3*(7090*d^3*g + e^3*x^2* 
(49*f + 27*g*x) + 4*d*e^2*x*(147*f + 71*g*x) + 2*d^2*e*(2107*f + 786*g*x)) 
 + 32*b*c^5*e*(-3054*d^5*g - 123*d^4*e*(35*f + 11*g*x) + 12*d^3*e^2*x*(91* 
f + 75*g*x) + 8*e^5*x^4*(91*f + 75*g*x) + 4*d*e^4*x^3*(707*f + 556*g*x) + 
2*d^2*e^3*x^2*(1911*f + 1409*g*x))))/((-2*c*d + b*e)^6*(d + e*x)*(-(b*e) + 
 c*(d - e*x)))) - (105*(14*c*e*f + 4*c*d*g - 9*b*e*g)*ArcTan[Sqrt[c*d - b* 
e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/((d + e*x)^(3/2)*(-(b*e) + c*(d - e*x 
))^(3/2))))/(107520*c^(11/2)*e^2)
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1221, 1134, 1160, 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.

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

\(\Big \downarrow \) 1221

\(\displaystyle \frac {(-9 b e g+4 c d g+14 c e f) \int (d+e x)^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{14 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {(-9 b e g+4 c d g+14 c e f) \left (\frac {7 (2 c d-b e) \int (d+e x) \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {(-9 b e g+4 c d g+14 c e f) \left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \int \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(-9 b e g+4 c d g+14 c e f) \left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \int \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(-9 b e g+4 c d g+14 c e f) \left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(-9 b e g+4 c d g+14 c e f) \left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right ) (-9 b e g+4 c d g+14 c e f)}{14 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}\)

Input:

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

Output:

-1/7*(g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(c*e^2) + 
 ((14*c*e*f + 4*c*d*g - 9*b*e*g)*(-1/6*((d + e*x)*(d*(c*d - b*e) - b*e^2*x 
 - c*e^2*x^2)^(5/2))/(c*e) + (7*(2*c*d - b*e)*(-1/5*(d*(c*d - b*e) - b*e^2 
*x - c*e^2*x^2)^(5/2)/(c*e) + ((2*c*d - b*e)*(((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)*Sqr 
t[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)))/(2*c)))/(12*c)))/(14*c*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 1134

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1)))   Int[(d + e*x)^ 
(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ 
c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 
*p]
 

rule 1160

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

rule 1221

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 
)/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c 
*f - b*g))/(c*e*(m + 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && NeQ[m + 2*p + 2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1976\) vs. \(2(327)=654\).

Time = 2.88 (sec) , antiderivative size = 1977, normalized size of antiderivative = 5.63

method	result	size
default	\(\text {Expression too large to display}\)	\(1977\)

Input:

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

Output:

d^2*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))))+e*(2*d*g+e*f)*(-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*d*e+c*d^2 
)^(1/2)))))+d*(d*g+2*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^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (327) = 654\).

Time = 1.74 (sec) , antiderivative size = 1877, normalized size of antiderivative = 5.35 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algo 
rithm="fricas")
 

Output:

[1/430080*(105*(14*(64*c^7*d^6*e - 192*b*c^6*d^5*e^2 + 240*b^2*c^5*d^4*e^3 
 - 160*b^3*c^4*d^3*e^4 + 60*b^4*c^3*d^2*e^5 - 12*b^5*c^2*d*e^6 + b^6*c*e^7 
)*f + (256*c^7*d^7 - 1344*b*c^6*d^6*e + 2688*b^2*c^5*d^5*e^2 - 2800*b^3*c^ 
4*d^4*e^3 + 1680*b^4*c^3*d^3*e^4 - 588*b^5*c^2*d^2*e^5 + 112*b^6*c*d*e^6 - 
 9*b^7*e^7)*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*(15360*c^7*e^6*g*x^6 + 1280*(14*c^7*e^6*f + (28*c^7*d*e 
^5 + 15*b*c^6*e^6)*g)*x^5 + 128*(14*(24*c^7*d*e^5 + 13*b*c^6*e^6)*f - (24* 
c^7*d^2*e^4 - 556*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*g)*x^4 - 16*(14*(20*c^7*d^2 
*e^4 - 404*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*f + (3920*c^7*d^3*e^3 - 5636*b*c^6 
*d^2*e^4 - 216*b^2*c^5*d*e^5 + 27*b^3*c^4*e^6)*g)*x^3 - 8*(14*(768*c^7*d^3 
*e^3 - 1092*b*c^6*d^2*e^4 - 60*b^2*c^5*d*e^5 + 7*b^3*c^4*e^6)*f + (4992*c^ 
7*d^4*e^2 - 3600*b*c^6*d^3*e^3 - 1932*b^2*c^5*d^2*e^4 + 568*b^3*c^4*d*e^5 
- 63*b^4*c^3*e^6)*g)*x^2 + 14*(3072*c^7*d^5*e - 9840*b*c^6*d^4*e^2 + 10464 
*b^2*c^5*d^3*e^3 - 4816*b^3*c^4*d^2*e^4 + 1120*b^4*c^3*d*e^5 - 105*b^5*c^2 
*e^6)*f + (27648*c^7*d^6 - 97728*b*c^6*d^5*e + 145776*b^2*c^5*d^4*e^2 - 11 
3440*b^3*c^4*d^3*e^3 + 47824*b^4*c^3*d^2*e^4 - 10500*b^5*c^2*d*e^5 + 945*b 
^6*c*e^6)*g - 2*(14*(2160*c^7*d^4*e^2 - 1248*b*c^6*d^3*e^3 - 1248*b^2*c^5* 
d^2*e^4 + 336*b^3*c^4*d*e^5 - 35*b^4*c^3*e^6)*f - (6720*c^7*d^5*e - 21648* 
b*c^6*d^4*e^2 + 24480*b^2*c^5*d^3*e^3 - 12576*b^3*c^4*d^2*e^4 + 3164*b^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5834 vs. \(2 (337) = 674\).

Time = 1.62 (sec) , antiderivative size = 5834, normalized size of antiderivative = 16.62 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**2*(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**4*g*x**6/ 
7 - x**5*(15*b*c*e**6*g/14 + 2*c**2*d*e**5*g + c**2*e**6*f)/(6*c*e**2) - x 
**4*(b**2*e**6*g + 6*b*c*d*e**5*g + 2*b*c*e**6*f - 11*b*(15*b*c*e**6*g/14 
+ 2*c**2*d*e**5*g + c**2*e**6*f)/(12*c) - c**2*d**2*e**4*g + 2*c**2*d*e**5 
*f + c*e**4*g*(-6*b*d*e + 6*c*d**2)/7)/(5*c*e**2) - x**3*(4*b**2*d*e**5*g 
+ b**2*e**6*f + 4*b*c*d**2*e**4*g + 6*b*c*d*e**5*f - 9*b*(b**2*e**6*g + 6* 
b*c*d*e**5*g + 2*b*c*e**6*f - 11*b*(15*b*c*e**6*g/14 + 2*c**2*d*e**5*g + c 
**2*e**6*f)/(12*c) - c**2*d**2*e**4*g + 2*c**2*d*e**5*f + c*e**4*g*(-6*b*d 
*e + 6*c*d**2)/7)/(10*c) - 4*c**2*d**3*e**3*g - c**2*d**2*e**4*f + (-5*b*d 
*e + 5*c*d**2)*(15*b*c*e**6*g/14 + 2*c**2*d*e**5*g + c**2*e**6*f)/(6*c*e** 
2))/(4*c*e**2) - x**2*(6*b**2*d**2*e**4*g + 4*b**2*d*e**5*f - 4*b*c*d**3*e 
**3*g + 4*b*c*d**2*e**4*f - 7*b*(4*b**2*d*e**5*g + b**2*e**6*f + 4*b*c*d** 
2*e**4*g + 6*b*c*d*e**5*f - 9*b*(b**2*e**6*g + 6*b*c*d*e**5*g + 2*b*c*e**6 
*f - 11*b*(15*b*c*e**6*g/14 + 2*c**2*d*e**5*g + c**2*e**6*f)/(12*c) - c**2 
*d**2*e**4*g + 2*c**2*d*e**5*f + c*e**4*g*(-6*b*d*e + 6*c*d**2)/7)/(10*c) 
- 4*c**2*d**3*e**3*g - c**2*d**2*e**4*f + (-5*b*d*e + 5*c*d**2)*(15*b*c*e* 
*6*g/14 + 2*c**2*d*e**5*g + c**2*e**6*f)/(6*c*e**2))/(8*c) - c**2*d**4*e** 
2*g - 4*c**2*d**3*e**3*f + (-4*b*d*e + 4*c*d**2)*(b**2*e**6*g + 6*b*c*d*e* 
*5*g + 2*b*c*e**6*f - 11*b*(15*b*c*e**6*g/14 + 2*c**2*d*e**5*g + c**2*e**6 
*f)/(12*c) - c**2*d**2*e**4*g + 2*c**2*d*e**5*f + c*e**4*g*(-6*b*d*e + ...
 

Maxima [F(-2)]

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

integrate((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algo 
rithm="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. 983 vs. \(2 (327) = 654\).

Time = 0.38 (sec) , antiderivative size = 983, normalized size of antiderivative = 2.80 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algo 
rithm="giac")
 

Output:

-1/107520*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(2*(8*(10*(12*c 
*e^4*g*x + (14*c^7*e^14*f + 28*c^7*d*e^13*g + 15*b*c^6*e^14*g)/(c^6*e^10)) 
*x + (336*c^7*d*e^13*f + 182*b*c^6*e^14*f - 24*c^7*d^2*e^12*g + 556*b*c^6* 
d*e^13*g + 3*b^2*c^5*e^14*g)/(c^6*e^10))*x - (280*c^7*d^2*e^12*f - 5656*b* 
c^6*d*e^13*f - 42*b^2*c^5*e^14*f + 3920*c^7*d^3*e^11*g - 5636*b*c^6*d^2*e^ 
12*g - 216*b^2*c^5*d*e^13*g + 27*b^3*c^4*e^14*g)/(c^6*e^10))*x - (10752*c^ 
7*d^3*e^11*f - 15288*b*c^6*d^2*e^12*f - 840*b^2*c^5*d*e^13*f + 98*b^3*c^4* 
e^14*f + 4992*c^7*d^4*e^10*g - 3600*b*c^6*d^3*e^11*g - 1932*b^2*c^5*d^2*e^ 
12*g + 568*b^3*c^4*d*e^13*g - 63*b^4*c^3*e^14*g)/(c^6*e^10))*x - (30240*c^ 
7*d^4*e^10*f - 17472*b*c^6*d^3*e^11*f - 17472*b^2*c^5*d^2*e^12*f + 4704*b^ 
3*c^4*d*e^13*f - 490*b^4*c^3*e^14*f - 6720*c^7*d^5*e^9*g + 21648*b*c^6*d^4 
*e^10*g - 24480*b^2*c^5*d^3*e^11*g + 12576*b^3*c^4*d^2*e^12*g - 3164*b^4*c 
^3*d*e^13*g + 315*b^5*c^2*e^14*g)/(c^6*e^10))*x + (43008*c^7*d^5*e^9*f - 1 
37760*b*c^6*d^4*e^10*f + 146496*b^2*c^5*d^3*e^11*f - 67424*b^3*c^4*d^2*e^1 
2*f + 15680*b^4*c^3*d*e^13*f - 1470*b^5*c^2*e^14*f + 27648*c^7*d^6*e^8*g - 
 97728*b*c^6*d^5*e^9*g + 145776*b^2*c^5*d^4*e^10*g - 113440*b^3*c^4*d^3*e^ 
11*g + 47824*b^4*c^3*d^2*e^12*g - 10500*b^5*c^2*d*e^13*g + 945*b^6*c*e^14* 
g)/(c^6*e^10)) - 1/2048*(896*c^7*d^6*e*f - 2688*b*c^6*d^5*e^2*f + 3360*b^2 
*c^5*d^4*e^3*f - 2240*b^3*c^4*d^3*e^4*f + 840*b^4*c^3*d^2*e^5*f - 168*b^5* 
c^2*d*e^6*f + 14*b^6*c*e^7*f + 256*c^7*d^7*g - 1344*b*c^6*d^6*e*g + 268...
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 4.96 (sec) , antiderivative size = 3367, normalized size of antiderivative = 9.59 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*( - 945*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c* 
d))*b**8*e**8*g + 13650*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( 
 - b*e + 2*c*d))*b**7*c*d*e**7*g + 1470*sqrt(c)*asinh((sqrt( - b*e + c*d - 
 c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**7*c*e**8*f - 85260*sqrt(c)*asinh((sqrt 
( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**6*c**2*d**2*e**6*g - 20 
580*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b** 
6*c**2*d*e**7*f + 299880*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt 
( - b*e + 2*c*d))*b**5*c**3*d**3*e**5*g + 123480*sqrt(c)*asinh((sqrt( - b* 
e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**5*c**3*d**2*e**6*f - 646800*s 
qrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c** 
4*d**4*e**4*g - 411600*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( 
- b*e + 2*c*d))*b**4*c**4*d**3*e**5*f + 870240*sqrt(c)*asinh((sqrt( - b*e 
+ c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**5*d**5*e**3*g + 823200*sqr 
t(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**5* 
d**4*e**4*f - 705600*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - 
b*e + 2*c*d))*b**2*c**6*d**6*e**2*g - 987840*sqrt(c)*asinh((sqrt( - b*e + 
c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**6*d**5*e**3*f + 309120*sqrt( 
c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**7*d**7* 
e*g + 658560*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2* 
c*d))*b*c**7*d**6*e**2*f - 53760*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e...