\(\int (d+e x)^2 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\) [137]

Optimal result

Integrand size = 44, antiderivative size = 277 \[ \int (d+e x)^2 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {(2 c d-b e)^2 (10 c e f+4 c d g-7 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^4 e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2}-\frac {(10 c e f+4 c d g-7 b e g) (16 c d-5 b e+6 c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{240 c^3 e^2}+\frac {(2 c d-b e)^4 (10 c e f+4 c d g-7 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{9/2} e^2} \] Output:

1/128*(-b*e+2*c*d)^2*(-7*b*e*g+4*c*d*g+10*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-1/5*g*(e*x+d)^2*(d*(-b*e+c*d)-b*e^2*x-c*e^2* 
x^2)^(3/2)/c/e^2-1/240*(-7*b*e*g+4*c*d*g+10*c*e*f)*(6*c*e*x-5*b*e+16*c*d)* 
(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^3/e^2+1/128*(-b*e+2*c*d)^4*(-7*b* 
e*g+4*c*d*g+10*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 = 1.33 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.26 \[ \int (d+e x)^2 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {(-2 c d+b e)^4 \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {\sqrt {c} \left (-105 b^4 e^4 g+10 b^3 c e^3 (15 e f+76 d g+7 e g x)-16 c^4 \left (56 d^4 g-20 d e^3 x^2 (4 f+3 g x)+10 d^3 e (8 f+3 g x)-6 e^4 x^3 (5 f+4 g x)-d^2 e^2 x (45 f+32 g x)\right )-4 b^2 c^2 e^2 \left (499 d^2 g+e^2 x (25 f+14 g x)+2 d e (125 f+54 g x)\right )+8 b c^3 e \left (274 d^3 g+2 e^3 x^2 (5 f+3 g x)+2 d e^2 x (35 f+18 g x)+d^2 e (285 f+109 g x)\right )\right )}{(-2 c d+b e)^4}-\frac {15 (10 c e f+4 c d g-7 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {d+e x} \sqrt {-b e+c (d-e x)}}\right )}{1920 c^{9/2} e^2} \] Input:

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]
 

Output:

((-2*c*d + b*e)^4*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((Sqrt[c]*(-105*b 
^4*e^4*g + 10*b^3*c*e^3*(15*e*f + 76*d*g + 7*e*g*x) - 16*c^4*(56*d^4*g - 2 
0*d*e^3*x^2*(4*f + 3*g*x) + 10*d^3*e*(8*f + 3*g*x) - 6*e^4*x^3*(5*f + 4*g* 
x) - d^2*e^2*x*(45*f + 32*g*x)) - 4*b^2*c^2*e^2*(499*d^2*g + e^2*x*(25*f + 
 14*g*x) + 2*d*e*(125*f + 54*g*x)) + 8*b*c^3*e*(274*d^3*g + 2*e^3*x^2*(5*f 
 + 3*g*x) + 2*d*e^2*x*(35*f + 18*g*x) + d^2*e*(285*f + 109*g*x))))/(-2*c*d 
 + b*e)^4 - (15*(10*c*e*f + 4*c*d*g - 7*b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e 
*x]/(Sqrt[c]*Sqrt[d + e*x])])/(Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)]))) 
/(1920*c^(9/2)*e^2)
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1221, 1134, 1160, 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.

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

\(\Big \downarrow \) 1221

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

\(\Big \downarrow \) 1134

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 217

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

Input:

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]
 

Output:

-1/5*(g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(c*e^2) + 
 ((10*c*e*f + 4*c*d*g - 7*b*e*g)*(-1/4*((d + e*x)*(d*(c*d - b*e) - b*e^2*x 
 - c*e^2*x^2)^(3/2))/(c*e) + (5*(2*c*d - b*e)*(-1/3*(d*(c*d - b*e) - b*e^2 
*x - c*e^2*x^2)^(3/2)/(c*e) + ((2*c*d - b*e)*(((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)))/ 
(2*c)))/(8*c)))/(10*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. \(1374\) vs. \(2(257)=514\).

Time = 2.74 (sec) , antiderivative size = 1375, normalized size of antiderivative = 4.96

method	result	size
default	\(\text {Expression too large to display}\)	\(1375\)

Input:

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,method=_RET 
URNVERBOSE)
 

Output:

d^2*f*(-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)* 
(-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-5/8*b/c*(-1/3*(-c*e^2 
*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-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))))+1/4*(-b*d*e+c*d^2)/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)*(-1/3*(-c*e^2*x^2-b*e^2*x-b 
*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-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*g*(-1/5*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2- 
7/10*b/c*(-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-5/8*b/c*(-1/ 
3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-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))))+1/4*(-b*d*e+c*d^2)/c/e^2*(-1/4*(-2*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (257) = 514\).

Time = 0.35 (sec) , antiderivative size = 1117, normalized size of antiderivative = 4.03 \[ \int (d+e x)^2 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo 
rithm="fricas")
 

Output:

[1/7680*(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)*sqrt(-c)*l 
og(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*(384*c 
^5*e^4*g*x^4 + 48*(10*c^5*e^4*f + (20*c^5*d*e^3 + b*c^4*e^4)*g)*x^3 + 8*(1 
0*(16*c^5*d*e^3 + b*c^4*e^4)*f + (64*c^5*d^2*e^2 + 36*b*c^4*d*e^3 - 7*b^2* 
c^3*e^4)*g)*x^2 - 10*(128*c^5*d^3*e - 228*b*c^4*d^2*e^2 + 100*b^2*c^3*d*e^ 
3 - 15*b^3*c^2*e^4)*f - (896*c^5*d^4 - 2192*b*c^4*d^3*e + 1996*b^2*c^3*d^2 
*e^2 - 760*b^3*c^2*d*e^3 + 105*b^4*c*e^4)*g + 2*(10*(36*c^5*d^2*e^2 + 28*b 
*c^4*d*e^3 - 5*b^2*c^3*e^4)*f - (240*c^5*d^3*e - 436*b*c^4*d^2*e^2 + 216*b 
^2*c^3*d*e^3 - 35*b^3*c^2*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b 
*d*e))/(c^5*e^2), -1/3840*(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)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c 
*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) - 2*(38 
4*c^5*e^4*g*x^4 + 48*(10*c^5*e^4*f + (20*c^5*d*e^3 + b*c^4*e^4)*g)*x^3 + 8 
*(10*(16*c^5*d*e^3 + b*c^4*e^4)*f + (64*c^5*d^2*e^2 + 36*b*c^4*d*e^3 - 7*b 
^2*c^3*e^4)*g)*x^2 - 10*(128*c^5*d^3*e - 228*b*c^4*d^2*e^2 + 100*b^2*c^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1608 vs. \(2 (265) = 530\).

Time = 1.58 (sec) , antiderivative size = 1608, normalized size of antiderivative = 5.81 \[ \int (d+e x)^2 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)*(e**2*g*x**4/5 - 
 x**3*(-b*e**4*g/10 - 2*c*d*e**3*g - c*e**4*f)/(4*c*e**2) - x**2*(-3*b*d*e 
**3*g - b*e**4*f - 7*b*(-b*e**4*g/10 - 2*c*d*e**3*g - c*e**4*f)/(8*c) - 2* 
c*d*e**3*f - e**2*g*(-4*b*d*e + 4*c*d**2)/5)/(3*c*e**2) - x*(-3*b*d**2*e** 
2*g - 3*b*d*e**3*f - 5*b*(-3*b*d*e**3*g - b*e**4*f - 7*b*(-b*e**4*g/10 - 2 
*c*d*e**3*g - c*e**4*f)/(8*c) - 2*c*d*e**3*f - e**2*g*(-4*b*d*e + 4*c*d**2 
)/5)/(6*c) + 2*c*d**3*e*g + (-3*b*d*e + 3*c*d**2)*(-b*e**4*g/10 - 2*c*d*e* 
*3*g - c*e**4*f)/(4*c*e**2))/(2*c*e**2) - (-b*d**3*e*g - 3*b*d**2*e**2*f - 
 3*b*(-3*b*d**2*e**2*g - 3*b*d*e**3*f - 5*b*(-3*b*d*e**3*g - b*e**4*f - 7* 
b*(-b*e**4*g/10 - 2*c*d*e**3*g - c*e**4*f)/(8*c) - 2*c*d*e**3*f - e**2*g*( 
-4*b*d*e + 4*c*d**2)/5)/(6*c) + 2*c*d**3*e*g + (-3*b*d*e + 3*c*d**2)*(-b*e 
**4*g/10 - 2*c*d*e**3*g - c*e**4*f)/(4*c*e**2))/(4*c) + c*d**4*g + 2*c*d** 
3*e*f + (-2*b*d*e + 2*c*d**2)*(-3*b*d*e**3*g - b*e**4*f - 7*b*(-b*e**4*g/1 
0 - 2*c*d*e**3*g - c*e**4*f)/(8*c) - 2*c*d*e**3*f - e**2*g*(-4*b*d*e + 4*c 
*d**2)/5)/(3*c*e**2))/(c*e**2)) + (-b*d**3*e*f - b*(-b*d**3*e*g - 3*b*d**2 
*e**2*f - 3*b*(-3*b*d**2*e**2*g - 3*b*d*e**3*f - 5*b*(-3*b*d*e**3*g - b*e* 
*4*f - 7*b*(-b*e**4*g/10 - 2*c*d*e**3*g - c*e**4*f)/(8*c) - 2*c*d*e**3*f - 
 e**2*g*(-4*b*d*e + 4*c*d**2)/5)/(6*c) + 2*c*d**3*e*g + (-3*b*d*e + 3*c*d* 
*2)*(-b*e**4*g/10 - 2*c*d*e**3*g - c*e**4*f)/(4*c*e**2))/(4*c) + c*d**4*g 
+ 2*c*d**3*e*f + (-2*b*d*e + 2*c*d**2)*(-3*b*d*e**3*g - b*e**4*f - 7*b*...
 

Maxima [F(-2)]

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

integrate((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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. 568 vs. \(2 (257) = 514\).

Time = 0.16 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.05 \[ \int (d+e x)^2 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {1}{1920} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, e^{2} g x + \frac {10 \, c^{4} e^{8} f + 20 \, c^{4} d e^{7} g + b c^{3} e^{8} g}{c^{4} e^{6}}\right )} x + \frac {160 \, c^{4} d e^{7} f + 10 \, b c^{3} e^{8} f + 64 \, c^{4} d^{2} e^{6} g + 36 \, b c^{3} d e^{7} g - 7 \, b^{2} c^{2} e^{8} g}{c^{4} e^{6}}\right )} x + \frac {360 \, c^{4} d^{2} e^{6} f + 280 \, b c^{3} d e^{7} f - 50 \, b^{2} c^{2} e^{8} f - 240 \, c^{4} d^{3} e^{5} g + 436 \, b c^{3} d^{2} e^{6} g - 216 \, b^{2} c^{2} d e^{7} g + 35 \, b^{3} c e^{8} g}{c^{4} e^{6}}\right )} x - \frac {1280 \, c^{4} d^{3} e^{5} f - 2280 \, b c^{3} d^{2} e^{6} f + 1000 \, b^{2} c^{2} d e^{7} f - 150 \, b^{3} c e^{8} f + 896 \, c^{4} d^{4} e^{4} g - 2192 \, b c^{3} d^{3} e^{5} g + 1996 \, b^{2} c^{2} d^{2} e^{6} g - 760 \, b^{3} c d e^{7} g + 105 \, b^{4} e^{8} g}{c^{4} e^{6}}\right )} - \frac {{\left (160 \, c^{5} d^{4} e f - 320 \, b c^{4} d^{3} e^{2} f + 240 \, b^{2} c^{3} d^{2} e^{3} f - 80 \, b^{3} c^{2} d e^{4} f + 10 \, b^{4} c e^{5} f + 64 \, c^{5} d^{5} g - 240 \, b c^{4} d^{4} e g + 320 \, b^{2} c^{3} d^{3} e^{2} g - 200 \, b^{3} c^{2} d^{2} e^{3} g + 60 \, b^{4} c d e^{4} g - 7 \, b^{5} e^{5} g\right )} \log \left ({\left | -b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} {\left | e \right |} \right |}\right )}{256 \, \sqrt {-c} c^{4} e {\left | e \right |}} \] Input:

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, algo 
rithm="giac")
 

Output:

1/1920*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(6*(8*e^2*g*x + (1 
0*c^4*e^8*f + 20*c^4*d*e^7*g + b*c^3*e^8*g)/(c^4*e^6))*x + (160*c^4*d*e^7* 
f + 10*b*c^3*e^8*f + 64*c^4*d^2*e^6*g + 36*b*c^3*d*e^7*g - 7*b^2*c^2*e^8*g 
)/(c^4*e^6))*x + (360*c^4*d^2*e^6*f + 280*b*c^3*d*e^7*f - 50*b^2*c^2*e^8*f 
 - 240*c^4*d^3*e^5*g + 436*b*c^3*d^2*e^6*g - 216*b^2*c^2*d*e^7*g + 35*b^3* 
c*e^8*g)/(c^4*e^6))*x - (1280*c^4*d^3*e^5*f - 2280*b*c^3*d^2*e^6*f + 1000* 
b^2*c^2*d*e^7*f - 150*b^3*c*e^8*f + 896*c^4*d^4*e^4*g - 2192*b*c^3*d^3*e^5 
*g + 1996*b^2*c^2*d^2*e^6*g - 760*b^3*c*d*e^7*g + 105*b^4*e^8*g)/(c^4*e^6) 
) - 1/256*(160*c^5*d^4*e*f - 320*b*c^4*d^3*e^2*f + 240*b^2*c^3*d^2*e^3*f - 
 80*b^3*c^2*d*e^4*f + 10*b^4*c*e^5*f + 64*c^5*d^5*g - 240*b*c^4*d^4*e*g + 
320*b^2*c^3*d^3*e^2*g - 200*b^3*c^2*d^2*e^3*g + 60*b^4*c*d*e^4*g - 7*b^5*e 
^5*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)*abs(e)))/(sqrt(-c)*c^4*e*abs(e))
 

Mupad [B] (verification not implemented)

Time = 8.08 (sec) , antiderivative size = 1732, normalized size of antiderivative = 6.25 \[ \int (d+e x)^2 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

d^2*f*(x/2 + b/(4*c))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - (f*x*( 
c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(4*c) - (g*x^2*(c*d^2 - c*e^2* 
x^2 - b*d*e - b*e^2*x)^(3/2))/(5*c) + (f*(c*d^2 - b*d*e)*((x/2 + b/(4*c))* 
(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - (log(b*e^2 - 2*(-c*e^2)^(1/2 
)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*((b^2*e^4)/4 + c*e^2 
*(c*d^2 - b*d*e)))/(2*(-c*e^2)^(3/2))))/(4*c) - (g*(2*c*d^2 - 2*b*d*e)*((l 
og(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e 
^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) - ((8*c*e 
^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x 
^2 - b*d*e - b*e^2*x)^(1/2))/(24*c^2*e^4)))/(5*c) - (7*b*e^2*g*((5*b*((log 
(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2 
*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) - ((8*c*e^2 
*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 
 - b*d*e - b*e^2*x)^(1/2))/(24*c^2*e^4)))/(8*c) + ((c*d^2 - b*d*e)*((x/2 + 
 b/(4*c))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - (log(b*e^2 - 2*(-c 
*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*((b^2*e^4) 
/4 + c*e^2*(c*d^2 - b*d*e)))/(2*(-c*e^2)^(3/2))))/(4*c*e^2) - (x*(c*d^2 - 
c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(4*c*e^2)))/(10*c) - (d^2*f*log(b*e^2 
- 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*((b 
^2*e^4)/4 + c*e^2*(c*d^2 - b*d*e)))/(2*(-c*e^2)^(3/2)) + (5*b*e^2*f*((l...
 

Reduce [B] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 1875, normalized size of antiderivative = 6.77 \[ \int (d+e x)^2 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*( - 105*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c* 
d))*b**6*e**6*g + 1110*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( 
- b*e + 2*c*d))*b**5*c*d*e**5*g + 150*sqrt(c)*asinh((sqrt( - b*e + c*d - c 
*e*x)*i)/sqrt( - b*e + 2*c*d))*b**5*c*e**6*f - 4800*sqrt(c)*asinh((sqrt( - 
 b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c**2*d**2*e**4*g - 1500* 
sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c* 
*2*d*e**5*f + 10800*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b 
*e + 2*c*d))*b**3*c**3*d**3*e**3*g + 6000*sqrt(c)*asinh((sqrt( - b*e + c*d 
 - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**3*d**2*e**4*f - 13200*sqrt(c)*a 
sinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**4*d**4*e 
**2*g - 12000*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2 
*c*d))*b**2*c**4*d**3*e**3*f + 8160*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e 
*x)*i)/sqrt( - b*e + 2*c*d))*b*c**5*d**5*e*g + 12000*sqrt(c)*asinh((sqrt( 
- b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**5*d**4*e**2*f - 1920*sq 
rt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**6*d**6 
*g - 4800*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d 
))*c**6*d**5*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*x)*b**4*c*e**4*g + 760*sqrt(d + e*x)*sqrt(b*e - 
2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*c**2*d*e**3*g 
+ 150*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e ...