\(\int (b+2 c x) (d+e x)^2 (a+b x+c x^2)^{3/2} \, dx\) [613]

Optimal result

Integrand size = 28, antiderivative size = 242 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^4}+\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^3}+\frac {2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}+\frac {\left (b^2-4 a c\right )^3 e (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{9/2}} \] Output:

-1/256*(-4*a*c+b^2)^2*e*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^4+1/9 
6*(-4*a*c+b^2)*e*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3+2/7*(e*x+d 
)^2*(c*x^2+b*x+a)^(5/2)+1/210*(24*c^2*d^2+7*b^2*e^2-2*c*e*(12*a*e+7*b*d)+1 
0*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(5/2)/c^2+1/512*(-4*a*c+b^2)^3*e*(-b*e 
+2*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)
 

Mathematica [A] (verified)

Time = 5.29 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.62 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (105 b^6 e^2-3072 a^3 c^3 e^2-70 b^5 c e (3 d+e x)+28 b^4 c^2 e x (5 d+2 e x)-16 b^3 c^3 e x^2 (7 d+3 e x)+512 c^6 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+256 b c^5 x^3 \left (84 d^2+133 d e x+55 e^2 x^2\right )+32 b^2 c^4 x^2 \left (336 d^2+483 d e x+188 e^2 x^2\right )+48 a^2 c^2 \left (77 b^2 e^2-2 b c e (77 d+19 e x)+4 c^2 \left (56 d^2+35 d e x+8 e^2 x^2\right )\right )+32 a c \left (-35 b^4 e^2+7 b^3 c e (10 d+3 e x)-3 b^2 c^2 e x (14 d+5 e x)+4 c^4 x^2 \left (168 d^2+245 d e x+96 e^2 x^2\right )+2 b c^3 x \left (336 d^2+399 d e x+139 e^2 x^2\right )\right )\right )-105 \left (b^2-4 a c\right )^3 e (-2 c d+b e) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{26880 c^{9/2}} \] Input:

Integrate[(b + 2*c*x)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2),x]
 

Output:

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(105*b^6*e^2 - 3072*a^3*c^3*e^2 - 70*b^5*c* 
e*(3*d + e*x) + 28*b^4*c^2*e*x*(5*d + 2*e*x) - 16*b^3*c^3*e*x^2*(7*d + 3*e 
*x) + 512*c^6*x^4*(21*d^2 + 35*d*e*x + 15*e^2*x^2) + 256*b*c^5*x^3*(84*d^2 
 + 133*d*e*x + 55*e^2*x^2) + 32*b^2*c^4*x^2*(336*d^2 + 483*d*e*x + 188*e^2 
*x^2) + 48*a^2*c^2*(77*b^2*e^2 - 2*b*c*e*(77*d + 19*e*x) + 4*c^2*(56*d^2 + 
 35*d*e*x + 8*e^2*x^2)) + 32*a*c*(-35*b^4*e^2 + 7*b^3*c*e*(10*d + 3*e*x) - 
 3*b^2*c^2*e*x*(14*d + 5*e*x) + 4*c^4*x^2*(168*d^2 + 245*d*e*x + 96*e^2*x^ 
2) + 2*b*c^3*x*(336*d^2 + 399*d*e*x + 139*e^2*x^2))) - 105*(b^2 - 4*a*c)^3 
*e*(-2*c*d + b*e)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])]) 
/(26880*c^(9/2))
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1236, 27, 1225, 1087, 1087, 1092, 219}

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.

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1225

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(2*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/7 + (2*(((24*c^2*d^2 + 7*b^2*e^2 - 
 2*c*e*(7*b*d + 12*a*e) + 10*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2)) 
/(60*c^2) + (7*(b^2 - 4*a*c)*e*(2*c*d - b*e)*(((b + 2*c*x)*(a + b*x + c*x^ 
2)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4 
*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2]) 
])/(8*c^(3/2))))/(16*c)))/(24*c^2)))/7
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[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]
 

rule 1236

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[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(528\) vs. \(2(216)=432\).

Time = 1.85 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.19

method	result	size
risch	\(-\frac {\left (-7680 c^{6} e^{2} x^{6}-14080 b \,c^{5} e^{2} x^{5}-17920 c^{6} d e \,x^{5}-12288 a \,c^{5} e^{2} x^{4}-6016 b^{2} c^{4} e^{2} x^{4}-34048 b \,c^{5} d e \,x^{4}-10752 c^{6} d^{2} x^{4}-8896 a b \,c^{4} e^{2} x^{3}-31360 a \,c^{5} d e \,x^{3}+48 b^{3} c^{3} e^{2} x^{3}-15456 b^{2} c^{4} d e \,x^{3}-21504 b \,c^{5} d^{2} x^{3}-1536 a^{2} c^{4} e^{2} x^{2}+480 a \,b^{2} c^{3} e^{2} x^{2}-25536 a b \,c^{4} d e \,x^{2}-21504 a \,c^{5} d^{2} x^{2}-56 b^{4} c^{2} e^{2} x^{2}+112 b^{3} c^{3} d e \,x^{2}-10752 b^{2} c^{4} d^{2} x^{2}+1824 a^{2} b \,c^{3} e^{2} x -6720 a^{2} c^{4} d e x -672 a \,b^{3} c^{2} e^{2} x +1344 a \,b^{2} c^{3} d e x -21504 a b \,c^{4} d^{2} x +70 b^{5} c \,e^{2} x -140 b^{4} c^{2} d e x +3072 a^{3} c^{3} e^{2}-3696 a^{2} b^{2} c^{2} e^{2}+7392 a^{2} b \,c^{3} d e -10752 a^{2} c^{4} d^{2}+1120 a \,b^{4} c \,e^{2}-2240 a \,b^{3} c^{2} d e -105 b^{6} e^{2}+210 b^{5} c d e \right ) \sqrt {c \,x^{2}+b x +a}}{26880 c^{4}}+\frac {e \left (64 a^{3} b \,c^{3} e -128 a^{3} c^{4} d -48 a^{2} b^{3} c^{2} e +96 a^{2} b^{2} c^{3} d +12 a \,b^{5} c e -24 c^{2} d a \,b^{4}-b^{7} e +2 b^{6} c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {9}{2}}}\)	\(529\)
default	\(\text {Expression too large to display}\)	\(945\)

Input:

int((2*c*x+b)*(e*x+d)^2*(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/26880/c^4*(-7680*c^6*e^2*x^6-14080*b*c^5*e^2*x^5-17920*c^6*d*e*x^5-1228 
8*a*c^5*e^2*x^4-6016*b^2*c^4*e^2*x^4-34048*b*c^5*d*e*x^4-10752*c^6*d^2*x^4 
-8896*a*b*c^4*e^2*x^3-31360*a*c^5*d*e*x^3+48*b^3*c^3*e^2*x^3-15456*b^2*c^4 
*d*e*x^3-21504*b*c^5*d^2*x^3-1536*a^2*c^4*e^2*x^2+480*a*b^2*c^3*e^2*x^2-25 
536*a*b*c^4*d*e*x^2-21504*a*c^5*d^2*x^2-56*b^4*c^2*e^2*x^2+112*b^3*c^3*d*e 
*x^2-10752*b^2*c^4*d^2*x^2+1824*a^2*b*c^3*e^2*x-6720*a^2*c^4*d*e*x-672*a*b 
^3*c^2*e^2*x+1344*a*b^2*c^3*d*e*x-21504*a*b*c^4*d^2*x+70*b^5*c*e^2*x-140*b 
^4*c^2*d*e*x+3072*a^3*c^3*e^2-3696*a^2*b^2*c^2*e^2+7392*a^2*b*c^3*d*e-1075 
2*a^2*c^4*d^2+1120*a*b^4*c*e^2-2240*a*b^3*c^2*d*e-105*b^6*e^2+210*b^5*c*d* 
e)*(c*x^2+b*x+a)^(1/2)+1/512*e*(64*a^3*b*c^3*e-128*a^3*c^4*d-48*a^2*b^3*c^ 
2*e+96*a^2*b^2*c^3*d+12*a*b^5*c*e-24*a*b^4*c^2*d-b^7*e+2*b^6*c*d)/c^(9/2)* 
ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (216) = 432\).

Time = 0.15 (sec) , antiderivative size = 1015, normalized size of antiderivative = 4.19 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((2*c*x+b)*(e*x+d)^2*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

[1/107520*(105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e 
 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^2)*sqrt(c)*log(-8* 
c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4* 
a*c) + 4*(7680*c^7*e^2*x^6 + 10752*a^2*c^5*d^2 + 1280*(14*c^7*d*e + 11*b*c 
^6*e^2)*x^5 + 128*(84*c^7*d^2 + 266*b*c^6*d*e + (47*b^2*c^5 + 96*a*c^6)*e^ 
2)*x^4 + 16*(1344*b*c^6*d^2 + 14*(69*b^2*c^5 + 140*a*c^6)*d*e - (3*b^3*c^4 
 - 556*a*b*c^5)*e^2)*x^3 - 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4) 
*d*e + (105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^3*c^4)*e^2 
+ 8*(1344*(b^2*c^5 + 2*a*c^6)*d^2 - 14*(b^3*c^4 - 228*a*b*c^5)*d*e + (7*b^ 
4*c^3 - 60*a*b^2*c^4 + 192*a^2*c^5)*e^2)*x^2 + 2*(10752*a*b*c^5*d^2 + 14*( 
5*b^4*c^3 - 48*a*b^2*c^4 + 240*a^2*c^5)*d*e - (35*b^5*c^2 - 336*a*b^3*c^3 
+ 912*a^2*b*c^4)*e^2)*x)*sqrt(c*x^2 + b*x + a))/c^5, -1/53760*(105*(2*(b^6 
*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e - (b^7 - 12*a*b^5*c + 
 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^2)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x 
+ a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(7680*c^7*e^2*x^6 + 
 10752*a^2*c^5*d^2 + 1280*(14*c^7*d*e + 11*b*c^6*e^2)*x^5 + 128*(84*c^7*d^ 
2 + 266*b*c^6*d*e + (47*b^2*c^5 + 96*a*c^6)*e^2)*x^4 + 16*(1344*b*c^6*d^2 
+ 14*(69*b^2*c^5 + 140*a*c^6)*d*e - (3*b^3*c^4 - 556*a*b*c^5)*e^2)*x^3 - 1 
4*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d*e + (105*b^6*c - 1120*a*b 
^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^3*c^4)*e^2 + 8*(1344*(b^2*c^5 + 2*a*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3582 vs. \(2 (233) = 466\).

Time = 1.12 (sec) , antiderivative size = 3582, normalized size of antiderivative = 14.80 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((2*c*x+b)*(e*x+d)**2*(c*x**2+b*x+a)**(3/2),x)
 

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(2*c**2*e**2*x**6/7 + x**5*(22*b*c**2*e* 
*2/7 + 4*c**3*d*e)/(6*c) + x**4*(16*a*c**2*e**2/7 + 4*b**2*c*e**2 + 10*b*c 
**2*d*e - 11*b*(22*b*c**2*e**2/7 + 4*c**3*d*e)/(12*c) + 2*c**3*d**2)/(5*c) 
 + x**3*(6*a*b*c*e**2 + 8*a*c**2*d*e - 5*a*(22*b*c**2*e**2/7 + 4*c**3*d*e) 
/(6*c) + b**3*e**2 + 8*b**2*c*d*e + 5*b*c**2*d**2 - 9*b*(16*a*c**2*e**2/7 
+ 4*b**2*c*e**2 + 10*b*c**2*d*e - 11*b*(22*b*c**2*e**2/7 + 4*c**3*d*e)/(12 
*c) + 2*c**3*d**2)/(10*c))/(4*c) + x**2*(2*a**2*c*e**2 + 2*a*b**2*e**2 + 1 
2*a*b*c*d*e + 4*a*c**2*d**2 - 4*a*(16*a*c**2*e**2/7 + 4*b**2*c*e**2 + 10*b 
*c**2*d*e - 11*b*(22*b*c**2*e**2/7 + 4*c**3*d*e)/(12*c) + 2*c**3*d**2)/(5* 
c) + 2*b**3*d*e + 4*b**2*c*d**2 - 7*b*(6*a*b*c*e**2 + 8*a*c**2*d*e - 5*a*( 
22*b*c**2*e**2/7 + 4*c**3*d*e)/(6*c) + b**3*e**2 + 8*b**2*c*d*e + 5*b*c**2 
*d**2 - 9*b*(16*a*c**2*e**2/7 + 4*b**2*c*e**2 + 10*b*c**2*d*e - 11*b*(22*b 
*c**2*e**2/7 + 4*c**3*d*e)/(12*c) + 2*c**3*d**2)/(10*c))/(8*c))/(3*c) + x* 
(a**2*b*e**2 + 4*a**2*c*d*e + 4*a*b**2*d*e + 6*a*b*c*d**2 - 3*a*(6*a*b*c*e 
**2 + 8*a*c**2*d*e - 5*a*(22*b*c**2*e**2/7 + 4*c**3*d*e)/(6*c) + b**3*e**2 
 + 8*b**2*c*d*e + 5*b*c**2*d**2 - 9*b*(16*a*c**2*e**2/7 + 4*b**2*c*e**2 + 
10*b*c**2*d*e - 11*b*(22*b*c**2*e**2/7 + 4*c**3*d*e)/(12*c) + 2*c**3*d**2) 
/(10*c))/(4*c) + b**3*d**2 - 5*b*(2*a**2*c*e**2 + 2*a*b**2*e**2 + 12*a*b*c 
*d*e + 4*a*c**2*d**2 - 4*a*(16*a*c**2*e**2/7 + 4*b**2*c*e**2 + 10*b*c**2*d 
*e - 11*b*(22*b*c**2*e**2/7 + 4*c**3*d*e)/(12*c) + 2*c**3*d**2)/(5*c) +...
 

Maxima [F(-2)]

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

integrate((2*c*x+b)*(e*x+d)^2*(c*x^2+b*x+a)^(3/2),x, algorithm="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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (216) = 432\).

Time = 0.20 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.21 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{26880} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (6 \, c^{2} e^{2} x + \frac {14 \, c^{8} d e + 11 \, b c^{7} e^{2}}{c^{6}}\right )} x + \frac {84 \, c^{8} d^{2} + 266 \, b c^{7} d e + 47 \, b^{2} c^{6} e^{2} + 96 \, a c^{7} e^{2}}{c^{6}}\right )} x + \frac {1344 \, b c^{7} d^{2} + 966 \, b^{2} c^{6} d e + 1960 \, a c^{7} d e - 3 \, b^{3} c^{5} e^{2} + 556 \, a b c^{6} e^{2}}{c^{6}}\right )} x + \frac {1344 \, b^{2} c^{6} d^{2} + 2688 \, a c^{7} d^{2} - 14 \, b^{3} c^{5} d e + 3192 \, a b c^{6} d e + 7 \, b^{4} c^{4} e^{2} - 60 \, a b^{2} c^{5} e^{2} + 192 \, a^{2} c^{6} e^{2}}{c^{6}}\right )} x + \frac {10752 \, a b c^{6} d^{2} + 70 \, b^{4} c^{4} d e - 672 \, a b^{2} c^{5} d e + 3360 \, a^{2} c^{6} d e - 35 \, b^{5} c^{3} e^{2} + 336 \, a b^{3} c^{4} e^{2} - 912 \, a^{2} b c^{5} e^{2}}{c^{6}}\right )} x + \frac {10752 \, a^{2} c^{6} d^{2} - 210 \, b^{5} c^{3} d e + 2240 \, a b^{3} c^{4} d e - 7392 \, a^{2} b c^{5} d e + 105 \, b^{6} c^{2} e^{2} - 1120 \, a b^{4} c^{3} e^{2} + 3696 \, a^{2} b^{2} c^{4} e^{2} - 3072 \, a^{3} c^{5} e^{2}}{c^{6}}\right )} - \frac {{\left (2 \, b^{6} c d e - 24 \, a b^{4} c^{2} d e + 96 \, a^{2} b^{2} c^{3} d e - 128 \, a^{3} c^{4} d e - b^{7} e^{2} + 12 \, a b^{5} c e^{2} - 48 \, a^{2} b^{3} c^{2} e^{2} + 64 \, a^{3} b c^{3} e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{512 \, c^{\frac {9}{2}}} \] Input:

integrate((2*c*x+b)*(e*x+d)^2*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

Output:

1/26880*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(6*c^2*e^2*x + (14*c^8*d*e + 
 11*b*c^7*e^2)/c^6)*x + (84*c^8*d^2 + 266*b*c^7*d*e + 47*b^2*c^6*e^2 + 96* 
a*c^7*e^2)/c^6)*x + (1344*b*c^7*d^2 + 966*b^2*c^6*d*e + 1960*a*c^7*d*e - 3 
*b^3*c^5*e^2 + 556*a*b*c^6*e^2)/c^6)*x + (1344*b^2*c^6*d^2 + 2688*a*c^7*d^ 
2 - 14*b^3*c^5*d*e + 3192*a*b*c^6*d*e + 7*b^4*c^4*e^2 - 60*a*b^2*c^5*e^2 + 
 192*a^2*c^6*e^2)/c^6)*x + (10752*a*b*c^6*d^2 + 70*b^4*c^4*d*e - 672*a*b^2 
*c^5*d*e + 3360*a^2*c^6*d*e - 35*b^5*c^3*e^2 + 336*a*b^3*c^4*e^2 - 912*a^2 
*b*c^5*e^2)/c^6)*x + (10752*a^2*c^6*d^2 - 210*b^5*c^3*d*e + 2240*a*b^3*c^4 
*d*e - 7392*a^2*b*c^5*d*e + 105*b^6*c^2*e^2 - 1120*a*b^4*c^3*e^2 + 3696*a^ 
2*b^2*c^4*e^2 - 3072*a^3*c^5*e^2)/c^6) - 1/512*(2*b^6*c*d*e - 24*a*b^4*c^2 
*d*e + 96*a^2*b^2*c^3*d*e - 128*a^3*c^4*d*e - b^7*e^2 + 12*a*b^5*c*e^2 - 4 
8*a^2*b^3*c^2*e^2 + 64*a^3*b*c^3*e^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))*sqrt(c) + b))/c^(9/2)
 

Mupad [F(-1)]

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

int((b + 2*c*x)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int((b + 2*c*x)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 2.40 (sec) , antiderivative size = 1180, normalized size of antiderivative = 4.88 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

int((2*c*x+b)*(e*x+d)^2*(c*x^2+b*x+a)^(3/2),x)
 

Output:

( - 6144*sqrt(a + b*x + c*x**2)*a**3*c**4*e**2 + 7392*sqrt(a + b*x + c*x** 
2)*a**2*b**2*c**3*e**2 - 14784*sqrt(a + b*x + c*x**2)*a**2*b*c**4*d*e - 36 
48*sqrt(a + b*x + c*x**2)*a**2*b*c**4*e**2*x + 21504*sqrt(a + b*x + c*x**2 
)*a**2*c**5*d**2 + 13440*sqrt(a + b*x + c*x**2)*a**2*c**5*d*e*x + 3072*sqr 
t(a + b*x + c*x**2)*a**2*c**5*e**2*x**2 - 2240*sqrt(a + b*x + c*x**2)*a*b* 
*4*c**2*e**2 + 4480*sqrt(a + b*x + c*x**2)*a*b**3*c**3*d*e + 1344*sqrt(a + 
 b*x + c*x**2)*a*b**3*c**3*e**2*x - 2688*sqrt(a + b*x + c*x**2)*a*b**2*c** 
4*d*e*x - 960*sqrt(a + b*x + c*x**2)*a*b**2*c**4*e**2*x**2 + 43008*sqrt(a 
+ b*x + c*x**2)*a*b*c**5*d**2*x + 51072*sqrt(a + b*x + c*x**2)*a*b*c**5*d* 
e*x**2 + 17792*sqrt(a + b*x + c*x**2)*a*b*c**5*e**2*x**3 + 43008*sqrt(a + 
b*x + c*x**2)*a*c**6*d**2*x**2 + 62720*sqrt(a + b*x + c*x**2)*a*c**6*d*e*x 
**3 + 24576*sqrt(a + b*x + c*x**2)*a*c**6*e**2*x**4 + 210*sqrt(a + b*x + c 
*x**2)*b**6*c*e**2 - 420*sqrt(a + b*x + c*x**2)*b**5*c**2*d*e - 140*sqrt(a 
 + b*x + c*x**2)*b**5*c**2*e**2*x + 280*sqrt(a + b*x + c*x**2)*b**4*c**3*d 
*e*x + 112*sqrt(a + b*x + c*x**2)*b**4*c**3*e**2*x**2 - 224*sqrt(a + b*x + 
 c*x**2)*b**3*c**4*d*e*x**2 - 96*sqrt(a + b*x + c*x**2)*b**3*c**4*e**2*x** 
3 + 21504*sqrt(a + b*x + c*x**2)*b**2*c**5*d**2*x**2 + 30912*sqrt(a + b*x 
+ c*x**2)*b**2*c**5*d*e*x**3 + 12032*sqrt(a + b*x + c*x**2)*b**2*c**5*e**2 
*x**4 + 43008*sqrt(a + b*x + c*x**2)*b*c**6*d**2*x**3 + 68096*sqrt(a + b*x 
 + c*x**2)*b*c**6*d*e*x**4 + 28160*sqrt(a + b*x + c*x**2)*b*c**6*e**2*x...