\(\int x^2 (A+B x) (a+b x+c x^2)^{3/2} \, dx\) [118]

Optimal result

Integrand size = 23, antiderivative size = 269 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}} \] Output:

1/1024*(-4*a*c+b^2)*(8*A*a*c^2-14*A*b^2*c-12*B*a*b*c+9*B*b^3)*(2*c*x+b)*(c 
*x^2+b*x+a)^(1/2)/c^5-1/384*(8*A*a*c^2-14*A*b^2*c-12*B*a*b*c+9*B*b^3)*(2*c 
*x+b)*(c*x^2+b*x+a)^(3/2)/c^4+1/7*B*x^2*(c*x^2+b*x+a)^(5/2)/c+1/840*(63*B* 
b^2-98*A*b*c-48*B*a*c-10*c*(-14*A*c+9*B*b)*x)*(c*x^2+b*x+a)^(5/2)/c^3-1/20 
48*(-4*a*c+b^2)^2*(8*A*a*c^2-14*A*b^2*c-12*B*a*b*c+9*B*b^3)*arctanh(1/2*(2 
*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)
 

Mathematica [A] (verified)

Time = 2.11 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.22 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^6 B-210 b^5 c (7 A+3 B x)+28 b^4 c (-270 a B+c x (35 A+18 B x))+48 b^2 c^2 \left (343 a^2 B+2 c^2 x^3 (7 A+4 B x)-2 a c x (63 A+31 B x)\right )+16 b^3 c^2 \left (-c x^2 (49 A+27 B x)+7 a (95 A+39 B x)\right )+32 b c^3 \left (6 a c x^2 (21 A+11 B x)-3 a^2 (189 A+73 B x)+8 c^2 x^4 (91 A+75 B x)\right )+64 c^3 \left (-96 a^3 B+40 c^3 x^5 (7 A+6 B x)+3 a^2 c x (35 A+16 B x)+2 a c^2 x^3 (245 A+192 B x)\right )\right )+105 \left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{215040 c^{11/2}} \] Input:

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

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^6*B - 210*b^5*c*(7*A + 3*B*x) + 28 
*b^4*c*(-270*a*B + c*x*(35*A + 18*B*x)) + 48*b^2*c^2*(343*a^2*B + 2*c^2*x^ 
3*(7*A + 4*B*x) - 2*a*c*x*(63*A + 31*B*x)) + 16*b^3*c^2*(-(c*x^2*(49*A + 2 
7*B*x)) + 7*a*(95*A + 39*B*x)) + 32*b*c^3*(6*a*c*x^2*(21*A + 11*B*x) - 3*a 
^2*(189*A + 73*B*x) + 8*c^2*x^4*(91*A + 75*B*x)) + 64*c^3*(-96*a^3*B + 40* 
c^3*x^5*(7*A + 6*B*x) + 3*a^2*c*x*(35*A + 16*B*x) + 2*a*c^2*x^3*(245*A + 1 
92*B*x))) + 105*(b^2 - 4*a*c)^2*(9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A 
*c^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(215040*c^(11/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int -\frac {1}{2} x (4 a B+(9 b B-14 A c) x) \left (c x^2+b x+a\right )^{3/2}dx}{7 c}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}-\frac {\int x (4 a B+(9 b B-14 A c) x) \left (c x^2+b x+a\right )^{3/2}dx}{14 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}-\frac {\frac {7 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{60 c^2}}{14 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}-\frac {\frac {7 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{60 c^2}}{14 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}-\frac {\frac {7 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{60 c^2}}{14 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}-\frac {\frac {7 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{60 c^2}}{14 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}-\frac {\frac {7 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{60 c^2}}{14 c}\)

Input:

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

Output:

(B*x^2*(a + b*x + c*x^2)^(5/2))/(7*c) - (-1/60*((63*b^2*B - 98*A*b*c - 48* 
a*B*c - 10*c*(9*b*B - 14*A*c)*x)*(a + b*x + c*x^2)^(5/2))/c^2 + (7*(9*b^3* 
B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(((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))/(14*c)
 

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 [A] (verified)

Time = 1.33 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.54

method	result	size
risch	\(-\frac {\left (-15360 B \,c^{6} x^{6}-17920 A \,c^{6} x^{5}-19200 B b \,c^{5} x^{5}-23296 A b \,c^{5} x^{4}-24576 B a \,c^{5} x^{4}-384 B \,b^{2} c^{4} x^{4}-31360 A a \,c^{5} x^{3}-672 A \,b^{2} c^{4} x^{3}-2112 B a b \,c^{4} x^{3}+432 B \,b^{3} c^{3} x^{3}-4032 A a b \,c^{4} x^{2}+784 A \,b^{3} c^{3} x^{2}-3072 B \,a^{2} c^{4} x^{2}+2976 B a \,b^{2} c^{3} x^{2}-504 B \,b^{4} c^{2} x^{2}-6720 A \,a^{2} c^{4} x +6048 A a \,b^{2} c^{3} x -980 A \,b^{4} c^{2} x +7008 B \,a^{2} b \,c^{3} x -4368 B a \,b^{3} c^{2} x +630 B \,b^{5} c x +18144 a^{2} A b \,c^{3}-10640 A a \,b^{3} c^{2}+1470 A \,b^{5} c +6144 B \,a^{3} c^{3}-16464 B \,a^{2} b^{2} c^{2}+7560 B a \,b^{4} c -945 B \,b^{6}\right ) \sqrt {c \,x^{2}+b x +a}}{107520 c^{5}}-\frac {\left (128 A \,a^{3} c^{4}-288 A \,a^{2} b^{2} c^{3}+120 A a \,b^{4} c^{2}-14 A \,b^{6} c -192 B \,a^{3} b \,c^{3}+240 B \,a^{2} b^{3} c^{2}-84 B a \,b^{5} c +9 B \,b^{7}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {11}{2}}}\)	\(413\)
default	\(A \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )+B \left (\frac {x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{7 c}-\frac {9 b \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )}{14 c}-\frac {2 a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{7 c}\right )\)	\(688\)

Input:

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

Output:

-1/107520/c^5*(-15360*B*c^6*x^6-17920*A*c^6*x^5-19200*B*b*c^5*x^5-23296*A* 
b*c^5*x^4-24576*B*a*c^5*x^4-384*B*b^2*c^4*x^4-31360*A*a*c^5*x^3-672*A*b^2* 
c^4*x^3-2112*B*a*b*c^4*x^3+432*B*b^3*c^3*x^3-4032*A*a*b*c^4*x^2+784*A*b^3* 
c^3*x^2-3072*B*a^2*c^4*x^2+2976*B*a*b^2*c^3*x^2-504*B*b^4*c^2*x^2-6720*A*a 
^2*c^4*x+6048*A*a*b^2*c^3*x-980*A*b^4*c^2*x+7008*B*a^2*b*c^3*x-4368*B*a*b^ 
3*c^2*x+630*B*b^5*c*x+18144*A*a^2*b*c^3-10640*A*a*b^3*c^2+1470*A*b^5*c+614 
4*B*a^3*c^3-16464*B*a^2*b^2*c^2+7560*B*a*b^4*c-945*B*b^6)*(c*x^2+b*x+a)^(1 
/2)-1/2048*(128*A*a^3*c^4-288*A*a^2*b^2*c^3+120*A*a*b^4*c^2-14*A*b^6*c-192 
*B*a^3*b*c^3+240*B*a^2*b^3*c^2-84*B*a*b^5*c+9*B*b^7)/c^(11/2)*ln((1/2*b+c* 
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 845, normalized size of antiderivative = 3.14 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/430080*(105*(9*B*b^7 + 128*A*a^3*c^4 - 96*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 
 + 120*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 14*(6*B*a*b^5 + A*b^6)*c)*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*(15360*B*c^7*x^6 + 945*B*b^6*c + 1280*(15*B*b*c^6 + 14*A*c^7) 
*x^5 - 96*(64*B*a^3 + 189*A*a^2*b)*c^4 + 128*(3*B*b^2*c^5 + 2*(96*B*a + 91 
*A*b)*c^6)*x^4 + 112*(147*B*a^2*b^2 + 95*A*a*b^3)*c^3 - 16*(27*B*b^3*c^4 - 
 1960*A*a*c^6 - 6*(22*B*a*b + 7*A*b^2)*c^5)*x^3 - 210*(36*B*a*b^4 + 7*A*b^ 
5)*c^2 + 8*(63*B*b^4*c^3 + 24*(16*B*a^2 + 21*A*a*b)*c^5 - 2*(186*B*a*b^2 + 
 49*A*b^3)*c^4)*x^2 - 2*(315*B*b^5*c^2 - 3360*A*a^2*c^5 + 48*(73*B*a^2*b + 
 63*A*a*b^2)*c^4 - 14*(156*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + 
a))/c^6, 1/215040*(105*(9*B*b^7 + 128*A*a^3*c^4 - 96*(2*B*a^3*b + 3*A*a^2* 
b^2)*c^3 + 120*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 14*(6*B*a*b^5 + A*b^6)*c)*sqr 
t(-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*(15360*B*c^7*x^6 + 945*B*b^6*c + 1280*(15*B*b*c^6 + 14*A*c^ 
7)*x^5 - 96*(64*B*a^3 + 189*A*a^2*b)*c^4 + 128*(3*B*b^2*c^5 + 2*(96*B*a + 
91*A*b)*c^6)*x^4 + 112*(147*B*a^2*b^2 + 95*A*a*b^3)*c^3 - 16*(27*B*b^3*c^4 
 - 1960*A*a*c^6 - 6*(22*B*a*b + 7*A*b^2)*c^5)*x^3 - 210*(36*B*a*b^4 + 7*A* 
b^5)*c^2 + 8*(63*B*b^4*c^3 + 24*(16*B*a^2 + 21*A*a*b)*c^5 - 2*(186*B*a*b^2 
 + 49*A*b^3)*c^4)*x^2 - 2*(315*B*b^5*c^2 - 3360*A*a^2*c^5 + 48*(73*B*a^2*b 
 + 63*A*a*b^2)*c^4 - 14*(156*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1911 vs. \(2 (280) = 560\).

Time = 0.70 (sec) , antiderivative size = 1911, normalized size of antiderivative = 7.10 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate(x**2*(B*x+A)*(c*x**2+b*x+a)**(3/2),x)
 

Output:

Piecewise(((-a*(A*a**2 - 3*a*(2*A*a*c + A*b**2 + 2*B*a*b - 5*a*(A*c**2 + 1 
5*B*b*c/14)/(6*c) - 9*b*(2*A*b*c + 8*B*a*c/7 + B*b**2 - 11*b*(A*c**2 + 15* 
B*b*c/14)/(12*c))/(10*c))/(4*c) - 5*b*(2*A*a*b + B*a**2 - 4*a*(2*A*b*c + 8 
*B*a*c/7 + B*b**2 - 11*b*(A*c**2 + 15*B*b*c/14)/(12*c))/(5*c) - 7*b*(2*A*a 
*c + A*b**2 + 2*B*a*b - 5*a*(A*c**2 + 15*B*b*c/14)/(6*c) - 9*b*(2*A*b*c + 
8*B*a*c/7 + B*b**2 - 11*b*(A*c**2 + 15*B*b*c/14)/(12*c))/(10*c))/(8*c))/(6 
*c))/(2*c) - b*(-2*a*(2*A*a*b + B*a**2 - 4*a*(2*A*b*c + 8*B*a*c/7 + B*b**2 
 - 11*b*(A*c**2 + 15*B*b*c/14)/(12*c))/(5*c) - 7*b*(2*A*a*c + A*b**2 + 2*B 
*a*b - 5*a*(A*c**2 + 15*B*b*c/14)/(6*c) - 9*b*(2*A*b*c + 8*B*a*c/7 + B*b** 
2 - 11*b*(A*c**2 + 15*B*b*c/14)/(12*c))/(10*c))/(8*c))/(3*c) - 3*b*(A*a**2 
 - 3*a*(2*A*a*c + A*b**2 + 2*B*a*b - 5*a*(A*c**2 + 15*B*b*c/14)/(6*c) - 9* 
b*(2*A*b*c + 8*B*a*c/7 + B*b**2 - 11*b*(A*c**2 + 15*B*b*c/14)/(12*c))/(10* 
c))/(4*c) - 5*b*(2*A*a*b + B*a**2 - 4*a*(2*A*b*c + 8*B*a*c/7 + B*b**2 - 11 
*b*(A*c**2 + 15*B*b*c/14)/(12*c))/(5*c) - 7*b*(2*A*a*c + A*b**2 + 2*B*a*b 
- 5*a*(A*c**2 + 15*B*b*c/14)/(6*c) - 9*b*(2*A*b*c + 8*B*a*c/7 + B*b**2 - 1 
1*b*(A*c**2 + 15*B*b*c/14)/(12*c))/(10*c))/(8*c))/(6*c))/(4*c))/(2*c))*Pie 
cewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - 
b**2/(4*c), 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), 
 True)) + sqrt(a + b*x + c*x**2)*(B*c*x**6/7 + x**5*(A*c**2 + 15*B*b*c/14) 
/(6*c) + x**4*(2*A*b*c + 8*B*a*c/7 + B*b**2 - 11*b*(A*c**2 + 15*B*b*c/1...
 

Maxima [F(-2)]

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

integrate(x^2*(B*x+A)*(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 [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.56 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{107520} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c x + \frac {15 \, B b c^{6} + 14 \, A c^{7}}{c^{6}}\right )} x + \frac {3 \, B b^{2} c^{5} + 192 \, B a c^{6} + 182 \, A b c^{6}}{c^{6}}\right )} x - \frac {27 \, B b^{3} c^{4} - 132 \, B a b c^{5} - 42 \, A b^{2} c^{5} - 1960 \, A a c^{6}}{c^{6}}\right )} x + \frac {63 \, B b^{4} c^{3} - 372 \, B a b^{2} c^{4} - 98 \, A b^{3} c^{4} + 384 \, B a^{2} c^{5} + 504 \, A a b c^{5}}{c^{6}}\right )} x - \frac {315 \, B b^{5} c^{2} - 2184 \, B a b^{3} c^{3} - 490 \, A b^{4} c^{3} + 3504 \, B a^{2} b c^{4} + 3024 \, A a b^{2} c^{4} - 3360 \, A a^{2} c^{5}}{c^{6}}\right )} x + \frac {945 \, B b^{6} c - 7560 \, B a b^{4} c^{2} - 1470 \, A b^{5} c^{2} + 16464 \, B a^{2} b^{2} c^{3} + 10640 \, A a b^{3} c^{3} - 6144 \, B a^{3} c^{4} - 18144 \, A a^{2} b c^{4}}{c^{6}}\right )} + \frac {{\left (9 \, B b^{7} - 84 \, B a b^{5} c - 14 \, A b^{6} c + 240 \, B a^{2} b^{3} c^{2} + 120 \, A a b^{4} c^{2} - 192 \, B a^{3} b c^{3} - 288 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \] Input:

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

Output:

1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*B*c*x + (15*B*b*c^6 + 1 
4*A*c^7)/c^6)*x + (3*B*b^2*c^5 + 192*B*a*c^6 + 182*A*b*c^6)/c^6)*x - (27*B 
*b^3*c^4 - 132*B*a*b*c^5 - 42*A*b^2*c^5 - 1960*A*a*c^6)/c^6)*x + (63*B*b^4 
*c^3 - 372*B*a*b^2*c^4 - 98*A*b^3*c^4 + 384*B*a^2*c^5 + 504*A*a*b*c^5)/c^6 
)*x - (315*B*b^5*c^2 - 2184*B*a*b^3*c^3 - 490*A*b^4*c^3 + 3504*B*a^2*b*c^4 
 + 3024*A*a*b^2*c^4 - 3360*A*a^2*c^5)/c^6)*x + (945*B*b^6*c - 7560*B*a*b^4 
*c^2 - 1470*A*b^5*c^2 + 16464*B*a^2*b^2*c^3 + 10640*A*a*b^3*c^3 - 6144*B*a 
^3*c^4 - 18144*A*a^2*b*c^4)/c^6) + 1/2048*(9*B*b^7 - 84*B*a*b^5*c - 14*A*b 
^6*c + 240*B*a^2*b^3*c^2 + 120*A*a*b^4*c^2 - 192*B*a^3*b*c^3 - 288*A*a^2*b 
^2*c^3 + 128*A*a^3*c^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt 
(c) + b))/c^(11/2)
 

Mupad [F(-1)]

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

int(x^2*(A + B*x)*(a + b*x + c*x^2)^(3/2),x)
 

Output:

int(x^2*(A + B*x)*(a + b*x + c*x^2)^(3/2), x)
 

Reduce [F]

\[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\int x^{2} \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}d x \] Input:

int(x^2*(B*x+A)*(c*x^2+b*x+a)^(3/2),x)
 

Output:

int(x^2*(B*x+A)*(c*x^2+b*x+a)^(3/2),x)