\(\int x^3 (A+B x) (a+b x+c x^2)^{3/2} \, dx\) [117]

Optimal result

Integrand size = 23, antiderivative size = 356 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {3 \left (b^2-4 a c\right ) \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{2048 c^5}-\frac {(11 b B-16 A c) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\left (231 b^3 B-336 A b^2 c-372 a b B c+256 a A c^2-10 c \left (33 b^2 B-48 A b c-28 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{4480 c^4}+\frac {3 \left (b^2-4 a c\right )^2 \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2}} \] Output:

-3/16384*(-4*a*c+b^2)*(64*A*a*b*c^2-48*A*b^3*c+16*B*a^2*c^2-72*B*a*b^2*c+3 
3*B*b^4)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^6+1/2048*(64*A*a*b*c^2-48*A*b^3*c 
+16*B*a^2*c^2-72*B*a*b^2*c+33*B*b^4)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^5-1/1 
12*(-16*A*c+11*B*b)*x^2*(c*x^2+b*x+a)^(5/2)/c^2+1/8*B*x^3*(c*x^2+b*x+a)^(5 
/2)/c-1/4480*(231*B*b^3-336*A*b^2*c-372*B*a*b*c+256*A*a*c^2-10*c*(-48*A*b* 
c-28*B*a*c+33*B*b^2)*x)*(c*x^2+b*x+a)^(5/2)/c^4+3/32768*(-4*a*c+b^2)^2*(64 
*A*a*b*c^2-48*A*b^3*c+16*B*a^2*c^2-72*B*a*b^2*c+33*B*b^4)*arctanh(1/2*(2*c 
*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(13/2)
 

Mathematica [A] (verified)

Time = 2.89 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.16 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^7 B+210 b^6 c (24 A+11 B x)+84 b^5 c (365 a B-2 c x (20 A+11 B x))-16 b^3 c^2 \left (5103 a^2 B+8 c^2 x^3 (18 A+11 B x)-52 a c x (28 A+15 B x)\right )+128 c^4 \left (80 c^3 x^6 (8 A+7 B x)+2 a^2 c x^2 (64 A+35 B x)+8 a c^2 x^4 (128 A+105 B x)-a^3 (256 A+105 B x)\right )+24 b^4 c^2 \left (2 c x^2 (56 A+33 B x)-7 a (240 A+107 B x)\right )+64 b c^3 \left (919 a^3 B+8 a c^2 x^3 (22 A+13 B x)+80 c^3 x^5 (20 A+17 B x)-2 a^2 c x (292 A+151 B x)\right )+32 b^2 c^3 \left (8 c^2 x^4 (8 A+5 B x)-4 a c x^2 (124 A+71 B x)+a^2 (2744 A+1181 B x)\right )\right )-105 \left (b^2-4 a c\right )^2 \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{1146880 c^{13/2}} \] Input:

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

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^7*B + 210*b^6*c*(24*A + 11*B*x) 
+ 84*b^5*c*(365*a*B - 2*c*x*(20*A + 11*B*x)) - 16*b^3*c^2*(5103*a^2*B + 8* 
c^2*x^3*(18*A + 11*B*x) - 52*a*c*x*(28*A + 15*B*x)) + 128*c^4*(80*c^3*x^6* 
(8*A + 7*B*x) + 2*a^2*c*x^2*(64*A + 35*B*x) + 8*a*c^2*x^4*(128*A + 105*B*x 
) - a^3*(256*A + 105*B*x)) + 24*b^4*c^2*(2*c*x^2*(56*A + 33*B*x) - 7*a*(24 
0*A + 107*B*x)) + 64*b*c^3*(919*a^3*B + 8*a*c^2*x^3*(22*A + 13*B*x) + 80*c 
^3*x^5*(20*A + 17*B*x) - 2*a^2*c*x*(292*A + 151*B*x)) + 32*b^2*c^3*(8*c^2* 
x^4*(8*A + 5*B*x) - 4*a*c*x^2*(124*A + 71*B*x) + a^2*(2744*A + 1181*B*x))) 
 - 105*(b^2 - 4*a*c)^2*(33*b^4*B - 48*A*b^3*c - 72*a*b^2*B*c + 64*a*A*b*c^ 
2 + 16*a^2*B*c^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(11468 
80*c^(13/2))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.87, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1236, 27, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int -\frac {1}{2} x^2 (6 a B+(11 b B-16 A c) x) \left (c x^2+b x+a\right )^{3/2}dx}{8 c}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\int x^2 (6 a B+(11 b B-16 A c) x) \left (c x^2+b x+a\right )^{3/2}dx}{16 c}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\frac {\int -\frac {1}{2} x \left (4 a (11 b B-16 A c)+3 \left (33 B b^2-48 A c b-28 a B c\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{7 c}+\frac {x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{7 c}}{16 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{7 c}-\frac {\int x \left (4 a (11 b B-16 A c)+3 \left (33 B b^2-48 A c b-28 a B c\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{14 c}}{16 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{7 c}-\frac {\frac {7 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c x \left (-28 a B c-48 A b c+33 b^2 B\right )+256 a A c^2-372 a b B c-336 A b^2 c+231 b^3 B\right )}{20 c^2}}{14 c}}{16 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{7 c}-\frac {\frac {7 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c x \left (-28 a B c-48 A b c+33 b^2 B\right )+256 a A c^2-372 a b B c-336 A b^2 c+231 b^3 B\right )}{20 c^2}}{14 c}}{16 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{7 c}-\frac {\frac {7 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c x \left (-28 a B c-48 A b c+33 b^2 B\right )+256 a A c^2-372 a b B c-336 A b^2 c+231 b^3 B\right )}{20 c^2}}{14 c}}{16 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{7 c}-\frac {\frac {7 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c x \left (-28 a B c-48 A b c+33 b^2 B\right )+256 a A c^2-372 a b B c-336 A b^2 c+231 b^3 B\right )}{20 c^2}}{14 c}}{16 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{7 c}-\frac {\frac {7 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c x \left (-28 a B c-48 A b c+33 b^2 B\right )+256 a A c^2-372 a b B c-336 A b^2 c+231 b^3 B\right )}{20 c^2}}{14 c}}{16 c}\)

Input:

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

Output:

(B*x^3*(a + b*x + c*x^2)^(5/2))/(8*c) - (((11*b*B - 16*A*c)*x^2*(a + b*x + 
 c*x^2)^(5/2))/(7*c) - (-1/20*((231*b^3*B - 336*A*b^2*c - 372*a*b*B*c + 25 
6*a*A*c^2 - 10*c*(33*b^2*B - 48*A*b*c - 28*a*B*c)*x)*(a + b*x + c*x^2)^(5/ 
2))/c^2 + (7*(33*b^4*B - 48*A*b^3*c - 72*a*b^2*B*c + 64*a*A*b*c^2 + 16*a^2 
*B*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)))/(8*c^2))/(1 
4*c))/(16*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 = 1.39 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.47

method	result	size
risch	\(-\frac {\left (-71680 B \,c^{7} x^{7}-81920 A \,c^{7} x^{6}-87040 B b \,c^{6} x^{6}-102400 A b \,c^{6} x^{5}-107520 B a \,c^{6} x^{5}-1280 B \,b^{2} c^{5} x^{5}-131072 A a \,c^{6} x^{4}-2048 A \,b^{2} c^{5} x^{4}-6656 B a b \,c^{5} x^{4}+1408 B \,b^{3} c^{4} x^{4}-11264 A a b \,c^{5} x^{3}+2304 A \,b^{3} c^{4} x^{3}-8960 B \,a^{2} c^{5} x^{3}+9088 B a \,b^{2} c^{4} x^{3}-1584 B \,b^{4} c^{3} x^{3}-16384 A \,a^{2} c^{5} x^{2}+15872 A a \,b^{2} c^{4} x^{2}-2688 A \,b^{4} c^{3} x^{2}+19328 B \,a^{2} b \,c^{4} x^{2}-12480 B a \,b^{3} c^{3} x^{2}+1848 B \,b^{5} c^{2} x^{2}+37376 A \,a^{2} b \,c^{4} x -23296 A a \,b^{3} c^{3} x +3360 A \,b^{5} c^{2} x +13440 B \,a^{3} c^{4} x -37792 B \,a^{2} b^{2} c^{3} x +17976 B a \,b^{4} c^{2} x -2310 B \,b^{6} c x +32768 A \,a^{3} c^{4}-87808 A \,a^{2} b^{2} c^{3}+40320 A a \,b^{4} c^{2}-5040 A \,b^{6} c -58816 B \,a^{3} b \,c^{3}+81648 B \,a^{2} b^{3} c^{2}-30660 B a \,b^{5} c +3465 B \,b^{7}\right ) \sqrt {c \,x^{2}+b x +a}}{573440 c^{6}}+\frac {3 \left (1024 A \,a^{3} b \,c^{4}-1280 A \,a^{2} b^{3} c^{3}+448 A a \,b^{5} c^{2}-48 A \,b^{7} c +256 B \,a^{4} c^{4}-1280 B \,a^{3} b^{2} c^{3}+1120 B \,a^{2} b^{4} c^{2}-336 B a \,b^{6} c +33 B \,b^{8}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32768 c^{\frac {13}{2}}}\)	\(525\)
default	\(\text {Expression too large to display}\)	\(1142\)

Input:

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

Output:

-1/573440/c^6*(-71680*B*c^7*x^7-81920*A*c^7*x^6-87040*B*b*c^6*x^6-102400*A 
*b*c^6*x^5-107520*B*a*c^6*x^5-1280*B*b^2*c^5*x^5-131072*A*a*c^6*x^4-2048*A 
*b^2*c^5*x^4-6656*B*a*b*c^5*x^4+1408*B*b^3*c^4*x^4-11264*A*a*b*c^5*x^3+230 
4*A*b^3*c^4*x^3-8960*B*a^2*c^5*x^3+9088*B*a*b^2*c^4*x^3-1584*B*b^4*c^3*x^3 
-16384*A*a^2*c^5*x^2+15872*A*a*b^2*c^4*x^2-2688*A*b^4*c^3*x^2+19328*B*a^2* 
b*c^4*x^2-12480*B*a*b^3*c^3*x^2+1848*B*b^5*c^2*x^2+37376*A*a^2*b*c^4*x-232 
96*A*a*b^3*c^3*x+3360*A*b^5*c^2*x+13440*B*a^3*c^4*x-37792*B*a^2*b^2*c^3*x+ 
17976*B*a*b^4*c^2*x-2310*B*b^6*c*x+32768*A*a^3*c^4-87808*A*a^2*b^2*c^3+403 
20*A*a*b^4*c^2-5040*A*b^6*c-58816*B*a^3*b*c^3+81648*B*a^2*b^3*c^2-30660*B* 
a*b^5*c+3465*B*b^7)*(c*x^2+b*x+a)^(1/2)+3/32768*(1024*A*a^3*b*c^4-1280*A*a 
^2*b^3*c^3+448*A*a*b^5*c^2-48*A*b^7*c+256*B*a^4*c^4-1280*B*a^3*b^2*c^3+112 
0*B*a^2*b^4*c^2-336*B*a*b^6*c+33*B*b^8)/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c 
*x^2+b*x+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1037, normalized size of antiderivative = 2.91 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2293760*(105*(33*B*b^8 + 256*(B*a^4 + 4*A*a^3*b)*c^4 - 1280*(B*a^3*b^2 
+ A*a^2*b^3)*c^3 + 224*(5*B*a^2*b^4 + 2*A*a*b^5)*c^2 - 48*(7*B*a*b^6 + A*b 
^7)*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*(71680*B*c^8*x^7 - 3465*B*b^7*c - 32768*A*a 
^3*c^5 + 5120*(17*B*b*c^7 + 16*A*c^8)*x^6 + 1280*(B*b^2*c^6 + 4*(21*B*a + 
20*A*b)*c^7)*x^5 + 64*(919*B*a^3*b + 1372*A*a^2*b^2)*c^4 - 128*(11*B*b^3*c 
^5 - 1024*A*a*c^7 - 4*(13*B*a*b + 4*A*b^2)*c^6)*x^4 - 1008*(81*B*a^2*b^3 + 
 40*A*a*b^4)*c^3 + 16*(99*B*b^4*c^4 + 16*(35*B*a^2 + 44*A*a*b)*c^6 - 8*(71 
*B*a*b^2 + 18*A*b^3)*c^5)*x^3 + 420*(73*B*a*b^5 + 12*A*b^6)*c^2 - 8*(231*B 
*b^5*c^3 - 2048*A*a^2*c^6 + 16*(151*B*a^2*b + 124*A*a*b^2)*c^5 - 24*(65*B* 
a*b^3 + 14*A*b^4)*c^4)*x^2 + 2*(1155*B*b^6*c^2 - 64*(105*B*a^3 + 292*A*a^2 
*b)*c^5 + 16*(1181*B*a^2*b^2 + 728*A*a*b^3)*c^4 - 84*(107*B*a*b^4 + 20*A*b 
^5)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^7, -1/1146880*(105*(33*B*b^8 + 256*(B 
*a^4 + 4*A*a^3*b)*c^4 - 1280*(B*a^3*b^2 + A*a^2*b^3)*c^3 + 224*(5*B*a^2*b^ 
4 + 2*A*a*b^5)*c^2 - 48*(7*B*a*b^6 + A*b^7)*c)*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*(71680*B* 
c^8*x^7 - 3465*B*b^7*c - 32768*A*a^3*c^5 + 5120*(17*B*b*c^7 + 16*A*c^8)*x^ 
6 + 1280*(B*b^2*c^6 + 4*(21*B*a + 20*A*b)*c^7)*x^5 + 64*(919*B*a^3*b + 137 
2*A*a^2*b^2)*c^4 - 128*(11*B*b^3*c^5 - 1024*A*a*c^7 - 4*(13*B*a*b + 4*A*b^ 
2)*c^6)*x^4 - 1008*(81*B*a^2*b^3 + 40*A*a*b^4)*c^3 + 16*(99*B*b^4*c^4 +...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3089 vs. \(2 (384) = 768\).

Time = 0.76 (sec) , antiderivative size = 3089, normalized size of antiderivative = 8.68 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise(((-a*(-3*a*(2*A*a*b + B*a**2 - 5*a*(2*A*b*c + 9*B*a*c/8 + B*b**2 
 - 13*b*(A*c**2 + 17*B*b*c/16)/(14*c))/(6*c) - 9*b*(2*A*a*c + A*b**2 + 2*B 
*a*b - 6*a*(A*c**2 + 17*B*b*c/16)/(7*c) - 11*b*(2*A*b*c + 9*B*a*c/8 + B*b* 
*2 - 13*b*(A*c**2 + 17*B*b*c/16)/(14*c))/(12*c))/(10*c))/(4*c) - 5*b*(A*a* 
*2 - 4*a*(2*A*a*c + A*b**2 + 2*B*a*b - 6*a*(A*c**2 + 17*B*b*c/16)/(7*c) - 
11*b*(2*A*b*c + 9*B*a*c/8 + B*b**2 - 13*b*(A*c**2 + 17*B*b*c/16)/(14*c))/( 
12*c))/(5*c) - 7*b*(2*A*a*b + B*a**2 - 5*a*(2*A*b*c + 9*B*a*c/8 + B*b**2 - 
 13*b*(A*c**2 + 17*B*b*c/16)/(14*c))/(6*c) - 9*b*(2*A*a*c + A*b**2 + 2*B*a 
*b - 6*a*(A*c**2 + 17*B*b*c/16)/(7*c) - 11*b*(2*A*b*c + 9*B*a*c/8 + B*b**2 
 - 13*b*(A*c**2 + 17*B*b*c/16)/(14*c))/(12*c))/(10*c))/(8*c))/(6*c))/(2*c) 
 - b*(-2*a*(A*a**2 - 4*a*(2*A*a*c + A*b**2 + 2*B*a*b - 6*a*(A*c**2 + 17*B* 
b*c/16)/(7*c) - 11*b*(2*A*b*c + 9*B*a*c/8 + B*b**2 - 13*b*(A*c**2 + 17*B*b 
*c/16)/(14*c))/(12*c))/(5*c) - 7*b*(2*A*a*b + B*a**2 - 5*a*(2*A*b*c + 9*B* 
a*c/8 + B*b**2 - 13*b*(A*c**2 + 17*B*b*c/16)/(14*c))/(6*c) - 9*b*(2*A*a*c 
+ A*b**2 + 2*B*a*b - 6*a*(A*c**2 + 17*B*b*c/16)/(7*c) - 11*b*(2*A*b*c + 9* 
B*a*c/8 + B*b**2 - 13*b*(A*c**2 + 17*B*b*c/16)/(14*c))/(12*c))/(10*c))/(8* 
c))/(3*c) - 3*b*(-3*a*(2*A*a*b + B*a**2 - 5*a*(2*A*b*c + 9*B*a*c/8 + B*b** 
2 - 13*b*(A*c**2 + 17*B*b*c/16)/(14*c))/(6*c) - 9*b*(2*A*a*c + A*b**2 + 2* 
B*a*b - 6*a*(A*c**2 + 17*B*b*c/16)/(7*c) - 11*b*(2*A*b*c + 9*B*a*c/8 + B*b 
**2 - 13*b*(A*c**2 + 17*B*b*c/16)/(14*c))/(12*c))/(10*c))/(4*c) - 5*b*(...
 

Maxima [F(-2)]

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

integrate(x^3*(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 = 522, normalized size of antiderivative = 1.47 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{573440} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, B c x + \frac {17 \, B b c^{7} + 16 \, A c^{8}}{c^{7}}\right )} x + \frac {B b^{2} c^{6} + 84 \, B a c^{7} + 80 \, A b c^{7}}{c^{7}}\right )} x - \frac {11 \, B b^{3} c^{5} - 52 \, B a b c^{6} - 16 \, A b^{2} c^{6} - 1024 \, A a c^{7}}{c^{7}}\right )} x + \frac {99 \, B b^{4} c^{4} - 568 \, B a b^{2} c^{5} - 144 \, A b^{3} c^{5} + 560 \, B a^{2} c^{6} + 704 \, A a b c^{6}}{c^{7}}\right )} x - \frac {231 \, B b^{5} c^{3} - 1560 \, B a b^{3} c^{4} - 336 \, A b^{4} c^{4} + 2416 \, B a^{2} b c^{5} + 1984 \, A a b^{2} c^{5} - 2048 \, A a^{2} c^{6}}{c^{7}}\right )} x + \frac {1155 \, B b^{6} c^{2} - 8988 \, B a b^{4} c^{3} - 1680 \, A b^{5} c^{3} + 18896 \, B a^{2} b^{2} c^{4} + 11648 \, A a b^{3} c^{4} - 6720 \, B a^{3} c^{5} - 18688 \, A a^{2} b c^{5}}{c^{7}}\right )} x - \frac {3465 \, B b^{7} c - 30660 \, B a b^{5} c^{2} - 5040 \, A b^{6} c^{2} + 81648 \, B a^{2} b^{3} c^{3} + 40320 \, A a b^{4} c^{3} - 58816 \, B a^{3} b c^{4} - 87808 \, A a^{2} b^{2} c^{4} + 32768 \, A a^{3} c^{5}}{c^{7}}\right )} - \frac {3 \, {\left (33 \, B b^{8} - 336 \, B a b^{6} c - 48 \, A b^{7} c + 1120 \, B a^{2} b^{4} c^{2} + 448 \, A a b^{5} c^{2} - 1280 \, B a^{3} b^{2} c^{3} - 1280 \, A a^{2} b^{3} c^{3} + 256 \, B a^{4} c^{4} + 1024 \, A a^{3} b c^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{32768 \, c^{\frac {13}{2}}} \] Input:

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

Output:

1/573440*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*(14*B*c*x + (17*B*b*c^7 
+ 16*A*c^8)/c^7)*x + (B*b^2*c^6 + 84*B*a*c^7 + 80*A*b*c^7)/c^7)*x - (11*B* 
b^3*c^5 - 52*B*a*b*c^6 - 16*A*b^2*c^6 - 1024*A*a*c^7)/c^7)*x + (99*B*b^4*c 
^4 - 568*B*a*b^2*c^5 - 144*A*b^3*c^5 + 560*B*a^2*c^6 + 704*A*a*b*c^6)/c^7) 
*x - (231*B*b^5*c^3 - 1560*B*a*b^3*c^4 - 336*A*b^4*c^4 + 2416*B*a^2*b*c^5 
+ 1984*A*a*b^2*c^5 - 2048*A*a^2*c^6)/c^7)*x + (1155*B*b^6*c^2 - 8988*B*a*b 
^4*c^3 - 1680*A*b^5*c^3 + 18896*B*a^2*b^2*c^4 + 11648*A*a*b^3*c^4 - 6720*B 
*a^3*c^5 - 18688*A*a^2*b*c^5)/c^7)*x - (3465*B*b^7*c - 30660*B*a*b^5*c^2 - 
 5040*A*b^6*c^2 + 81648*B*a^2*b^3*c^3 + 40320*A*a*b^4*c^3 - 58816*B*a^3*b* 
c^4 - 87808*A*a^2*b^2*c^4 + 32768*A*a^3*c^5)/c^7) - 3/32768*(33*B*b^8 - 33 
6*B*a*b^6*c - 48*A*b^7*c + 1120*B*a^2*b^4*c^2 + 448*A*a*b^5*c^2 - 1280*B*a 
^3*b^2*c^3 - 1280*A*a^2*b^3*c^3 + 256*B*a^4*c^4 + 1024*A*a^3*b*c^4)*log(ab 
s(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(13/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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