\(\int x^3 (A+B x) \sqrt {a+b x+c x^2} \, dx\) [105]

Optimal result

Integrand size = 23, antiderivative size = 280 \[ \int x^3 (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {\left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}-\frac {(3 b B-4 A c) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (105 b^3 B-140 A b^2 c-196 a b B c+128 a A c^2-6 c \left (21 b^2 B-28 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}-\frac {\left (b^2-4 a c\right ) \left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}} \] Output:

1/512*(48*A*a*b*c^2-28*A*b^3*c+16*B*a^2*c^2-56*B*a*b^2*c+21*B*b^4)*(2*c*x+ 
b)*(c*x^2+b*x+a)^(1/2)/c^5-1/20*(-4*A*c+3*B*b)*x^2*(c*x^2+b*x+a)^(3/2)/c^2 
+1/6*B*x^3*(c*x^2+b*x+a)^(3/2)/c-1/960*(105*B*b^3-140*A*b^2*c-196*B*a*b*c+ 
128*A*a*c^2-6*c*(-28*A*b*c-20*B*a*c+21*B*b^2)*x)*(c*x^2+b*x+a)^(3/2)/c^4-1 
/1024*(-4*a*c+b^2)*(48*A*a*b*c^2-28*A*b^3*c+16*B*a^2*c^2-56*B*a*b^2*c+21*B 
*b^4)*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 = 1.44 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.97 \[ \int x^3 (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (315 b^5 B-210 b^4 c (2 A+B x)+56 b^3 c (-30 a B+c x (5 A+3 B x))-32 c^3 \left (-8 c^2 x^4 (6 A+5 B x)-2 a c x^2 (8 A+5 B x)+a^2 (32 A+15 B x)\right )+16 b c^2 \left (113 a^2 B+4 c^2 x^3 (3 A+2 B x)-2 a c x (29 A+17 B x)\right )+16 b^2 c^2 \left (-c x^2 (14 A+9 B x)+a (115 A+56 B x)\right )\right )+15 \left (b^2-4 a c\right ) \left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{15360 c^{11/2}} \] Input:

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

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(315*b^5*B - 210*b^4*c*(2*A + B*x) + 56*b 
^3*c*(-30*a*B + c*x*(5*A + 3*B*x)) - 32*c^3*(-8*c^2*x^4*(6*A + 5*B*x) - 2* 
a*c*x^2*(8*A + 5*B*x) + a^2*(32*A + 15*B*x)) + 16*b*c^2*(113*a^2*B + 4*c^2 
*x^3*(3*A + 2*B*x) - 2*a*c*x*(29*A + 17*B*x)) + 16*b^2*c^2*(-(c*x^2*(14*A 
+ 9*B*x)) + a*(115*A + 56*B*x))) + 15*(b^2 - 4*a*c)*(21*b^4*B - 28*A*b^3*c 
 - 56*a*b^2*B*c + 48*a*A*b*c^2 + 16*a^2*B*c^2)*Log[b + 2*c*x - 2*Sqrt[c]*S 
qrt[a + x*(b + c*x)]])/(15360*c^(11/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1236, 27, 1236, 27, 1225, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int -\frac {3}{2} x^2 (2 a B+(3 b B-4 A c) x) \sqrt {c x^2+b x+a}dx}{6 c}+\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\int x^2 (2 a B+(3 b B-4 A c) x) \sqrt {c x^2+b x+a}dx}{4 c}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\frac {\int -\frac {1}{2} x \left (4 a (3 b B-4 A c)+\left (21 B b^2-28 A c b-20 a B c\right ) x\right ) \sqrt {c x^2+b x+a}dx}{5 c}+\frac {x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{5 c}}{4 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{5 c}-\frac {\int x \left (4 a (3 b B-4 A c)+\left (21 B b^2-28 A c b-20 a B c\right ) x\right ) \sqrt {c x^2+b x+a}dx}{10 c}}{4 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{5 c}-\frac {\frac {5 \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \int \sqrt {c x^2+b x+a}dx}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c x \left (-20 a B c-28 A b c+21 b^2 B\right )+128 a A c^2-196 a b B c-140 A b^2 c+105 b^3 B\right )}{24 c^2}}{10 c}}{4 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{5 c}-\frac {\frac {5 \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c x \left (-20 a B c-28 A b c+21 b^2 B\right )+128 a A c^2-196 a b B c-140 A b^2 c+105 b^3 B\right )}{24 c^2}}{10 c}}{4 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{5 c}-\frac {\frac {5 \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c x \left (-20 a B c-28 A b c+21 b^2 B\right )+128 a A c^2-196 a b B c-140 A b^2 c+105 b^3 B\right )}{24 c^2}}{10 c}}{4 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{5 c}-\frac {\frac {5 \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c x \left (-20 a B c-28 A b c+21 b^2 B\right )+128 a A c^2-196 a b B c-140 A b^2 c+105 b^3 B\right )}{24 c^2}}{10 c}}{4 c}\)

Input:

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

Output:

(B*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) - (((3*b*B - 4*A*c)*x^2*(a + b*x + c 
*x^2)^(3/2))/(5*c) - (-1/24*((105*b^3*B - 140*A*b^2*c - 196*a*b*B*c + 128* 
a*A*c^2 - 6*c*(21*b^2*B - 28*A*b*c - 20*a*B*c)*x)*(a + b*x + c*x^2)^(3/2)) 
/c^2 + (5*(21*b^4*B - 28*A*b^3*c - 56*a*b^2*B*c + 48*a*A*b*c^2 + 16*a^2*B* 
c^2)*(((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^2))/(10 
*c))/(4*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 = 1.24 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {\left (-1280 B \,c^{5} x^{5}-1536 A \,c^{5} x^{4}-128 B b \,c^{4} x^{4}-192 A b \,c^{4} x^{3}-320 B a \,c^{4} x^{3}+144 B \,b^{2} c^{3} x^{3}-512 A a \,c^{4} x^{2}+224 A \,b^{2} c^{3} x^{2}+544 B a b \,c^{3} x^{2}-168 B \,b^{3} c^{2} x^{2}+928 A a b \,c^{3} x -280 A \,b^{3} c^{2} x +480 B \,a^{2} c^{3} x -896 B a \,b^{2} c^{2} x +210 B \,b^{4} c x +1024 a^{2} A \,c^{3}-1840 A a \,b^{2} c^{2}+420 A \,b^{4} c -1808 B \,a^{2} b \,c^{2}+1680 B a \,b^{3} c -315 b^{5} B \right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{5}}+\frac {\left (192 a^{2} A b \,c^{3}-160 A a \,b^{3} c^{2}+28 A \,b^{5} c +64 B \,a^{3} c^{3}-240 B \,a^{2} b^{2} c^{2}+140 B a \,b^{4} c -21 B \,b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {11}{2}}}\)	\(316\)
default	\(A \left (\frac {x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{5 c}-\frac {7 b \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \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 )}{2 c}\right )}{8 c}-\frac {a \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 )}{4 c}\right )}{10 c}-\frac {2 a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \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 )}{2 c}\right )}{5 c}\right )+B \left (\frac {x^{3} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{6 c}-\frac {3 b \left (\frac {x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{5 c}-\frac {7 b \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \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 )}{2 c}\right )}{8 c}-\frac {a \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 )}{4 c}\right )}{10 c}-\frac {2 a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \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 )}{2 c}\right )}{5 c}\right )}{4 c}-\frac {a \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \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 )}{2 c}\right )}{8 c}-\frac {a \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 )}{4 c}\right )}{2 c}\right )\)	\(830\)

Input:

int(x^3*(B*x+A)*(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/7680*(-1280*B*c^5*x^5-1536*A*c^5*x^4-128*B*b*c^4*x^4-192*A*b*c^4*x^3-32 
0*B*a*c^4*x^3+144*B*b^2*c^3*x^3-512*A*a*c^4*x^2+224*A*b^2*c^3*x^2+544*B*a* 
b*c^3*x^2-168*B*b^3*c^2*x^2+928*A*a*b*c^3*x-280*A*b^3*c^2*x+480*B*a^2*c^3* 
x-896*B*a*b^2*c^2*x+210*B*b^4*c*x+1024*A*a^2*c^3-1840*A*a*b^2*c^2+420*A*b^ 
4*c-1808*B*a^2*b*c^2+1680*B*a*b^3*c-315*B*b^5)/c^5*(c*x^2+b*x+a)^(1/2)+1/1 
024*(192*A*a^2*b*c^3-160*A*a*b^3*c^2+28*A*b^5*c+64*B*a^3*c^3-240*B*a^2*b^2 
*c^2+140*B*a*b^4*c-21*B*b^6)/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 = 667, normalized size of antiderivative = 2.38 \[ \int x^3 (A+B x) \sqrt {a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

integrate(x^3*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/30720*(15*(21*B*b^6 - 64*(B*a^3 + 3*A*a^2*b)*c^3 + 80*(3*B*a^2*b^2 + 2 
*A*a*b^3)*c^2 - 28*(5*B*a*b^4 + A*b^5)*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*(1280*B* 
c^6*x^5 + 315*B*b^5*c - 1024*A*a^2*c^4 + 128*(B*b*c^5 + 12*A*c^6)*x^4 + 16 
*(113*B*a^2*b + 115*A*a*b^2)*c^3 - 16*(9*B*b^2*c^4 - 4*(5*B*a + 3*A*b)*c^5 
)*x^3 - 420*(4*B*a*b^3 + A*b^4)*c^2 + 8*(21*B*b^3*c^3 + 64*A*a*c^5 - 4*(17 
*B*a*b + 7*A*b^2)*c^4)*x^2 - 2*(105*B*b^4*c^2 + 16*(15*B*a^2 + 29*A*a*b)*c 
^4 - 28*(16*B*a*b^2 + 5*A*b^3)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/15360 
*(15*(21*B*b^6 - 64*(B*a^3 + 3*A*a^2*b)*c^3 + 80*(3*B*a^2*b^2 + 2*A*a*b^3) 
*c^2 - 28*(5*B*a*b^4 + A*b^5)*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*(1280*B*c^6*x^5 + 315*B 
*b^5*c - 1024*A*a^2*c^4 + 128*(B*b*c^5 + 12*A*c^6)*x^4 + 16*(113*B*a^2*b + 
 115*A*a*b^2)*c^3 - 16*(9*B*b^2*c^4 - 4*(5*B*a + 3*A*b)*c^5)*x^3 - 420*(4* 
B*a*b^3 + A*b^4)*c^2 + 8*(21*B*b^3*c^3 + 64*A*a*c^5 - 4*(17*B*a*b + 7*A*b^ 
2)*c^4)*x^2 - 2*(105*B*b^4*c^2 + 16*(15*B*a^2 + 29*A*a*b)*c^4 - 28*(16*B*a 
*b^2 + 5*A*b^3)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 831 vs. \(2 (299) = 598\).

Time = 0.67 (sec) , antiderivative size = 831, normalized size of antiderivative = 2.97 \[ \int x^3 (A+B x) \sqrt {a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate(x**3*(B*x+A)*(c*x**2+b*x+a)**(1/2),x)
 

Output:

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

Maxima [F(-2)]

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

integrate(x^3*(B*x+A)*(c*x^2+b*x+a)^(1/2),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 = 0.27 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.15 \[ \int x^3 (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B x + \frac {B b c^{4} + 12 \, A c^{5}}{c^{5}}\right )} x - \frac {9 \, B b^{2} c^{3} - 20 \, B a c^{4} - 12 \, A b c^{4}}{c^{5}}\right )} x + \frac {21 \, B b^{3} c^{2} - 68 \, B a b c^{3} - 28 \, A b^{2} c^{3} + 64 \, A a c^{4}}{c^{5}}\right )} x - \frac {105 \, B b^{4} c - 448 \, B a b^{2} c^{2} - 140 \, A b^{3} c^{2} + 240 \, B a^{2} c^{3} + 464 \, A a b c^{3}}{c^{5}}\right )} x + \frac {315 \, B b^{5} - 1680 \, B a b^{3} c - 420 \, A b^{4} c + 1808 \, B a^{2} b c^{2} + 1840 \, A a b^{2} c^{2} - 1024 \, A a^{2} c^{3}}{c^{5}}\right )} + \frac {{\left (21 \, B b^{6} - 140 \, B a b^{4} c - 28 \, A b^{5} c + 240 \, B a^{2} b^{2} c^{2} + 160 \, A a b^{3} c^{2} - 64 \, B a^{3} c^{3} - 192 \, A a^{2} b c^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {11}{2}}} \] Input:

integrate(x^3*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*B*x + (B*b*c^4 + 12*A*c^5)/c^ 
5)*x - (9*B*b^2*c^3 - 20*B*a*c^4 - 12*A*b*c^4)/c^5)*x + (21*B*b^3*c^2 - 68 
*B*a*b*c^3 - 28*A*b^2*c^3 + 64*A*a*c^4)/c^5)*x - (105*B*b^4*c - 448*B*a*b^ 
2*c^2 - 140*A*b^3*c^2 + 240*B*a^2*c^3 + 464*A*a*b*c^3)/c^5)*x + (315*B*b^5 
 - 1680*B*a*b^3*c - 420*A*b^4*c + 1808*B*a^2*b*c^2 + 1840*A*a*b^2*c^2 - 10 
24*A*a^2*c^3)/c^5) + 1/1024*(21*B*b^6 - 140*B*a*b^4*c - 28*A*b^5*c + 240*B 
*a^2*b^2*c^2 + 160*A*a*b^3*c^2 - 64*B*a^3*c^3 - 192*A*a^2*b*c^3)*log(abs(2 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(11/2)
 

Mupad [B] (verification not implemented)

Time = 11.90 (sec) , antiderivative size = 781, normalized size of antiderivative = 2.79 \[ \int x^3 (A+B x) \sqrt {a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

int(x^3*(A + B*x)*(a + b*x + c*x^2)^(1/2),x)
 

Output:

(A*x^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (B*x^3*(a + b*x + c*x^2)^(3/2))/(6 
*c) - (2*A*a*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 
 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + 
c*x^2)^(1/2))/(24*c^2)))/(5*c) + (7*A*b*((5*b*((log((b + 2*c*x)/c^(1/2) + 
2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^ 
2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + 
b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + 
(log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3 
/2))))/(4*c)))/(10*c) + (B*a*((5*b*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x 
+ c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 
+ 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + b*x + c*x^2 
)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + 
 c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c 
)))/(2*c) - (3*B*b*((7*b*((5*b*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c* 
x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2* 
b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3 
/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x 
)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/ 
(10*c) - (2*a*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 
- 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*...
 

Reduce [F]

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

int(x^3*(B*x+A)*(c*x^2+b*x+a)^(1/2),x)
 

Output:

int(x^3*(B*x+A)*(c*x^2+b*x+a)^(1/2),x)