\(\int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx\) [104]

Optimal result

Integrand size = 23, antiderivative size = 367 \[ \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx=-\frac {\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^6}+\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac {\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{13/2}} \] Output:

-1/1024*(-32*A*a^2*c^3+112*A*a*b^2*c^2-42*A*b^4*c+80*B*a^2*b*c^2-120*B*a*b 
^3*c+33*B*b^5)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^6+1/280*(-42*A*b*c-32*B*a*c 
+33*B*b^2)*x^2*(c*x^2+b*x+a)^(3/2)/c^3-1/84*(-14*A*c+11*B*b)*x^3*(c*x^2+b* 
x+a)^(3/2)/c^2+1/7*B*x^4*(c*x^2+b*x+a)^(3/2)/c+1/13440*(1155*B*b^4-1470*A* 
b^3*c-3276*B*a*b^2*c+2744*A*a*b*c^2+1024*B*a^2*c^2-6*c*(280*A*a*c^2-294*A* 
b^2*c-444*B*a*b*c+231*B*b^3)*x)*(c*x^2+b*x+a)^(3/2)/c^5+1/2048*(-4*a*c+b^2 
)*(-32*A*a^2*c^3+112*A*a*b^2*c^2-42*A*b^4*c+80*B*a^2*b*c^2-120*B*a*b^3*c+3 
3*B*b^5)*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 = 2.07 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.95 \[ \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^6 B+210 b^5 c (21 A+11 B x)+84 b^4 c (260 a B-c x (35 A+22 B x))+48 b^3 c^2 \left (-14 a (35 A+18 B x)+c x^2 (49 A+33 B x)\right )+64 c^3 \left (128 a^3 B+40 c^3 x^5 (7 A+6 B x)+2 a c^2 x^3 (35 A+24 B x)-a^2 c x (105 A+64 B x)\right )-16 b^2 c^2 \left (2163 a^2 B+2 c^2 x^3 (63 A+44 B x)-2 a c x (392 A+243 B x)\right )+32 b c^3 \left (8 c^2 x^4 (7 A+5 B x)-2 a c x^2 (119 A+79 B x)+a^2 (791 A+397 B x)\right )\right )-105 \left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{215040 c^{13/2}} \] Input:

Integrate[x^4*(A + B*x)*Sqrt[a + b*x + c*x^2],x]
 

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^6*B + 210*b^5*c*(21*A + 11*B*x) 
+ 84*b^4*c*(260*a*B - c*x*(35*A + 22*B*x)) + 48*b^3*c^2*(-14*a*(35*A + 18* 
B*x) + c*x^2*(49*A + 33*B*x)) + 64*c^3*(128*a^3*B + 40*c^3*x^5*(7*A + 6*B* 
x) + 2*a*c^2*x^3*(35*A + 24*B*x) - a^2*c*x*(105*A + 64*B*x)) - 16*b^2*c^2* 
(2163*a^2*B + 2*c^2*x^3*(63*A + 44*B*x) - 2*a*c*x*(392*A + 243*B*x)) + 32* 
b*c^3*(8*c^2*x^4*(7*A + 5*B*x) - 2*a*c*x^2*(119*A + 79*B*x) + a^2*(791*A + 
 397*B*x))) - 105*(b^2 - 4*a*c)*(33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c + 1 
12*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*Log[b + 2*c*x - 2*Sqrt[c]* 
Sqrt[a + x*(b + c*x)]])/(215040*c^(13/2))
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1236, 27, 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.

\(\displaystyle \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int -\frac {1}{2} x^3 (8 a B+(11 b B-14 A c) x) \sqrt {c x^2+b x+a}dx}{7 c}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {\int -\frac {3}{2} x^2 \left (2 a (11 b B-14 A c)+\left (33 B b^2-42 A c b-32 a B c\right ) x\right ) \sqrt {c x^2+b x+a}dx}{6 c}+\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{6 c}}{14 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{6 c}-\frac {\int x^2 \left (2 a (11 b B-14 A c)+\left (33 B b^2-42 A c b-32 a B c\right ) x\right ) \sqrt {c x^2+b x+a}dx}{4 c}}{14 c}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{6 c}-\frac {\frac {\int -\frac {1}{2} x \left (4 a \left (33 B b^2-42 A c b-32 a B c\right )+\left (231 B b^3-294 A c b^2-444 a B c b+280 a A c^2\right ) x\right ) \sqrt {c x^2+b x+a}dx}{5 c}+\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{5 c}}{4 c}}{14 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{5 c}-\frac {\int x \left (4 a \left (33 B b^2-42 A c b-32 a B c\right )+\left (231 B b^3-294 A c b^2-444 a B c b+280 a A c^2\right ) x\right ) \sqrt {c x^2+b x+a}dx}{10 c}}{4 c}}{14 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{5 c}-\frac {\frac {35 \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \int \sqrt {c x^2+b x+a}dx}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{24 c^2}}{10 c}}{4 c}}{14 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{5 c}-\frac {\frac {35 \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{24 c^2}}{10 c}}{4 c}}{14 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{5 c}-\frac {\frac {35 \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{24 c^2}}{10 c}}{4 c}}{14 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{6 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{5 c}-\frac {\frac {35 \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{24 c^2}}{10 c}}{4 c}}{14 c}\)

Input:

Int[x^4*(A + B*x)*Sqrt[a + b*x + c*x^2],x]
 

Output:

(B*x^4*(a + b*x + c*x^2)^(3/2))/(7*c) - (((11*b*B - 14*A*c)*x^3*(a + b*x + 
 c*x^2)^(3/2))/(6*c) - (((33*b^2*B - 42*A*b*c - 32*a*B*c)*x^2*(a + b*x + c 
*x^2)^(3/2))/(5*c) - (-1/24*((1155*b^4*B - 1470*A*b^3*c - 3276*a*b^2*B*c + 
 2744*a*A*b*c^2 + 1024*a^2*B*c^2 - 6*c*(231*b^3*B - 294*A*b^2*c - 444*a*b* 
B*c + 280*a*A*c^2)*x)*(a + b*x + c*x^2)^(3/2))/c^2 + (35*(33*b^5*B - 42*A* 
b^4*c - 120*a*b^3*B*c + 112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*( 
((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))/(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 = 1.37 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.13

method	result	size
risch	\(\frac {\left (15360 B \,c^{6} x^{6}+17920 A \,c^{6} x^{5}+1280 B b \,c^{5} x^{5}+1792 A b \,c^{5} x^{4}+3072 B a \,c^{5} x^{4}-1408 B \,b^{2} c^{4} x^{4}+4480 A a \,c^{5} x^{3}-2016 A \,b^{2} c^{4} x^{3}-5056 B a b \,c^{4} x^{3}+1584 B \,b^{3} c^{3} x^{3}-7616 A a b \,c^{4} x^{2}+2352 A \,b^{3} c^{3} x^{2}-4096 B \,a^{2} c^{4} x^{2}+7776 B a \,b^{2} c^{3} x^{2}-1848 B \,b^{4} c^{2} x^{2}-6720 A \,a^{2} c^{4} x +12544 A a \,b^{2} c^{3} x -2940 A \,b^{4} c^{2} x +12704 B \,a^{2} b \,c^{3} x -12096 B a \,b^{3} c^{2} x +2310 B \,b^{5} c x +25312 a^{2} A b \,c^{3}-23520 A a \,b^{3} c^{2}+4410 A \,b^{5} c +8192 B \,a^{3} c^{3}-34608 B \,a^{2} b^{2} c^{2}+21840 B a \,b^{4} c -3465 B \,b^{6}\right ) \sqrt {c \,x^{2}+b x +a}}{107520 c^{6}}+\frac {\left (128 A \,a^{3} c^{4}-480 A \,a^{2} b^{2} c^{3}+280 A a \,b^{4} c^{2}-42 A \,b^{6} c -320 B \,a^{3} b \,c^{3}+560 B \,a^{2} b^{3} c^{2}-252 B a \,b^{5} c +33 B \,b^{7}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {13}{2}}}\)	\(413\)
default	\(\text {Expression too large to display}\)	\(1383\)

Input:

int(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/107520*(15360*B*c^6*x^6+17920*A*c^6*x^5+1280*B*b*c^5*x^5+1792*A*b*c^5*x^ 
4+3072*B*a*c^5*x^4-1408*B*b^2*c^4*x^4+4480*A*a*c^5*x^3-2016*A*b^2*c^4*x^3- 
5056*B*a*b*c^4*x^3+1584*B*b^3*c^3*x^3-7616*A*a*b*c^4*x^2+2352*A*b^3*c^3*x^ 
2-4096*B*a^2*c^4*x^2+7776*B*a*b^2*c^3*x^2-1848*B*b^4*c^2*x^2-6720*A*a^2*c^ 
4*x+12544*A*a*b^2*c^3*x-2940*A*b^4*c^2*x+12704*B*a^2*b*c^3*x-12096*B*a*b^3 
*c^2*x+2310*B*b^5*c*x+25312*A*a^2*b*c^3-23520*A*a*b^3*c^2+4410*A*b^5*c+819 
2*B*a^3*c^3-34608*B*a^2*b^2*c^2+21840*B*a*b^4*c-3465*B*b^6)/c^6*(c*x^2+b*x 
+a)^(1/2)+1/2048*(128*A*a^3*c^4-480*A*a^2*b^2*c^3+280*A*a*b^4*c^2-42*A*b^6 
*c-320*B*a^3*b*c^3+560*B*a^2*b^3*c^2-252*B*a*b^5*c+33*B*b^7)/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 = 0.15 (sec) , antiderivative size = 843, normalized size of antiderivative = 2.30 \[ \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/430080*(105*(33*B*b^7 + 128*A*a^3*c^4 - 160*(2*B*a^3*b + 3*A*a^2*b^2)*c 
^3 + 280*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 42*(6*B*a*b^5 + A*b^6)*c)*sqrt(c)*l 
og(-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 - 3465*B*b^6*c + 1280*(B*b*c^6 + 14*A*c^7) 
*x^5 + 32*(256*B*a^3 + 791*A*a^2*b)*c^4 - 128*(11*B*b^2*c^5 - 2*(12*B*a + 
7*A*b)*c^6)*x^4 - 336*(103*B*a^2*b^2 + 70*A*a*b^3)*c^3 + 16*(99*B*b^3*c^4 
+ 280*A*a*c^6 - 2*(158*B*a*b + 63*A*b^2)*c^5)*x^3 + 210*(104*B*a*b^4 + 21* 
A*b^5)*c^2 - 8*(231*B*b^4*c^3 + 8*(64*B*a^2 + 119*A*a*b)*c^5 - 6*(162*B*a* 
b^2 + 49*A*b^3)*c^4)*x^2 + 2*(1155*B*b^5*c^2 - 3360*A*a^2*c^5 + 16*(397*B* 
a^2*b + 392*A*a*b^2)*c^4 - 42*(144*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 
+ b*x + a))/c^7, -1/215040*(105*(33*B*b^7 + 128*A*a^3*c^4 - 160*(2*B*a^3*b 
 + 3*A*a^2*b^2)*c^3 + 280*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 42*(6*B*a*b^5 + A* 
b^6)*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*(15360*B*c^7*x^6 - 3465*B*b^6*c + 1280*(B*b*c^6 
+ 14*A*c^7)*x^5 + 32*(256*B*a^3 + 791*A*a^2*b)*c^4 - 128*(11*B*b^2*c^5 - 2 
*(12*B*a + 7*A*b)*c^6)*x^4 - 336*(103*B*a^2*b^2 + 70*A*a*b^3)*c^3 + 16*(99 
*B*b^3*c^4 + 280*A*a*c^6 - 2*(158*B*a*b + 63*A*b^2)*c^5)*x^3 + 210*(104*B* 
a*b^4 + 21*A*b^5)*c^2 - 8*(231*B*b^4*c^3 + 8*(64*B*a^2 + 119*A*a*b)*c^5 - 
6*(162*B*a*b^2 + 49*A*b^3)*c^4)*x^2 + 2*(1155*B*b^5*c^2 - 3360*A*a^2*c^5 + 
 16*(397*B*a^2*b + 392*A*a*b^2)*c^4 - 42*(144*B*a*b^3 + 35*A*b^4)*c^3)*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1292 vs. \(2 (401) = 802\).

Time = 0.70 (sec) , antiderivative size = 1292, normalized size of antiderivative = 3.52 \[ \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate(x**4*(B*x+A)*(c*x**2+b*x+a)**(1/2),x)
 

Output:

Piecewise(((-a*(-3*a*(A*a - 5*a*(A*c + B*b/14)/(6*c) - 9*b*(A*b + B*a/7 - 
11*b*(A*c + B*b/14)/(12*c))/(10*c))/(4*c) - 5*b*(-4*a*(A*b + B*a/7 - 11*b* 
(A*c + B*b/14)/(12*c))/(5*c) - 7*b*(A*a - 5*a*(A*c + B*b/14)/(6*c) - 9*b*( 
A*b + B*a/7 - 11*b*(A*c + B*b/14)/(12*c))/(10*c))/(8*c))/(6*c))/(2*c) - b* 
(-2*a*(-4*a*(A*b + B*a/7 - 11*b*(A*c + B*b/14)/(12*c))/(5*c) - 7*b*(A*a - 
5*a*(A*c + B*b/14)/(6*c) - 9*b*(A*b + B*a/7 - 11*b*(A*c + B*b/14)/(12*c))/ 
(10*c))/(8*c))/(3*c) - 3*b*(-3*a*(A*a - 5*a*(A*c + B*b/14)/(6*c) - 9*b*(A* 
b + B*a/7 - 11*b*(A*c + B*b/14)/(12*c))/(10*c))/(4*c) - 5*b*(-4*a*(A*b + B 
*a/7 - 11*b*(A*c + B*b/14)/(12*c))/(5*c) - 7*b*(A*a - 5*a*(A*c + B*b/14)/( 
6*c) - 9*b*(A*b + B*a/7 - 11*b*(A*c + B*b/14)/(12*c))/(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**6/7 + x**5*(A*c 
+ B*b/14)/(6*c) + x**4*(A*b + B*a/7 - 11*b*(A*c + B*b/14)/(12*c))/(5*c) + 
x**3*(A*a - 5*a*(A*c + B*b/14)/(6*c) - 9*b*(A*b + B*a/7 - 11*b*(A*c + B*b/ 
14)/(12*c))/(10*c))/(4*c) + x**2*(-4*a*(A*b + B*a/7 - 11*b*(A*c + B*b/14)/ 
(12*c))/(5*c) - 7*b*(A*a - 5*a*(A*c + B*b/14)/(6*c) - 9*b*(A*b + B*a/7 - 1 
1*b*(A*c + B*b/14)/(12*c))/(10*c))/(8*c))/(3*c) + x*(-3*a*(A*a - 5*a*(A*c 
+ B*b/14)/(6*c) - 9*b*(A*b + B*a/7 - 11*b*(A*c + B*b/14)/(12*c))/(10*c))/( 
4*c) - 5*b*(-4*a*(A*b + B*a/7 - 11*b*(A*c + B*b/14)/(12*c))/(5*c) - 7*b...
 

Maxima [F(-2)]

Exception generated. \[ \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^4*(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 = 412, normalized size of antiderivative = 1.12 \[ \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {1}{107520} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B x + \frac {B b c^{5} + 14 \, A c^{6}}{c^{6}}\right )} x - \frac {11 \, B b^{2} c^{4} - 24 \, B a c^{5} - 14 \, A b c^{5}}{c^{6}}\right )} x + \frac {99 \, B b^{3} c^{3} - 316 \, B a b c^{4} - 126 \, A b^{2} c^{4} + 280 \, A a c^{5}}{c^{6}}\right )} x - \frac {231 \, B b^{4} c^{2} - 972 \, B a b^{2} c^{3} - 294 \, A b^{3} c^{3} + 512 \, B a^{2} c^{4} + 952 \, A a b c^{4}}{c^{6}}\right )} x + \frac {1155 \, B b^{5} c - 6048 \, B a b^{3} c^{2} - 1470 \, A b^{4} c^{2} + 6352 \, B a^{2} b c^{3} + 6272 \, A a b^{2} c^{3} - 3360 \, A a^{2} c^{4}}{c^{6}}\right )} x - \frac {3465 \, B b^{6} - 21840 \, B a b^{4} c - 4410 \, A b^{5} c + 34608 \, B a^{2} b^{2} c^{2} + 23520 \, A a b^{3} c^{2} - 8192 \, B a^{3} c^{3} - 25312 \, A a^{2} b c^{3}}{c^{6}}\right )} - \frac {{\left (33 \, B b^{7} - 252 \, B a b^{5} c - 42 \, A b^{6} c + 560 \, B a^{2} b^{3} c^{2} + 280 \, A a b^{4} c^{2} - 320 \, B a^{3} b c^{3} - 480 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {13}{2}}} \] Input:

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*B*x + (B*b*c^5 + 14*A*c 
^6)/c^6)*x - (11*B*b^2*c^4 - 24*B*a*c^5 - 14*A*b*c^5)/c^6)*x + (99*B*b^3*c 
^3 - 316*B*a*b*c^4 - 126*A*b^2*c^4 + 280*A*a*c^5)/c^6)*x - (231*B*b^4*c^2 
- 972*B*a*b^2*c^3 - 294*A*b^3*c^3 + 512*B*a^2*c^4 + 952*A*a*b*c^4)/c^6)*x 
+ (1155*B*b^5*c - 6048*B*a*b^3*c^2 - 1470*A*b^4*c^2 + 6352*B*a^2*b*c^3 + 6 
272*A*a*b^2*c^3 - 3360*A*a^2*c^4)/c^6)*x - (3465*B*b^6 - 21840*B*a*b^4*c - 
 4410*A*b^5*c + 34608*B*a^2*b^2*c^2 + 23520*A*a*b^3*c^2 - 8192*B*a^3*c^3 - 
 25312*A*a^2*b*c^3)/c^6) - 1/2048*(33*B*b^7 - 252*B*a*b^5*c - 42*A*b^6*c + 
 560*B*a^2*b^3*c^2 + 280*A*a*b^4*c^2 - 320*B*a^3*b*c^3 - 480*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^(13/2)
 

Mupad [B] (verification not implemented)

Time = 13.89 (sec) , antiderivative size = 992, normalized size of antiderivative = 2.70 \[ \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

int(x^4*(A + B*x)*(a + b*x + c*x^2)^(1/2),x)
 

Output:

(8*B*a^3*(a + b*x + c*x^2)^(1/2))/(105*c^3) - (33*B*b^6*(a + b*x + c*x^2)^ 
(1/2))/(1024*c^6) + (A*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) + (B*x^4*(a + b* 
x + c*x^2)^(3/2))/(7*c) + (33*B*b^7*log(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1 
/2) + 2*c*x))/(2048*c^(13/2)) + (A*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) + (lo 
g((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*A*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*x + c*x^2)^(1/2))/(24*c^2)))/(5*c) + (x^2*(a + b*x + c*x^2)^(3/2))/(5 
*c)))/(4*c) - (5*B*a^3*b*log(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1/2) + 2*c*x 
))/(32*c^(7/2)) - (63*B*a*b^5*log(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1/2) + 
2*c*x))/(512*c^(11/2)) + (35*B*a^2*b^3*log(b + 2*c^(1/2)*(a + b*x + c*x^2) 
^(1/2) + 2*c*x))/(128*c^(9/2)) + (13*B*a*b^4*(a + b*x + c*x^2)^(1/2))/(...
 

Reduce [F]

\[ \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx=\int x^{4} \left (B x +A \right ) \sqrt {c \,x^{2}+b x +a}d x \] Input:

int(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x)
 

Output:

int(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x)