\(\int x^2 (A+B x) (a+b x+c x^2)^{5/2} \, dx\) [131]

Optimal result

Integrand size = 23, antiderivative size = 333 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {5 \left (b^2-4 a c\right )^2 \left (11 b^3 B-18 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}+\frac {5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}-\frac {\left (11 b^3 B-18 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {5 \left (b^2-4 a c\right )^3 \left (11 b^3 B-18 A b^2 c-12 a b B c+8 a A c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}} \] Output:

-5/32768*(-4*a*c+b^2)^2*(8*A*a*c^2-18*A*b^2*c-12*B*a*b*c+11*B*b^3)*(2*c*x+ 
b)*(c*x^2+b*x+a)^(1/2)/c^6+5/12288*(-4*a*c+b^2)*(8*A*a*c^2-18*A*b^2*c-12*B 
*a*b*c+11*B*b^3)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^5-1/768*(8*A*a*c^2-18*A*b 
^2*c-12*B*a*b*c+11*B*b^3)*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^4+1/9*B*x^2*(c*x 
^2+b*x+a)^(7/2)/c+1/2016*(99*B*b^2-162*A*b*c-64*B*a*c-14*c*(-18*A*c+11*B*b 
)*x)*(c*x^2+b*x+a)^(7/2)/c^3+5/65536*(-4*a*c+b^2)^3*(8*A*a*c^2-18*A*b^2*c- 
12*B*a*b*c+11*B*b^3)*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 = 4.06 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.45 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^8 B+210 b^7 c (27 A+11 B x)+84 b^6 c (485 a B-c x (45 A+22 B x))+72 b^5 c^2 \left (2 c x^2 (21 A+11 B x)-7 a (125 A+49 B x)\right )+128 b c^4 \left (6 a^2 c x^2 (87 A+41 B x)-13 a^3 (153 A+53 B x)+24 a c^2 x^4 (307 A+251 B x)+16 c^3 x^6 (297 A+259 B x)\right )+192 b^2 c^3 \left (1221 a^3 B+4 a c^2 x^3 (27 A+14 B x)+8 c^3 x^5 (243 A+206 B x)-a^2 c x (597 A+266 B x)\right )+256 c^4 \left (-256 a^4 B+112 c^4 x^7 (9 A+8 B x)+a^3 c x (315 A+128 B x)+8 a c^3 x^5 (357 A+304 B x)+6 a^2 c^2 x^3 (413 A+320 B x)\right )-16 b^4 c^2 \left (10143 a^2 B+2 c^2 x^3 (81 A+44 B x)-3 a c x (791 A+372 B x)\right )+32 b^3 c^3 \left (8 c^2 x^4 (9 A+5 B x)-4 a c x^2 (213 A+107 B x)+3 a^2 (2359 A+879 B x)\right )\right )-315 \left (b^2-4 a c\right )^3 \left (11 b^3 B-18 A b^2 c-12 a b B c+8 a A c^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{4128768 c^{13/2}} \] Input:

Integrate[x^2*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^8*B + 210*b^7*c*(27*A + 11*B*x) 
+ 84*b^6*c*(485*a*B - c*x*(45*A + 22*B*x)) + 72*b^5*c^2*(2*c*x^2*(21*A + 1 
1*B*x) - 7*a*(125*A + 49*B*x)) + 128*b*c^4*(6*a^2*c*x^2*(87*A + 41*B*x) - 
13*a^3*(153*A + 53*B*x) + 24*a*c^2*x^4*(307*A + 251*B*x) + 16*c^3*x^6*(297 
*A + 259*B*x)) + 192*b^2*c^3*(1221*a^3*B + 4*a*c^2*x^3*(27*A + 14*B*x) + 8 
*c^3*x^5*(243*A + 206*B*x) - a^2*c*x*(597*A + 266*B*x)) + 256*c^4*(-256*a^ 
4*B + 112*c^4*x^7*(9*A + 8*B*x) + a^3*c*x*(315*A + 128*B*x) + 8*a*c^3*x^5* 
(357*A + 304*B*x) + 6*a^2*c^2*x^3*(413*A + 320*B*x)) - 16*b^4*c^2*(10143*a 
^2*B + 2*c^2*x^3*(81*A + 44*B*x) - 3*a*c*x*(791*A + 372*B*x)) + 32*b^3*c^3 
*(8*c^2*x^4*(9*A + 5*B*x) - 4*a*c*x^2*(213*A + 107*B*x) + 3*a^2*(2359*A + 
879*B*x))) - 315*(b^2 - 4*a*c)^3*(11*b^3*B - 18*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)]])/(4128768*c^(13/2 
))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1236, 27, 1225, 1087, 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.

\(\displaystyle \int x^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int -\frac {1}{2} x (4 a B+(11 b B-18 A c) x) \left (c x^2+b x+a\right )^{5/2}dx}{9 c}+\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}-\frac {\int x (4 a B+(11 b B-18 A c) x) \left (c x^2+b x+a\right )^{5/2}dx}{18 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}-\frac {\frac {9 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \int \left (c x^2+b x+a\right )^{5/2}dx}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{112 c^2}}{18 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}-\frac {\frac {9 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{112 c^2}}{18 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}-\frac {\frac {9 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{112 c^2}}{18 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}-\frac {\frac {9 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{112 c^2}}{18 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}-\frac {\frac {9 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{112 c^2}}{18 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}-\frac {\frac {9 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{112 c^2}}{18 c}\)

Input:

Int[x^2*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(B*x^2*(a + b*x + c*x^2)^(7/2))/(9*c) - (-1/112*((99*b^2*B - 162*A*b*c - 6 
4*a*B*c - 14*c*(11*b*B - 18*A*c)*x)*(a + b*x + c*x^2)^(7/2))/c^2 + (9*(11* 
b^3*B - 18*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(((b + 2*c*x)*(a + b*x + c*x^ 
2)^(5/2))/(12*c) - (5*(b^2 - 4*a*c)*(((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)))/(32*c^2))/(18*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(651\) vs. \(2(303)=606\).

Time = 1.29 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.96

method	result	size
risch	\(-\frac {\left (-229376 B \,c^{8} x^{8}-258048 A \,c^{8} x^{7}-530432 B b \,c^{7} x^{7}-608256 A b \,c^{7} x^{6}-622592 B a \,c^{7} x^{6}-316416 B \,b^{2} c^{6} x^{6}-731136 A a \,c^{7} x^{5}-373248 A \,b^{2} c^{6} x^{5}-771072 B a b \,c^{6} x^{5}-1280 B \,b^{3} c^{5} x^{5}-943104 A a b \,c^{6} x^{4}-2304 A \,b^{3} c^{5} x^{4}-491520 B \,a^{2} c^{6} x^{4}-10752 B a \,b^{2} c^{5} x^{4}+1408 B \,b^{4} c^{4} x^{4}-634368 A \,a^{2} c^{6} x^{3}-20736 A a \,b^{2} c^{5} x^{3}+2592 A \,b^{4} c^{4} x^{3}-31488 B \,a^{2} b \,c^{5} x^{3}+13696 B a \,b^{3} c^{4} x^{3}-1584 B \,b^{5} c^{3} x^{3}-66816 A \,a^{2} b \,c^{5} x^{2}+27264 A a \,b^{3} c^{4} x^{2}-3024 A \,b^{5} c^{3} x^{2}-32768 B \,a^{3} c^{5} x^{2}+51072 B \,a^{2} b^{2} c^{4} x^{2}-17856 B a \,b^{4} c^{3} x^{2}+1848 B \,b^{6} c^{2} x^{2}-80640 A \,a^{3} c^{5} x +114624 A \,a^{2} b^{2} c^{4} x -37968 A a \,b^{4} c^{3} x +3780 A \,b^{6} c^{2} x +88192 B \,a^{3} b \,c^{4} x -84384 B \,a^{2} b^{3} c^{3} x +24696 B a \,b^{5} c^{2} x -2310 B \,b^{7} c x +254592 A \,a^{3} b \,c^{4}-226464 A \,a^{2} b^{3} c^{3}+63000 A a \,b^{5} c^{2}-5670 A \,b^{7} c +65536 B \,a^{4} c^{4}-234432 B \,a^{3} b^{2} c^{3}+162288 B \,a^{2} b^{4} c^{2}-40740 B a \,b^{6} c +3465 B \,b^{8}\right ) \sqrt {c \,x^{2}+b x +a}}{2064384 c^{6}}-\frac {5 \left (512 A \,a^{4} c^{5}-1536 A \,a^{3} b^{2} c^{4}+960 A \,a^{2} b^{4} c^{3}-224 A a \,b^{6} c^{2}+18 A \,b^{8} c -768 B \,a^{4} b \,c^{4}+1280 B \,a^{3} b^{3} c^{3}-672 B \,a^{2} b^{5} c^{2}+144 B a \,b^{7} c -11 B \,b^{9}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{65536 c^{\frac {13}{2}}}\)	\(652\)
default	\(A \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \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 )}{24 c}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \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 )}{24 c}\right )}{8 c}\right )+B \left (\frac {x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{9 c}-\frac {11 b \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \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 )}{24 c}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \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 )}{24 c}\right )}{8 c}\right )}{18 c}-\frac {2 a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \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 )}{24 c}\right )}{2 c}\right )}{9 c}\right )\)	\(883\)

Input:

int(x^2*(B*x+A)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2064384/c^6*(-229376*B*c^8*x^8-258048*A*c^8*x^7-530432*B*b*c^7*x^7-6082 
56*A*b*c^7*x^6-622592*B*a*c^7*x^6-316416*B*b^2*c^6*x^6-731136*A*a*c^7*x^5- 
373248*A*b^2*c^6*x^5-771072*B*a*b*c^6*x^5-1280*B*b^3*c^5*x^5-943104*A*a*b* 
c^6*x^4-2304*A*b^3*c^5*x^4-491520*B*a^2*c^6*x^4-10752*B*a*b^2*c^5*x^4+1408 
*B*b^4*c^4*x^4-634368*A*a^2*c^6*x^3-20736*A*a*b^2*c^5*x^3+2592*A*b^4*c^4*x 
^3-31488*B*a^2*b*c^5*x^3+13696*B*a*b^3*c^4*x^3-1584*B*b^5*c^3*x^3-66816*A* 
a^2*b*c^5*x^2+27264*A*a*b^3*c^4*x^2-3024*A*b^5*c^3*x^2-32768*B*a^3*c^5*x^2 
+51072*B*a^2*b^2*c^4*x^2-17856*B*a*b^4*c^3*x^2+1848*B*b^6*c^2*x^2-80640*A* 
a^3*c^5*x+114624*A*a^2*b^2*c^4*x-37968*A*a*b^4*c^3*x+3780*A*b^6*c^2*x+8819 
2*B*a^3*b*c^4*x-84384*B*a^2*b^3*c^3*x+24696*B*a*b^5*c^2*x-2310*B*b^7*c*x+2 
54592*A*a^3*b*c^4-226464*A*a^2*b^3*c^3+63000*A*a*b^5*c^2-5670*A*b^7*c+6553 
6*B*a^4*c^4-234432*B*a^3*b^2*c^3+162288*B*a^2*b^4*c^2-40740*B*a*b^6*c+3465 
*B*b^8)*(c*x^2+b*x+a)^(1/2)-5/65536*(512*A*a^4*c^5-1536*A*a^3*b^2*c^4+960* 
A*a^2*b^4*c^3-224*A*a*b^6*c^2+18*A*b^8*c-768*B*a^4*b*c^4+1280*B*a^3*b^3*c^ 
3-672*B*a^2*b^5*c^2+144*B*a*b^7*c-11*B*b^9)/c^(13/2)*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. 630 vs. \(2 (303) = 606\).

Time = 0.22 (sec) , antiderivative size = 1263, normalized size of antiderivative = 3.79 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate(x^2*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[-1/8257536*(315*(11*B*b^9 - 512*A*a^4*c^5 + 768*(B*a^4*b + 2*A*a^3*b^2)*c 
^4 - 320*(4*B*a^3*b^3 + 3*A*a^2*b^4)*c^3 + 224*(3*B*a^2*b^5 + A*a*b^6)*c^2 
 - 18*(8*B*a*b^7 + A*b^8)*c)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sq 
rt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(229376*B*c^9*x^8 - 3 
465*B*b^8*c + 14336*(37*B*b*c^8 + 18*A*c^9)*x^7 + 1024*(309*B*b^2*c^7 + 2* 
(304*B*a + 297*A*b)*c^8)*x^6 - 128*(512*B*a^4 + 1989*A*a^3*b)*c^5 + 256*(5 
*B*b^3*c^6 + 2856*A*a*c^8 + 6*(502*B*a*b + 243*A*b^2)*c^7)*x^5 + 96*(2442* 
B*a^3*b^2 + 2359*A*a^2*b^3)*c^4 - 128*(11*B*b^4*c^5 - 24*(160*B*a^2 + 307* 
A*a*b)*c^7 - 6*(14*B*a*b^2 + 3*A*b^3)*c^6)*x^4 - 504*(322*B*a^2*b^4 + 125* 
A*a*b^5)*c^3 + 16*(99*B*b^5*c^4 + 39648*A*a^2*c^7 + 48*(41*B*a^2*b + 27*A* 
a*b^2)*c^6 - 2*(428*B*a*b^3 + 81*A*b^4)*c^5)*x^3 + 210*(194*B*a*b^6 + 27*A 
*b^7)*c^2 - 8*(231*B*b^6*c^3 - 32*(128*B*a^3 + 261*A*a^2*b)*c^6 + 48*(133* 
B*a^2*b^2 + 71*A*a*b^3)*c^5 - 18*(124*B*a*b^4 + 21*A*b^5)*c^4)*x^2 + 2*(11 
55*B*b^7*c^2 + 40320*A*a^3*c^6 - 32*(1378*B*a^3*b + 1791*A*a^2*b^2)*c^5 + 
24*(1758*B*a^2*b^3 + 791*A*a*b^4)*c^4 - 126*(98*B*a*b^5 + 15*A*b^6)*c^3)*x 
)*sqrt(c*x^2 + b*x + a))/c^7, -1/4128768*(315*(11*B*b^9 - 512*A*a^4*c^5 + 
768*(B*a^4*b + 2*A*a^3*b^2)*c^4 - 320*(4*B*a^3*b^3 + 3*A*a^2*b^4)*c^3 + 22 
4*(3*B*a^2*b^5 + A*a*b^6)*c^2 - 18*(8*B*a*b^7 + A*b^8)*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*(229376*B*c^9*x^8 - 3465*B*b^8*c + 14336*(37*B*b*c^8 + 18*A*c^9)*x^7 ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6552 vs. \(2 (355) = 710\).

Time = 0.94 (sec) , antiderivative size = 6552, normalized size of antiderivative = 19.68 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate(x**2*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)
 

Output:

Piecewise(((-a*(A*a**3 - 3*a*(3*A*a**2*c + 3*A*a*b**2 + 3*B*a**2*b - 5*a*( 
3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 7*a*(A*c**3 + 37*B*b*c**2/1 
8)/(8*c) - 13*b*(3*A*b*c**2 + 19*B*a*c**2/9 + 3*B*b**2*c - 15*b*(A*c**3 + 
37*B*b*c**2/18)/(16*c))/(14*c))/(6*c) - 9*b*(6*A*a*b*c + A*b**3 + 3*B*a**2 
*c + 3*B*a*b**2 - 6*a*(3*A*b*c**2 + 19*B*a*c**2/9 + 3*B*b**2*c - 15*b*(A*c 
**3 + 37*B*b*c**2/18)/(16*c))/(7*c) - 11*b*(3*A*a*c**2 + 3*A*b**2*c + 6*B* 
a*b*c + B*b**3 - 7*a*(A*c**3 + 37*B*b*c**2/18)/(8*c) - 13*b*(3*A*b*c**2 + 
19*B*a*c**2/9 + 3*B*b**2*c - 15*b*(A*c**3 + 37*B*b*c**2/18)/(16*c))/(14*c) 
)/(12*c))/(10*c))/(4*c) - 5*b*(3*A*a**2*b + B*a**3 - 4*a*(6*A*a*b*c + A*b* 
*3 + 3*B*a**2*c + 3*B*a*b**2 - 6*a*(3*A*b*c**2 + 19*B*a*c**2/9 + 3*B*b**2* 
c - 15*b*(A*c**3 + 37*B*b*c**2/18)/(16*c))/(7*c) - 11*b*(3*A*a*c**2 + 3*A* 
b**2*c + 6*B*a*b*c + B*b**3 - 7*a*(A*c**3 + 37*B*b*c**2/18)/(8*c) - 13*b*( 
3*A*b*c**2 + 19*B*a*c**2/9 + 3*B*b**2*c - 15*b*(A*c**3 + 37*B*b*c**2/18)/( 
16*c))/(14*c))/(12*c))/(5*c) - 7*b*(3*A*a**2*c + 3*A*a*b**2 + 3*B*a**2*b - 
 5*a*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 7*a*(A*c**3 + 37*B*b* 
c**2/18)/(8*c) - 13*b*(3*A*b*c**2 + 19*B*a*c**2/9 + 3*B*b**2*c - 15*b*(A*c 
**3 + 37*B*b*c**2/18)/(16*c))/(14*c))/(6*c) - 9*b*(6*A*a*b*c + A*b**3 + 3* 
B*a**2*c + 3*B*a*b**2 - 6*a*(3*A*b*c**2 + 19*B*a*c**2/9 + 3*B*b**2*c - 15* 
b*(A*c**3 + 37*B*b*c**2/18)/(16*c))/(7*c) - 11*b*(3*A*a*c**2 + 3*A*b**2*c 
+ 6*B*a*b*c + B*b**3 - 7*a*(A*c**3 + 37*B*b*c**2/18)/(8*c) - 13*b*(3*A*...
 

Maxima [F(-2)]

Exception generated. \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(B*x+A)*(c*x^2+b*x+a)^(5/2),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. 641 vs. \(2 (303) = 606\).

Time = 0.27 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.92 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {1}{2064384} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, B c^{2} x + \frac {37 \, B b c^{9} + 18 \, A c^{10}}{c^{8}}\right )} x + \frac {309 \, B b^{2} c^{8} + 608 \, B a c^{9} + 594 \, A b c^{9}}{c^{8}}\right )} x + \frac {5 \, B b^{3} c^{7} + 3012 \, B a b c^{8} + 1458 \, A b^{2} c^{8} + 2856 \, A a c^{9}}{c^{8}}\right )} x - \frac {11 \, B b^{4} c^{6} - 84 \, B a b^{2} c^{7} - 18 \, A b^{3} c^{7} - 3840 \, B a^{2} c^{8} - 7368 \, A a b c^{8}}{c^{8}}\right )} x + \frac {99 \, B b^{5} c^{5} - 856 \, B a b^{3} c^{6} - 162 \, A b^{4} c^{6} + 1968 \, B a^{2} b c^{7} + 1296 \, A a b^{2} c^{7} + 39648 \, A a^{2} c^{8}}{c^{8}}\right )} x - \frac {231 \, B b^{6} c^{4} - 2232 \, B a b^{4} c^{5} - 378 \, A b^{5} c^{5} + 6384 \, B a^{2} b^{2} c^{6} + 3408 \, A a b^{3} c^{6} - 4096 \, B a^{3} c^{7} - 8352 \, A a^{2} b c^{7}}{c^{8}}\right )} x + \frac {1155 \, B b^{7} c^{3} - 12348 \, B a b^{5} c^{4} - 1890 \, A b^{6} c^{4} + 42192 \, B a^{2} b^{3} c^{5} + 18984 \, A a b^{4} c^{5} - 44096 \, B a^{3} b c^{6} - 57312 \, A a^{2} b^{2} c^{6} + 40320 \, A a^{3} c^{7}}{c^{8}}\right )} x - \frac {3465 \, B b^{8} c^{2} - 40740 \, B a b^{6} c^{3} - 5670 \, A b^{7} c^{3} + 162288 \, B a^{2} b^{4} c^{4} + 63000 \, A a b^{5} c^{4} - 234432 \, B a^{3} b^{2} c^{5} - 226464 \, A a^{2} b^{3} c^{5} + 65536 \, B a^{4} c^{6} + 254592 \, A a^{3} b c^{6}}{c^{8}}\right )} - \frac {5 \, {\left (11 \, B b^{9} - 144 \, B a b^{7} c - 18 \, A b^{8} c + 672 \, B a^{2} b^{5} c^{2} + 224 \, A a b^{6} c^{2} - 1280 \, B a^{3} b^{3} c^{3} - 960 \, A a^{2} b^{4} c^{3} + 768 \, B a^{4} b c^{4} + 1536 \, A a^{3} b^{2} c^{4} - 512 \, A a^{4} c^{5}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{65536 \, c^{\frac {13}{2}}} \] Input:

integrate(x^2*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

1/2064384*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*B*c^2*x + (37*B* 
b*c^9 + 18*A*c^10)/c^8)*x + (309*B*b^2*c^8 + 608*B*a*c^9 + 594*A*b*c^9)/c^ 
8)*x + (5*B*b^3*c^7 + 3012*B*a*b*c^8 + 1458*A*b^2*c^8 + 2856*A*a*c^9)/c^8) 
*x - (11*B*b^4*c^6 - 84*B*a*b^2*c^7 - 18*A*b^3*c^7 - 3840*B*a^2*c^8 - 7368 
*A*a*b*c^8)/c^8)*x + (99*B*b^5*c^5 - 856*B*a*b^3*c^6 - 162*A*b^4*c^6 + 196 
8*B*a^2*b*c^7 + 1296*A*a*b^2*c^7 + 39648*A*a^2*c^8)/c^8)*x - (231*B*b^6*c^ 
4 - 2232*B*a*b^4*c^5 - 378*A*b^5*c^5 + 6384*B*a^2*b^2*c^6 + 3408*A*a*b^3*c 
^6 - 4096*B*a^3*c^7 - 8352*A*a^2*b*c^7)/c^8)*x + (1155*B*b^7*c^3 - 12348*B 
*a*b^5*c^4 - 1890*A*b^6*c^4 + 42192*B*a^2*b^3*c^5 + 18984*A*a*b^4*c^5 - 44 
096*B*a^3*b*c^6 - 57312*A*a^2*b^2*c^6 + 40320*A*a^3*c^7)/c^8)*x - (3465*B* 
b^8*c^2 - 40740*B*a*b^6*c^3 - 5670*A*b^7*c^3 + 162288*B*a^2*b^4*c^4 + 6300 
0*A*a*b^5*c^4 - 234432*B*a^3*b^2*c^5 - 226464*A*a^2*b^3*c^5 + 65536*B*a^4* 
c^6 + 254592*A*a^3*b*c^6)/c^8) - 5/65536*(11*B*b^9 - 144*B*a*b^7*c - 18*A* 
b^8*c + 672*B*a^2*b^5*c^2 + 224*A*a*b^6*c^2 - 1280*B*a^3*b^3*c^3 - 960*A*a 
^2*b^4*c^3 + 768*B*a^4*b*c^4 + 1536*A*a^3*b^2*c^4 - 512*A*a^4*c^5)*log(abs 
(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(13/2)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int x^2\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \] Input:

int(x^2*(A + B*x)*(a + b*x + c*x^2)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int(x^2*(A + B*x)*(a + b*x + c*x^2)^(5/2), x)
 

Reduce [F]

\[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int x^{2} \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}d x \] Input:

int(x^2*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)
 

Output:

int(x^2*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)