\(\int x^2 (A+B x) (b x+c x^2)^{5/2} \, dx\) [127]

Optimal result

Integrand size = 22, antiderivative size = 332 \[ \int x^2 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=-\frac {5 b^7 (11 b B-18 A c) \sqrt {b x+c x^2}}{32768 c^6}+\frac {5 b^6 (11 b B-18 A c) x \sqrt {b x+c x^2}}{49152 c^5}-\frac {b^5 (11 b B-18 A c) x^2 \sqrt {b x+c x^2}}{12288 c^4}+\frac {b^4 (11 b B-18 A c) x^3 \sqrt {b x+c x^2}}{14336 c^3}-\frac {b^3 (11 b B-18 A c) x^4 \sqrt {b x+c x^2}}{16128 c^2}-\frac {9 b^2 (11 b B-18 A c) x^5 \sqrt {b x+c x^2}}{896 c}-\frac {11}{672} b (11 b B-18 A c) x^6 \sqrt {b x+c x^2}-\frac {1}{144} c (11 b B-18 A c) x^7 \sqrt {b x+c x^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {5 b^8 (11 b B-18 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{32768 c^{13/2}} \] Output:

-5/32768*b^7*(-18*A*c+11*B*b)*(c*x^2+b*x)^(1/2)/c^6+5/49152*b^6*(-18*A*c+1 
1*B*b)*x*(c*x^2+b*x)^(1/2)/c^5-1/12288*b^5*(-18*A*c+11*B*b)*x^2*(c*x^2+b*x 
)^(1/2)/c^4+1/14336*b^4*(-18*A*c+11*B*b)*x^3*(c*x^2+b*x)^(1/2)/c^3-1/16128 
*b^3*(-18*A*c+11*B*b)*x^4*(c*x^2+b*x)^(1/2)/c^2-9/896*b^2*(-18*A*c+11*B*b) 
*x^5*(c*x^2+b*x)^(1/2)/c-11/672*b*(-18*A*c+11*B*b)*x^6*(c*x^2+b*x)^(1/2)-1 
/144*c*(-18*A*c+11*B*b)*x^7*(c*x^2+b*x)^(1/2)+1/9*B*x^2*(c*x^2+b*x)^(7/2)/ 
c+5/32768*b^8*(-18*A*c+11*B*b)*arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/c^(13/ 
2)
 

Mathematica [A] (verified)

Time = 1.98 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.83 \[ \int x^2 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (-3465 b^8 B+256 b^3 c^5 x^4 (9 A+5 B x)+28672 c^8 x^7 (9 A+8 B x)+144 b^5 c^3 x^2 (21 A+11 B x)+210 b^7 c (27 A+11 B x)-84 b^6 c^2 x (45 A+22 B x)-32 b^4 c^4 x^3 (81 A+44 B x)+1536 b^2 c^6 x^5 (243 A+206 B x)+2048 b c^7 x^6 (297 A+259 B x)\right )+11340 A b^8 c \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+6930 b^9 B \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{2064384 c^{13/2} \sqrt {x (b+c x)}} \] Input:

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

Output:

(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(-3465*b^8*B + 256*b 
^3*c^5*x^4*(9*A + 5*B*x) + 28672*c^8*x^7*(9*A + 8*B*x) + 144*b^5*c^3*x^2*( 
21*A + 11*B*x) + 210*b^7*c*(27*A + 11*B*x) - 84*b^6*c^2*x*(45*A + 22*B*x) 
- 32*b^4*c^4*x^3*(81*A + 44*B*x) + 1536*b^2*c^6*x^5*(243*A + 206*B*x) + 20 
48*b*c^7*x^6*(297*A + 259*B*x)) + 11340*A*b^8*c*ArcTanh[(Sqrt[c]*Sqrt[x])/ 
(Sqrt[b] - Sqrt[b + c*x])] + 6930*b^9*B*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b 
] + Sqrt[b + c*x])]))/(2064384*c^(13/2)*Sqrt[x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.70, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1221, 1134, 1160, 1087, 1087, 1087, 1091, 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.

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

\(\Big \downarrow \) 1221

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

\(\Big \downarrow \) 1134

\(\displaystyle \frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {(11 b B-18 A c) \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \int x \left (c x^2+b x\right )^{5/2}dx}{16 c}\right )}{18 c}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {(11 b B-18 A c) \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \left (\frac {\left (b x+c x^2\right )^{7/2}}{7 c}-\frac {b \int \left (c x^2+b x\right )^{5/2}dx}{2 c}\right )}{16 c}\right )}{18 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {(11 b B-18 A c) \left (\frac {x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {9 b \left (\frac {\left (b x+c x^2\right )^{7/2}}{7 c}-\frac {b \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac {5 b^2 \int \left (c x^2+b x\right )^{3/2}dx}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\)

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1091

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(B*x^2*(b*x + c*x^2)^(7/2))/(9*c) - ((11*b*B - 18*A*c)*((x*(b*x + c*x^2)^( 
7/2))/(8*c) - (9*b*((b*x + c*x^2)^(7/2)/(7*c) - (b*(((b + 2*c*x)*(b*x + c* 
x^2)^(5/2))/(12*c) - (5*b^2*(((b + 2*c*x)*(b*x + c*x^2)^(3/2))/(8*c) - (3* 
b^2*(((b + 2*c*x)*Sqrt[b*x + c*x^2])/(4*c) - (b^2*ArcTanh[(Sqrt[c]*x)/Sqrt 
[b*x + c*x^2]])/(4*c^(3/2))))/(16*c)))/(24*c)))/(2*c)))/(16*c)))/(18*c)
 

Defintions of rubi rules used

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 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1134

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1)))   Int[(d + e*x)^ 
(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ 
c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 
*p]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1221

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[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c 
*f - b*g))/(c*e*(m + 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && NeQ[m + 2*p + 2, 0]
 

Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.73

method	result	size
risch	\(\frac {\left (229376 B \,c^{8} x^{8}+258048 A \,c^{8} x^{7}+530432 B b \,c^{7} x^{7}+608256 A b \,c^{7} x^{6}+316416 B \,b^{2} c^{6} x^{6}+373248 A \,b^{2} c^{6} x^{5}+1280 B \,b^{3} c^{5} x^{5}+2304 A \,b^{3} c^{5} x^{4}-1408 B \,b^{4} c^{4} x^{4}-2592 A \,b^{4} c^{4} x^{3}+1584 B \,b^{5} c^{3} x^{3}+3024 A \,b^{5} c^{3} x^{2}-1848 B \,b^{6} c^{2} x^{2}-3780 A \,b^{6} c^{2} x +2310 B \,b^{7} c x +5670 A \,b^{7} c -3465 B \,b^{8}\right ) x \left (c x +b \right )}{2064384 c^{6} \sqrt {x \left (c x +b \right )}}-\frac {5 b^{8} \left (18 A c -11 B b \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{65536 c^{\frac {13}{2}}}\)	\(241\)
default	\(A \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )+B \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 c}-\frac {11 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )\)	\(360\)

Input:

int(x^2*(B*x+A)*(c*x^2+b*x)^(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+608256 
*A*b*c^7*x^6+316416*B*b^2*c^6*x^6+373248*A*b^2*c^6*x^5+1280*B*b^3*c^5*x^5+ 
2304*A*b^3*c^5*x^4-1408*B*b^4*c^4*x^4-2592*A*b^4*c^4*x^3+1584*B*b^5*c^3*x^ 
3+3024*A*b^5*c^3*x^2-1848*B*b^6*c^2*x^2-3780*A*b^6*c^2*x+2310*B*b^7*c*x+56 
70*A*b^7*c-3465*B*b^8)*x*(c*x+b)/(x*(c*x+b))^(1/2)-5/65536*b^8*(18*A*c-11* 
B*b)/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.50 \[ \int x^2 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\left [-\frac {315 \, {\left (11 \, B b^{9} - 18 \, A b^{8} c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (229376 \, B c^{9} x^{8} - 3465 \, B b^{8} c + 5670 \, A b^{7} c^{2} + 14336 \, {\left (37 \, B b c^{8} + 18 \, A c^{9}\right )} x^{7} + 3072 \, {\left (103 \, B b^{2} c^{7} + 198 \, A b c^{8}\right )} x^{6} + 256 \, {\left (5 \, B b^{3} c^{6} + 1458 \, A b^{2} c^{7}\right )} x^{5} - 128 \, {\left (11 \, B b^{4} c^{5} - 18 \, A b^{3} c^{6}\right )} x^{4} + 144 \, {\left (11 \, B b^{5} c^{4} - 18 \, A b^{4} c^{5}\right )} x^{3} - 168 \, {\left (11 \, B b^{6} c^{3} - 18 \, A b^{5} c^{4}\right )} x^{2} + 210 \, {\left (11 \, B b^{7} c^{2} - 18 \, A b^{6} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{4128768 \, c^{7}}, -\frac {315 \, {\left (11 \, B b^{9} - 18 \, A b^{8} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right ) - {\left (229376 \, B c^{9} x^{8} - 3465 \, B b^{8} c + 5670 \, A b^{7} c^{2} + 14336 \, {\left (37 \, B b c^{8} + 18 \, A c^{9}\right )} x^{7} + 3072 \, {\left (103 \, B b^{2} c^{7} + 198 \, A b c^{8}\right )} x^{6} + 256 \, {\left (5 \, B b^{3} c^{6} + 1458 \, A b^{2} c^{7}\right )} x^{5} - 128 \, {\left (11 \, B b^{4} c^{5} - 18 \, A b^{3} c^{6}\right )} x^{4} + 144 \, {\left (11 \, B b^{5} c^{4} - 18 \, A b^{4} c^{5}\right )} x^{3} - 168 \, {\left (11 \, B b^{6} c^{3} - 18 \, A b^{5} c^{4}\right )} x^{2} + 210 \, {\left (11 \, B b^{7} c^{2} - 18 \, A b^{6} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{2064384 \, c^{7}}\right ] \] Input:

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

Output:

[-1/4128768*(315*(11*B*b^9 - 18*A*b^8*c)*sqrt(c)*log(2*c*x + b - 2*sqrt(c* 
x^2 + b*x)*sqrt(c)) - 2*(229376*B*c^9*x^8 - 3465*B*b^8*c + 5670*A*b^7*c^2 
+ 14336*(37*B*b*c^8 + 18*A*c^9)*x^7 + 3072*(103*B*b^2*c^7 + 198*A*b*c^8)*x 
^6 + 256*(5*B*b^3*c^6 + 1458*A*b^2*c^7)*x^5 - 128*(11*B*b^4*c^5 - 18*A*b^3 
*c^6)*x^4 + 144*(11*B*b^5*c^4 - 18*A*b^4*c^5)*x^3 - 168*(11*B*b^6*c^3 - 18 
*A*b^5*c^4)*x^2 + 210*(11*B*b^7*c^2 - 18*A*b^6*c^3)*x)*sqrt(c*x^2 + b*x))/ 
c^7, -1/2064384*(315*(11*B*b^9 - 18*A*b^8*c)*sqrt(-c)*arctan(sqrt(c*x^2 + 
b*x)*sqrt(-c)/(c*x + b)) - (229376*B*c^9*x^8 - 3465*B*b^8*c + 5670*A*b^7*c 
^2 + 14336*(37*B*b*c^8 + 18*A*c^9)*x^7 + 3072*(103*B*b^2*c^7 + 198*A*b*c^8 
)*x^6 + 256*(5*B*b^3*c^6 + 1458*A*b^2*c^7)*x^5 - 128*(11*B*b^4*c^5 - 18*A* 
b^3*c^6)*x^4 + 144*(11*B*b^5*c^4 - 18*A*b^4*c^5)*x^3 - 168*(11*B*b^6*c^3 - 
 18*A*b^5*c^4)*x^2 + 210*(11*B*b^7*c^2 - 18*A*b^6*c^3)*x)*sqrt(c*x^2 + b*x 
))/c^7]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (323) = 646\).

Time = 0.58 (sec) , antiderivative size = 741, normalized size of antiderivative = 2.23 \[ \int x^2 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((-63*b**5*(A*b**3 - 11*b*(3*A*b**2*c + B*b**3 - 13*b*(3*A*b*c**2 
 + 3*B*b**2*c - 15*b*(A*c**3 + 37*B*b*c**2/18)/(16*c))/(14*c))/(12*c))*Pie 
cewise((log(b + 2*sqrt(c)*sqrt(b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(b**2/c, 
0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True))/(256 
*c**5) + sqrt(b*x + c*x**2)*(B*c**2*x**8/9 + 63*b**4*(A*b**3 - 11*b*(3*A*b 
**2*c + B*b**3 - 13*b*(3*A*b*c**2 + 3*B*b**2*c - 15*b*(A*c**3 + 37*B*b*c** 
2/18)/(16*c))/(14*c))/(12*c))/(128*c**5) - 21*b**3*x*(A*b**3 - 11*b*(3*A*b 
**2*c + B*b**3 - 13*b*(3*A*b*c**2 + 3*B*b**2*c - 15*b*(A*c**3 + 37*B*b*c** 
2/18)/(16*c))/(14*c))/(12*c))/(64*c**4) + 21*b**2*x**2*(A*b**3 - 11*b*(3*A 
*b**2*c + B*b**3 - 13*b*(3*A*b*c**2 + 3*B*b**2*c - 15*b*(A*c**3 + 37*B*b*c 
**2/18)/(16*c))/(14*c))/(12*c))/(80*c**3) - 9*b*x**3*(A*b**3 - 11*b*(3*A*b 
**2*c + B*b**3 - 13*b*(3*A*b*c**2 + 3*B*b**2*c - 15*b*(A*c**3 + 37*B*b*c** 
2/18)/(16*c))/(14*c))/(12*c))/(40*c**2) + x**7*(A*c**3 + 37*B*b*c**2/18)/( 
8*c) + x**6*(3*A*b*c**2 + 3*B*b**2*c - 15*b*(A*c**3 + 37*B*b*c**2/18)/(16* 
c))/(7*c) + x**5*(3*A*b**2*c + B*b**3 - 13*b*(3*A*b*c**2 + 3*B*b**2*c - 15 
*b*(A*c**3 + 37*B*b*c**2/18)/(16*c))/(14*c))/(6*c) + x**4*(A*b**3 - 11*b*( 
3*A*b**2*c + B*b**3 - 13*b*(3*A*b*c**2 + 3*B*b**2*c - 15*b*(A*c**3 + 37*B* 
b*c**2/18)/(16*c))/(14*c))/(12*c))/(5*c)), Ne(c, 0)), (2*(A*(b*x)**(11/2)/ 
11 + B*(b*x)**(13/2)/(13*b))/b**3, Ne(b, 0)), (0, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.22 \[ \int x^2 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B x^{2}}{9 \, c} - \frac {55 \, \sqrt {c x^{2} + b x} B b^{7} x}{16384 \, c^{5}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{5} x}{6144 \, c^{4}} + \frac {45 \, \sqrt {c x^{2} + b x} A b^{6} x}{8192 \, c^{4}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{3} x}{384 \, c^{3}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{4} x}{1024 \, c^{3}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b x}{144 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{2} x}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} A x}{8 \, c} + \frac {55 \, B b^{9} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{65536 \, c^{\frac {13}{2}}} - \frac {45 \, A b^{8} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} - \frac {55 \, \sqrt {c x^{2} + b x} B b^{8}}{32768 \, c^{6}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{6}}{12288 \, c^{5}} + \frac {45 \, \sqrt {c x^{2} + b x} A b^{7}}{16384 \, c^{5}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{4}}{768 \, c^{4}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{5}}{2048 \, c^{4}} + \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b^{2}}{224 \, c^{3}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{3}}{128 \, c^{3}} - \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} A b}{112 \, c^{2}} \] Input:

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

Output:

1/9*(c*x^2 + b*x)^(7/2)*B*x^2/c - 55/16384*sqrt(c*x^2 + b*x)*B*b^7*x/c^5 + 
 55/6144*(c*x^2 + b*x)^(3/2)*B*b^5*x/c^4 + 45/8192*sqrt(c*x^2 + b*x)*A*b^6 
*x/c^4 - 11/384*(c*x^2 + b*x)^(5/2)*B*b^3*x/c^3 - 15/1024*(c*x^2 + b*x)^(3 
/2)*A*b^4*x/c^3 - 11/144*(c*x^2 + b*x)^(7/2)*B*b*x/c^2 + 3/64*(c*x^2 + b*x 
)^(5/2)*A*b^2*x/c^2 + 1/8*(c*x^2 + b*x)^(7/2)*A*x/c + 55/65536*B*b^9*log(2 
*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(13/2) - 45/32768*A*b^8*log(2*c* 
x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) - 55/32768*sqrt(c*x^2 + b*x) 
*B*b^8/c^6 + 55/12288*(c*x^2 + b*x)^(3/2)*B*b^6/c^5 + 45/16384*sqrt(c*x^2 
+ b*x)*A*b^7/c^5 - 11/768*(c*x^2 + b*x)^(5/2)*B*b^4/c^4 - 15/2048*(c*x^2 + 
 b*x)^(3/2)*A*b^5/c^4 + 11/224*(c*x^2 + b*x)^(7/2)*B*b^2/c^3 + 3/128*(c*x^ 
2 + b*x)^(5/2)*A*b^3/c^3 - 9/112*(c*x^2 + b*x)^(7/2)*A*b/c^2
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.84 \[ \int x^2 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{2064384} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, B c^{2} x + \frac {37 \, B b c^{9} + 18 \, A c^{10}}{c^{8}}\right )} x + \frac {3 \, {\left (103 \, B b^{2} c^{8} + 198 \, A b c^{9}\right )}}{c^{8}}\right )} x + \frac {5 \, B b^{3} c^{7} + 1458 \, A b^{2} c^{8}}{c^{8}}\right )} x - \frac {11 \, B b^{4} c^{6} - 18 \, A b^{3} c^{7}}{c^{8}}\right )} x + \frac {9 \, {\left (11 \, B b^{5} c^{5} - 18 \, A b^{4} c^{6}\right )}}{c^{8}}\right )} x - \frac {21 \, {\left (11 \, B b^{6} c^{4} - 18 \, A b^{5} c^{5}\right )}}{c^{8}}\right )} x + \frac {105 \, {\left (11 \, B b^{7} c^{3} - 18 \, A b^{6} c^{4}\right )}}{c^{8}}\right )} x - \frac {315 \, {\left (11 \, B b^{8} c^{2} - 18 \, A b^{7} c^{3}\right )}}{c^{8}}\right )} - \frac {5 \, {\left (11 \, B b^{9} - 18 \, A b^{8} c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{65536 \, c^{\frac {13}{2}}} \] Input:

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

Output:

1/2064384*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(4*(14*(16*B*c^2*x + (37*B*b*c^ 
9 + 18*A*c^10)/c^8)*x + 3*(103*B*b^2*c^8 + 198*A*b*c^9)/c^8)*x + (5*B*b^3* 
c^7 + 1458*A*b^2*c^8)/c^8)*x - (11*B*b^4*c^6 - 18*A*b^3*c^7)/c^8)*x + 9*(1 
1*B*b^5*c^5 - 18*A*b^4*c^6)/c^8)*x - 21*(11*B*b^6*c^4 - 18*A*b^5*c^5)/c^8) 
*x + 105*(11*B*b^7*c^3 - 18*A*b^6*c^4)/c^8)*x - 315*(11*B*b^8*c^2 - 18*A*b 
^7*c^3)/c^8) - 5/65536*(11*B*b^9 - 18*A*b^8*c)*log(abs(2*(sqrt(c)*x - sqrt 
(c*x^2 + b*x))*sqrt(c) + b))/c^(13/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.11 \[ \int x^2 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx=\frac {5670 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{7} c^{2}-3780 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{6} c^{3} x +3024 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{5} c^{4} x^{2}-2592 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{4} c^{5} x^{3}+2304 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{3} c^{6} x^{4}+373248 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{2} c^{7} x^{5}+608256 \sqrt {x}\, \sqrt {c x +b}\, a b \,c^{8} x^{6}+258048 \sqrt {x}\, \sqrt {c x +b}\, a \,c^{9} x^{7}-3465 \sqrt {x}\, \sqrt {c x +b}\, b^{9} c +2310 \sqrt {x}\, \sqrt {c x +b}\, b^{8} c^{2} x -1848 \sqrt {x}\, \sqrt {c x +b}\, b^{7} c^{3} x^{2}+1584 \sqrt {x}\, \sqrt {c x +b}\, b^{6} c^{4} x^{3}-1408 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c^{5} x^{4}+1280 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{6} x^{5}+316416 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{7} x^{6}+530432 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{8} x^{7}+229376 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{9} x^{8}-5670 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) a \,b^{8} c +3465 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{10}}{2064384 c^{7}} \] Input:

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

Output:

(5670*sqrt(x)*sqrt(b + c*x)*a*b**7*c**2 - 3780*sqrt(x)*sqrt(b + c*x)*a*b** 
6*c**3*x + 3024*sqrt(x)*sqrt(b + c*x)*a*b**5*c**4*x**2 - 2592*sqrt(x)*sqrt 
(b + c*x)*a*b**4*c**5*x**3 + 2304*sqrt(x)*sqrt(b + c*x)*a*b**3*c**6*x**4 + 
 373248*sqrt(x)*sqrt(b + c*x)*a*b**2*c**7*x**5 + 608256*sqrt(x)*sqrt(b + c 
*x)*a*b*c**8*x**6 + 258048*sqrt(x)*sqrt(b + c*x)*a*c**9*x**7 - 3465*sqrt(x 
)*sqrt(b + c*x)*b**9*c + 2310*sqrt(x)*sqrt(b + c*x)*b**8*c**2*x - 1848*sqr 
t(x)*sqrt(b + c*x)*b**7*c**3*x**2 + 1584*sqrt(x)*sqrt(b + c*x)*b**6*c**4*x 
**3 - 1408*sqrt(x)*sqrt(b + c*x)*b**5*c**5*x**4 + 1280*sqrt(x)*sqrt(b + c* 
x)*b**4*c**6*x**5 + 316416*sqrt(x)*sqrt(b + c*x)*b**3*c**7*x**6 + 530432*s 
qrt(x)*sqrt(b + c*x)*b**2*c**8*x**7 + 229376*sqrt(x)*sqrt(b + c*x)*b*c**9* 
x**8 - 5670*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*a*b**8* 
c + 3465*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*b**10)/(20 
64384*c**7)