\(\int x^{5/2} (A+B x) (a+b x+c x^2)^3 \, dx\) [75]

Optimal result

Integrand size = 23, antiderivative size = 182 \[ \int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{9} a^2 (3 A b+a B) x^{9/2}+\frac {6}{11} a \left (a b B+A \left (b^2+a c\right )\right ) x^{11/2}+\frac {2}{13} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{13/2}+\frac {2}{15} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{15/2}+\frac {6}{17} c \left (b^2 B+A b c+a B c\right ) x^{17/2}+\frac {2}{19} c^2 (3 b B+A c) x^{19/2}+\frac {2}{21} B c^3 x^{21/2} \] Output:

2/7*a^3*A*x^(7/2)+2/9*a^2*(3*A*b+B*a)*x^(9/2)+6/11*a*(a*b*B+A*(a*c+b^2))*x 
^(11/2)+2/13*(3*a*B*(a*c+b^2)+A*(6*a*b*c+b^3))*x^(13/2)+2/15*(3*A*a*c^2+3* 
A*b^2*c+6*B*a*b*c+B*b^3)*x^(15/2)+6/17*c*(A*b*c+B*a*c+B*b^2)*x^(17/2)+2/19 
*c^2*(A*c+3*B*b)*x^(19/2)+2/21*B*c^3*x^(21/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.98 \[ \int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2 x^{7/2} \left (230945 a^3 (9 A+7 B x)+33915 a^2 x (13 A (11 b+9 c x)+9 B x (13 b+11 c x))+1197 a x^2 \left (17 A \left (195 b^2+330 b c x+143 c^2 x^2\right )+11 B x \left (255 b^2+442 b c x+195 c^2 x^2\right )\right )+33 x^3 \left (21 A \left (1615 b^3+4199 b^2 c x+3705 b c^2 x^2+1105 c^3 x^3\right )+13 B x \left (2261 b^3+5985 b^2 c x+5355 b c^2 x^2+1615 c^3 x^3\right )\right )\right )}{14549535} \] Input:

Integrate[x^(5/2)*(A + B*x)*(a + b*x + c*x^2)^3,x]
 

Output:

(2*x^(7/2)*(230945*a^3*(9*A + 7*B*x) + 33915*a^2*x*(13*A*(11*b + 9*c*x) + 
9*B*x*(13*b + 11*c*x)) + 1197*a*x^2*(17*A*(195*b^2 + 330*b*c*x + 143*c^2*x 
^2) + 11*B*x*(255*b^2 + 442*b*c*x + 195*c^2*x^2)) + 33*x^3*(21*A*(1615*b^3 
 + 4199*b^2*c*x + 3705*b*c^2*x^2 + 1105*c^3*x^3) + 13*B*x*(2261*b^3 + 5985 
*b^2*c*x + 5355*b*c^2*x^2 + 1615*c^3*x^3))))/14549535
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1195, 2009}

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.

\(\displaystyle \int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (a^3 A x^{5/2}+a^2 x^{7/2} (a B+3 A b)+3 c x^{15/2} \left (a B c+A b c+b^2 B\right )+3 a x^{9/2} \left (A \left (a c+b^2\right )+a b B\right )+x^{13/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+x^{11/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+c^2 x^{17/2} (A c+3 b B)+B c^3 x^{19/2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2}{7} a^3 A x^{7/2}+\frac {2}{9} a^2 x^{9/2} (a B+3 A b)+\frac {6}{17} c x^{17/2} \left (a B c+A b c+b^2 B\right )+\frac {6}{11} a x^{11/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac {2}{15} x^{15/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {2}{13} x^{13/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {2}{19} c^2 x^{19/2} (A c+3 b B)+\frac {2}{21} B c^3 x^{21/2}\)

Input:

Int[x^(5/2)*(A + B*x)*(a + b*x + c*x^2)^3,x]
 

Output:

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*(a*b*B + A*(b 
^2 + a*c))*x^(11/2))/11 + (2*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(13 
/2))/13 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(15/2))/15 + (6 
*c*(b^2*B + A*b*c + a*B*c)*x^(17/2))/17 + (2*c^2*(3*b*B + A*c)*x^(19/2))/1 
9 + (2*B*c^3*x^(21/2))/21
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.05

method	result	size
gosper	\(\frac {2 x^{\frac {7}{2}} \left (692835 B \,c^{3} x^{7}+765765 A \,c^{3} x^{6}+2297295 B b \,c^{2} x^{6}+2567565 A b \,c^{2} x^{5}+2567565 B a \,c^{2} x^{5}+2567565 B \,b^{2} c \,x^{5}+2909907 A a \,c^{2} x^{4}+2909907 A \,b^{2} c \,x^{4}+5819814 B a b c \,x^{4}+969969 B \,b^{3} x^{4}+6715170 A a b c \,x^{3}+1119195 A \,b^{3} x^{3}+3357585 B \,a^{2} c \,x^{3}+3357585 B a \,b^{2} x^{3}+3968055 A \,a^{2} c \,x^{2}+3968055 A a \,b^{2} x^{2}+3968055 B \,a^{2} b \,x^{2}+4849845 A \,a^{2} b x +1616615 B \,a^{3} x +2078505 a^{3} A \right )}{14549535}\)	\(192\)
trager	\(\frac {2 x^{\frac {7}{2}} \left (692835 B \,c^{3} x^{7}+765765 A \,c^{3} x^{6}+2297295 B b \,c^{2} x^{6}+2567565 A b \,c^{2} x^{5}+2567565 B a \,c^{2} x^{5}+2567565 B \,b^{2} c \,x^{5}+2909907 A a \,c^{2} x^{4}+2909907 A \,b^{2} c \,x^{4}+5819814 B a b c \,x^{4}+969969 B \,b^{3} x^{4}+6715170 A a b c \,x^{3}+1119195 A \,b^{3} x^{3}+3357585 B \,a^{2} c \,x^{3}+3357585 B a \,b^{2} x^{3}+3968055 A \,a^{2} c \,x^{2}+3968055 A a \,b^{2} x^{2}+3968055 B \,a^{2} b \,x^{2}+4849845 A \,a^{2} b x +1616615 B \,a^{3} x +2078505 a^{3} A \right )}{14549535}\)	\(192\)
risch	\(\frac {2 x^{\frac {7}{2}} \left (692835 B \,c^{3} x^{7}+765765 A \,c^{3} x^{6}+2297295 B b \,c^{2} x^{6}+2567565 A b \,c^{2} x^{5}+2567565 B a \,c^{2} x^{5}+2567565 B \,b^{2} c \,x^{5}+2909907 A a \,c^{2} x^{4}+2909907 A \,b^{2} c \,x^{4}+5819814 B a b c \,x^{4}+969969 B \,b^{3} x^{4}+6715170 A a b c \,x^{3}+1119195 A \,b^{3} x^{3}+3357585 B \,a^{2} c \,x^{3}+3357585 B a \,b^{2} x^{3}+3968055 A \,a^{2} c \,x^{2}+3968055 A a \,b^{2} x^{2}+3968055 B \,a^{2} b \,x^{2}+4849845 A \,a^{2} b x +1616615 B \,a^{3} x +2078505 a^{3} A \right )}{14549535}\)	\(192\)
orering	\(\frac {2 x^{\frac {7}{2}} \left (692835 B \,c^{3} x^{7}+765765 A \,c^{3} x^{6}+2297295 B b \,c^{2} x^{6}+2567565 A b \,c^{2} x^{5}+2567565 B a \,c^{2} x^{5}+2567565 B \,b^{2} c \,x^{5}+2909907 A a \,c^{2} x^{4}+2909907 A \,b^{2} c \,x^{4}+5819814 B a b c \,x^{4}+969969 B \,b^{3} x^{4}+6715170 A a b c \,x^{3}+1119195 A \,b^{3} x^{3}+3357585 B \,a^{2} c \,x^{3}+3357585 B a \,b^{2} x^{3}+3968055 A \,a^{2} c \,x^{2}+3968055 A a \,b^{2} x^{2}+3968055 B \,a^{2} b \,x^{2}+4849845 A \,a^{2} b x +1616615 B \,a^{3} x +2078505 a^{3} A \right )}{14549535}\)	\(192\)
derivativedivides	\(\frac {2 B \,c^{3} x^{\frac {21}{2}}}{21}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 A b \,c^{2}+B \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (A \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (A \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 B \,a^{2} b \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}\)	\(226\)
default	\(\frac {2 B \,c^{3} x^{\frac {21}{2}}}{21}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 A b \,c^{2}+B \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (A \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (A \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 B \,a^{2} b \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}\)	\(226\)

Input:

int(x^(5/2)*(B*x+A)*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2/14549535*x^(7/2)*(692835*B*c^3*x^7+765765*A*c^3*x^6+2297295*B*b*c^2*x^6+ 
2567565*A*b*c^2*x^5+2567565*B*a*c^2*x^5+2567565*B*b^2*c*x^5+2909907*A*a*c^ 
2*x^4+2909907*A*b^2*c*x^4+5819814*B*a*b*c*x^4+969969*B*b^3*x^4+6715170*A*a 
*b*c*x^3+1119195*A*b^3*x^3+3357585*B*a^2*c*x^3+3357585*B*a*b^2*x^3+3968055 
*A*a^2*c*x^2+3968055*A*a*b^2*x^2+3968055*B*a^2*b*x^2+4849845*A*a^2*b*x+161 
6615*B*a^3*x+2078505*A*a^3)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.94 \[ \int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2}{14549535} \, {\left (692835 \, B c^{3} x^{10} + 765765 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + 2567565 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{8} + 969969 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{7} + 2078505 \, A a^{3} x^{3} + 1119195 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + 3968055 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + 1616615 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt {x} \] Input:

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

2/14549535*(692835*B*c^3*x^10 + 765765*(3*B*b*c^2 + A*c^3)*x^9 + 2567565*( 
B*b^2*c + (B*a + A*b)*c^2)*x^8 + 969969*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + 
A*b^2)*c)*x^7 + 2078505*A*a^3*x^3 + 1119195*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 
+ 2*A*a*b)*c)*x^6 + 3968055*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^5 + 1616615*(B 
*a^3 + 3*A*a^2*b)*x^4)*sqrt(x)
 

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.62 \[ \int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2 A a^{3} x^{\frac {7}{2}}}{7} + \frac {2 A a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 A a^{2} c x^{\frac {11}{2}}}{11} + \frac {6 A a b^{2} x^{\frac {11}{2}}}{11} + \frac {12 A a b c x^{\frac {13}{2}}}{13} + \frac {2 A a c^{2} x^{\frac {15}{2}}}{5} + \frac {2 A b^{3} x^{\frac {13}{2}}}{13} + \frac {2 A b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 A b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 A c^{3} x^{\frac {19}{2}}}{19} + \frac {2 B a^{3} x^{\frac {9}{2}}}{9} + \frac {6 B a^{2} b x^{\frac {11}{2}}}{11} + \frac {6 B a^{2} c x^{\frac {13}{2}}}{13} + \frac {6 B a b^{2} x^{\frac {13}{2}}}{13} + \frac {4 B a b c x^{\frac {15}{2}}}{5} + \frac {6 B a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B b^{3} x^{\frac {15}{2}}}{15} + \frac {6 B b^{2} c x^{\frac {17}{2}}}{17} + \frac {6 B b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 B c^{3} x^{\frac {21}{2}}}{21} \] Input:

integrate(x**(5/2)*(B*x+A)*(c*x**2+b*x+a)**3,x)
 

Output:

2*A*a**3*x**(7/2)/7 + 2*A*a**2*b*x**(9/2)/3 + 6*A*a**2*c*x**(11/2)/11 + 6* 
A*a*b**2*x**(11/2)/11 + 12*A*a*b*c*x**(13/2)/13 + 2*A*a*c**2*x**(15/2)/5 + 
 2*A*b**3*x**(13/2)/13 + 2*A*b**2*c*x**(15/2)/5 + 6*A*b*c**2*x**(17/2)/17 
+ 2*A*c**3*x**(19/2)/19 + 2*B*a**3*x**(9/2)/9 + 6*B*a**2*b*x**(11/2)/11 + 
6*B*a**2*c*x**(13/2)/13 + 6*B*a*b**2*x**(13/2)/13 + 4*B*a*b*c*x**(15/2)/5 
+ 6*B*a*c**2*x**(17/2)/17 + 2*B*b**3*x**(15/2)/15 + 6*B*b**2*c*x**(17/2)/1 
7 + 6*B*b*c**2*x**(19/2)/19 + 2*B*c**3*x**(21/2)/21
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.91 \[ \int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {2}{19} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {19}{2}} + \frac {6}{17} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{\frac {17}{2}} + \frac {2}{15} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac {15}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} + \frac {2}{13} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac {13}{2}} + \frac {6}{11} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {9}{2}} \] Input:

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x+a)^3,x, algorithm="maxima")
 

Output:

2/21*B*c^3*x^(21/2) + 2/19*(3*B*b*c^2 + A*c^3)*x^(19/2) + 6/17*(B*b^2*c + 
(B*a + A*b)*c^2)*x^(17/2) + 2/15*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)* 
c)*x^(15/2) + 2/7*A*a^3*x^(7/2) + 2/13*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A 
*a*b)*c)*x^(13/2) + 6/11*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^(11/2) + 2/9*(B*a 
^3 + 3*A*a^2*b)*x^(9/2)
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.06 \[ \int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B b c^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B b^{2} c x^{\frac {17}{2}} + \frac {6}{17} \, B a c^{2} x^{\frac {17}{2}} + \frac {6}{17} \, A b c^{2} x^{\frac {17}{2}} + \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {4}{5} \, B a b c x^{\frac {15}{2}} + \frac {2}{5} \, A b^{2} c x^{\frac {15}{2}} + \frac {2}{5} \, A a c^{2} x^{\frac {15}{2}} + \frac {6}{13} \, B a b^{2} x^{\frac {13}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} + \frac {6}{13} \, B a^{2} c x^{\frac {13}{2}} + \frac {12}{13} \, A a b c x^{\frac {13}{2}} + \frac {6}{11} \, B a^{2} b x^{\frac {11}{2}} + \frac {6}{11} \, A a b^{2} x^{\frac {11}{2}} + \frac {6}{11} \, A a^{2} c x^{\frac {11}{2}} + \frac {2}{9} \, B a^{3} x^{\frac {9}{2}} + \frac {2}{3} \, A a^{2} b x^{\frac {9}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \] Input:

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x+a)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

2/21*B*c^3*x^(21/2) + 6/19*B*b*c^2*x^(19/2) + 2/19*A*c^3*x^(19/2) + 6/17*B 
*b^2*c*x^(17/2) + 6/17*B*a*c^2*x^(17/2) + 6/17*A*b*c^2*x^(17/2) + 2/15*B*b 
^3*x^(15/2) + 4/5*B*a*b*c*x^(15/2) + 2/5*A*b^2*c*x^(15/2) + 2/5*A*a*c^2*x^ 
(15/2) + 6/13*B*a*b^2*x^(13/2) + 2/13*A*b^3*x^(13/2) + 6/13*B*a^2*c*x^(13/ 
2) + 12/13*A*a*b*c*x^(13/2) + 6/11*B*a^2*b*x^(11/2) + 6/11*A*a*b^2*x^(11/2 
) + 6/11*A*a^2*c*x^(11/2) + 2/9*B*a^3*x^(9/2) + 2/3*A*a^2*b*x^(9/2) + 2/7* 
A*a^3*x^(7/2)
 

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.93 \[ \int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx=x^{13/2}\,\left (\frac {6\,B\,c\,a^2}{13}+\frac {6\,B\,a\,b^2}{13}+\frac {12\,A\,c\,a\,b}{13}+\frac {2\,A\,b^3}{13}\right )+x^{15/2}\,\left (\frac {2\,B\,b^3}{15}+\frac {2\,A\,b^2\,c}{5}+\frac {4\,B\,a\,b\,c}{5}+\frac {2\,A\,a\,c^2}{5}\right )+x^{9/2}\,\left (\frac {2\,B\,a^3}{9}+\frac {2\,A\,b\,a^2}{3}\right )+x^{19/2}\,\left (\frac {2\,A\,c^3}{19}+\frac {6\,B\,b\,c^2}{19}\right )+x^{11/2}\,\left (\frac {6\,B\,a^2\,b}{11}+\frac {6\,A\,c\,a^2}{11}+\frac {6\,A\,a\,b^2}{11}\right )+x^{17/2}\,\left (\frac {6\,B\,b^2\,c}{17}+\frac {6\,A\,b\,c^2}{17}+\frac {6\,B\,a\,c^2}{17}\right )+\frac {2\,A\,a^3\,x^{7/2}}{7}+\frac {2\,B\,c^3\,x^{21/2}}{21} \] Input:

int(x^(5/2)*(A + B*x)*(a + b*x + c*x^2)^3,x)
 

Output:

x^(13/2)*((2*A*b^3)/13 + (6*B*a*b^2)/13 + (6*B*a^2*c)/13 + (12*A*a*b*c)/13 
) + x^(15/2)*((2*B*b^3)/15 + (2*A*a*c^2)/5 + (2*A*b^2*c)/5 + (4*B*a*b*c)/5 
) + x^(9/2)*((2*B*a^3)/9 + (2*A*a^2*b)/3) + x^(19/2)*((2*A*c^3)/19 + (6*B* 
b*c^2)/19) + x^(11/2)*((6*A*a*b^2)/11 + (6*A*a^2*c)/11 + (6*B*a^2*b)/11) + 
 x^(17/2)*((6*A*b*c^2)/17 + (6*B*a*c^2)/17 + (6*B*b^2*c)/17) + (2*A*a^3*x^ 
(7/2))/7 + (2*B*c^3*x^(21/2))/21
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.75 \[ \int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \sqrt {x}\, x^{3} \left (692835 b \,c^{3} x^{7}+765765 a \,c^{3} x^{6}+2297295 b^{2} c^{2} x^{6}+5135130 a b \,c^{2} x^{5}+2567565 b^{3} c \,x^{5}+2909907 a^{2} c^{2} x^{4}+8729721 a \,b^{2} c \,x^{4}+969969 b^{4} x^{4}[phone] a^{2} b c \,x^{3}+4476780 a \,b^{3} x^{3}+3968055 a^{3} c \,x^{2}+7936110 a^{2} b^{2} x^{2}+6466460 a^{3} b x +2078505 a^{4}\right )}{14549535} \] Input:

int(x^(5/2)*(B*x+A)*(c*x^2+b*x+a)^3,x)
 

Output:

(2*sqrt(x)*x**3*(2078505*a**4 + 6466460*a**3*b*x + 3968055*a**3*c*x**2 + 7 
936110*a**2*b**2*x**2 + 10072755*a**2*b*c*x**3 + 2909907*a**2*c**2*x**4 + 
4476780*a*b**3*x**3 + 8729721*a*b**2*c*x**4 + 5135130*a*b*c**2*x**5 + 7657 
65*a*c**3*x**6 + 969969*b**4*x**4 + 2567565*b**3*c*x**5 + 2297295*b**2*c** 
2*x**6 + 692835*b*c**3*x**7))/14549535