\(\int (b d+2 c d x)^5 (a+b x+c x^2)^3 \, dx\) [29]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^3 \, dx=-\frac {\left (b^2-4 a c\right )^3 d^5 (b+2 c x)^6}{768 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^5 (b+2 c x)^8}{1024 c^4}-\frac {3 \left (b^2-4 a c\right ) d^5 (b+2 c x)^{10}}{1280 c^4}+\frac {d^5 (b+2 c x)^{12}}{1536 c^4} \] Output:

-1/768*(-4*a*c+b^2)^3*d^5*(2*c*x+b)^6/c^4+3/1024*(-4*a*c+b^2)^2*d^5*(2*c*x 
+b)^8/c^4-3/1280*(-4*a*c+b^2)*d^5*(2*c*x+b)^10/c^4+1/1536*d^5*(2*c*x+b)^12 
/c^4
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(224\) vs. \(2(101)=202\).

Time = 0.08 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.22 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{60} d^5 x (b+c x) \left (20 a^3 \left (3 b^4+12 b^3 c x+28 b^2 c^2 x^2+32 b c^3 x^3+16 c^4 x^4\right )+12 a x^2 (b+c x)^2 \left (5 b^4+30 b^3 c x+78 b^2 c^2 x^2+96 b c^3 x^3+48 c^4 x^4\right )+x^3 (b+c x)^3 \left (15 b^4+96 b^3 c x+256 b^2 c^2 x^2+320 b c^3 x^3+160 c^4 x^4\right )+30 a^2 x \left (3 b^5+19 b^4 c x+56 b^3 c^2 x^2+88 b^2 c^3 x^3+72 b c^4 x^4+24 c^5 x^5\right )\right ) \] Input:

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

Output:

(d^5*x*(b + c*x)*(20*a^3*(3*b^4 + 12*b^3*c*x + 28*b^2*c^2*x^2 + 32*b*c^3*x 
^3 + 16*c^4*x^4) + 12*a*x^2*(b + c*x)^2*(5*b^4 + 30*b^3*c*x + 78*b^2*c^2*x 
^2 + 96*b*c^3*x^3 + 48*c^4*x^4) + x^3*(b + c*x)^3*(15*b^4 + 96*b^3*c*x + 2 
56*b^2*c^2*x^2 + 320*b*c^3*x^3 + 160*c^4*x^4) + 30*a^2*x*(3*b^5 + 19*b^4*c 
*x + 56*b^3*c^2*x^2 + 88*b^2*c^3*x^3 + 72*b*c^4*x^4 + 24*c^5*x^5)))/60
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1107, 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.

\(\displaystyle \int \left (a+b x+c x^2\right )^3 (b d+2 c d x)^5 \, dx\)

\(\Big \downarrow \) 1107

\(\displaystyle \int \left (\frac {3 \left (4 a c-b^2\right ) (b d+2 c d x)^9}{64 c^3 d^4}+\frac {3 \left (4 a c-b^2\right )^2 (b d+2 c d x)^7}{64 c^3 d^2}+\frac {\left (4 a c-b^2\right )^3 (b d+2 c d x)^5}{64 c^3}+\frac {(b d+2 c d x)^{11}}{64 c^3 d^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 d^5 \left (b^2-4 a c\right ) (b+2 c x)^{10}}{1280 c^4}+\frac {3 d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^8}{1024 c^4}-\frac {d^5 \left (b^2-4 a c\right )^3 (b+2 c x)^6}{768 c^4}+\frac {d^5 (b+2 c x)^{12}}{1536 c^4}\)

Input:

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

Output:

-1/768*((b^2 - 4*a*c)^3*d^5*(b + 2*c*x)^6)/c^4 + (3*(b^2 - 4*a*c)^2*d^5*(b 
 + 2*c*x)^8)/(1024*c^4) - (3*(b^2 - 4*a*c)*d^5*(b + 2*c*x)^10)/(1280*c^4) 
+ (d^5*(b + 2*c*x)^12)/(1536*c^4)
 

Defintions of rubi rules used

rule 1107

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; F 
reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] &&  !(EqQ[ 
m, 3] && NeQ[p, 1])
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(337\) vs. \(2(93)=186\).

Time = 0.77 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.35

method	result	size
gosper	\(\frac {x \left (160 c^{8} x^{11}+960 b \,c^{7} x^{10}+576 x^{9} a \,c^{7}+2496 x^{9} c^{6} b^{2}+2880 x^{8} a b \,c^{6}+3680 x^{8} b^{3} c^{5}+720 x^{7} a^{2} c^{6}+6120 x^{7} a \,b^{2} c^{5}+3375 x^{7} b^{4} c^{4}+2880 a^{2} b \,c^{5} x^{6}+7200 a \,b^{3} c^{4} x^{6}+1980 b^{5} c^{3} x^{6}+320 x^{5} a^{3} c^{5}+4800 x^{5} a^{2} b^{2} c^{4}+5100 a \,b^{4} c^{3} x^{5}+730 b^{6} c^{2} x^{5}+960 x^{4} a^{3} b \,c^{4}+4320 x^{4} a^{2} b^{3} c^{3}+2196 x^{4} a \,b^{5} c^{2}+156 x^{4} b^{7} c +1200 x^{3} a^{3} b^{2} c^{3}+2250 x^{3} a^{2} b^{4} c^{2}+540 x^{3} a \,b^{6} c +15 x^{3} b^{8}+800 x^{2} a^{3} b^{3} c^{2}+660 x^{2} a^{2} b^{5} c +60 x^{2} a \,b^{7}+300 x \,a^{3} b^{4} c +90 a^{2} b^{6} x +60 a^{3} b^{5}\right ) d^{5}}{60}\)	\(338\)
orering	\(\frac {x \left (160 c^{8} x^{11}+960 b \,c^{7} x^{10}+576 x^{9} a \,c^{7}+2496 x^{9} c^{6} b^{2}+2880 x^{8} a b \,c^{6}+3680 x^{8} b^{3} c^{5}+720 x^{7} a^{2} c^{6}+6120 x^{7} a \,b^{2} c^{5}+3375 x^{7} b^{4} c^{4}+2880 a^{2} b \,c^{5} x^{6}+7200 a \,b^{3} c^{4} x^{6}+1980 b^{5} c^{3} x^{6}+320 x^{5} a^{3} c^{5}+4800 x^{5} a^{2} b^{2} c^{4}+5100 a \,b^{4} c^{3} x^{5}+730 b^{6} c^{2} x^{5}+960 x^{4} a^{3} b \,c^{4}+4320 x^{4} a^{2} b^{3} c^{3}+2196 x^{4} a \,b^{5} c^{2}+156 x^{4} b^{7} c +1200 x^{3} a^{3} b^{2} c^{3}+2250 x^{3} a^{2} b^{4} c^{2}+540 x^{3} a \,b^{6} c +15 x^{3} b^{8}+800 x^{2} a^{3} b^{3} c^{2}+660 x^{2} a^{2} b^{5} c +60 x^{2} a \,b^{7}+300 x \,a^{3} b^{4} c +90 a^{2} b^{6} x +60 a^{3} b^{5}\right ) \left (2 c d x +b d \right )^{5}}{60 \left (2 c x +b \right )^{5}}\)	\(354\)
norman	\(\left (\frac {48}{5} a \,c^{7} d^{5}+\frac {208}{5} b^{2} d^{5} c^{6}\right ) x^{10}+\left (48 a b \,c^{6} d^{5}+\frac {184}{3} b^{3} d^{5} c^{5}\right ) x^{9}+\left (5 b^{4} d^{5} c \,a^{3}+\frac {3}{2} b^{6} d^{5} a^{2}\right ) x^{2}+\left (12 a^{2} c^{6} d^{5}+102 a \,b^{2} c^{5} d^{5}+\frac {225}{4} b^{4} d^{5} c^{4}\right ) x^{8}+\left (\frac {40}{3} b^{3} d^{5} c^{2} a^{3}+11 b^{5} d^{5} c \,a^{2}+b^{7} d^{5} a \right ) x^{3}+\left (\frac {16}{3} d^{5} c^{5} a^{3}+80 b^{2} d^{5} c^{4} a^{2}+85 a \,b^{4} c^{3} d^{5}+\frac {73}{6} b^{6} d^{5} c^{2}\right ) x^{6}+\left (20 a^{3} b^{2} c^{3} d^{5}+\frac {75}{2} b^{4} d^{5} c^{2} a^{2}+9 a \,b^{6} c \,d^{5}+\frac {1}{4} b^{8} d^{5}\right ) x^{4}+\left (16 b \,d^{5} c^{4} a^{3}+72 b^{3} d^{5} c^{3} a^{2}+\frac {183}{5} a \,b^{5} c^{2} d^{5}+\frac {13}{5} b^{7} c \,d^{5}\right ) x^{5}+\left (48 d^{5} c^{5} a^{2} b +120 a \,b^{3} c^{4} d^{5}+33 b^{5} d^{5} c^{3}\right ) x^{7}+b^{5} d^{5} a^{3} x +\frac {8 c^{8} d^{5} x^{12}}{3}+16 b \,d^{5} c^{7} x^{11}\)	\(389\)
risch	\(\frac {208}{5} d^{5} x^{10} c^{6} b^{2}+\frac {3}{2} d^{5} x^{2} a^{2} b^{6}+\frac {184}{3} d^{5} x^{9} b^{3} c^{5}+d^{5} x^{3} a \,b^{7}+\frac {16}{3} d^{5} x^{6} a^{3} c^{5}+12 d^{5} x^{8} a^{2} c^{6}+\frac {225}{4} d^{5} x^{8} c^{4} b^{4}+16 b \,d^{5} c^{7} x^{11}+33 d^{5} b^{5} c^{3} x^{7}+\frac {73}{6} d^{5} b^{6} c^{2} x^{6}+\frac {13}{5} d^{5} x^{5} b^{7} c +\frac {48}{5} d^{5} x^{10} a \,c^{7}+b^{5} d^{5} a^{3} x +16 d^{5} x^{5} a^{3} b \,c^{4}+72 d^{5} x^{5} a^{2} b^{3} c^{3}+\frac {183}{5} d^{5} x^{5} a \,b^{5} c^{2}+20 d^{5} x^{4} a^{3} b^{2} c^{3}+\frac {75}{2} d^{5} x^{4} a^{2} b^{4} c^{2}+9 d^{5} x^{4} a \,b^{6} c +\frac {40}{3} d^{5} x^{3} a^{3} b^{3} c^{2}+11 d^{5} x^{3} a^{2} b^{5} c +5 d^{5} x^{2} a^{3} b^{4} c +48 d^{5} a^{2} b \,c^{5} x^{7}+120 d^{5} a \,b^{3} c^{4} x^{7}+48 d^{5} x^{9} a b \,c^{6}+102 d^{5} x^{8} a \,b^{2} c^{5}+80 d^{5} x^{6} a^{2} b^{2} c^{4}+85 d^{5} x^{6} a \,b^{4} c^{3}+\frac {8}{3} c^{8} d^{5} x^{12}+\frac {1}{4} d^{5} x^{4} b^{8}\)	\(425\)
parallelrisch	\(\frac {208}{5} d^{5} x^{10} c^{6} b^{2}+\frac {3}{2} d^{5} x^{2} a^{2} b^{6}+\frac {184}{3} d^{5} x^{9} b^{3} c^{5}+d^{5} x^{3} a \,b^{7}+\frac {16}{3} d^{5} x^{6} a^{3} c^{5}+12 d^{5} x^{8} a^{2} c^{6}+\frac {225}{4} d^{5} x^{8} c^{4} b^{4}+16 b \,d^{5} c^{7} x^{11}+33 d^{5} b^{5} c^{3} x^{7}+\frac {73}{6} d^{5} b^{6} c^{2} x^{6}+\frac {13}{5} d^{5} x^{5} b^{7} c +\frac {48}{5} d^{5} x^{10} a \,c^{7}+b^{5} d^{5} a^{3} x +16 d^{5} x^{5} a^{3} b \,c^{4}+72 d^{5} x^{5} a^{2} b^{3} c^{3}+\frac {183}{5} d^{5} x^{5} a \,b^{5} c^{2}+20 d^{5} x^{4} a^{3} b^{2} c^{3}+\frac {75}{2} d^{5} x^{4} a^{2} b^{4} c^{2}+9 d^{5} x^{4} a \,b^{6} c +\frac {40}{3} d^{5} x^{3} a^{3} b^{3} c^{2}+11 d^{5} x^{3} a^{2} b^{5} c +5 d^{5} x^{2} a^{3} b^{4} c +48 d^{5} a^{2} b \,c^{5} x^{7}+120 d^{5} a \,b^{3} c^{4} x^{7}+48 d^{5} x^{9} a b \,c^{6}+102 d^{5} x^{8} a \,b^{2} c^{5}+80 d^{5} x^{6} a^{2} b^{2} c^{4}+85 d^{5} x^{6} a \,b^{4} c^{3}+\frac {8}{3} c^{8} d^{5} x^{12}+\frac {1}{4} d^{5} x^{4} b^{8}\)	\(425\)
default	\(\frac {8 c^{8} d^{5} x^{12}}{3}+16 b \,d^{5} c^{7} x^{11}+\frac {\left (320 b^{2} d^{5} c^{6}+32 d^{5} c^{5} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{10}}{10}+\frac {\left (280 b^{3} d^{5} c^{5}+80 b \,d^{5} c^{4} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+32 d^{5} c^{5} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{9}}{9}+\frac {\left (130 b^{4} d^{5} c^{4}+80 b^{2} d^{5} c^{3} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+80 b \,d^{5} c^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+32 d^{5} c^{5} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{8}}{8}+\frac {\left (31 b^{5} d^{5} c^{3}+40 b^{3} d^{5} c^{2} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+80 b^{2} d^{5} c^{3} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+80 b \,d^{5} c^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+96 d^{5} c^{5} a^{2} b \right ) x^{7}}{7}+\frac {\left (3 b^{6} d^{5} c^{2}+10 b^{4} d^{5} c \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+40 b^{3} d^{5} c^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+80 b^{2} d^{5} c^{3} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+240 b^{2} d^{5} c^{4} a^{2}+32 d^{5} c^{5} a^{3}\right ) x^{6}}{6}+\frac {\left (b^{5} d^{5} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+10 b^{4} d^{5} c \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+40 b^{3} d^{5} c^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+240 b^{3} d^{5} c^{3} a^{2}+80 b \,d^{5} c^{4} a^{3}\right ) x^{5}}{5}+\frac {\left (b^{5} d^{5} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+10 b^{4} d^{5} c \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+120 b^{4} d^{5} c^{2} a^{2}+80 a^{3} b^{2} c^{3} d^{5}\right ) x^{4}}{4}+\frac {\left (b^{5} d^{5} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+30 b^{5} d^{5} c \,a^{2}+40 b^{3} d^{5} c^{2} a^{3}\right ) x^{3}}{3}+\frac {\left (10 b^{4} d^{5} c \,a^{3}+3 b^{6} d^{5} a^{2}\right ) x^{2}}{2}+b^{5} d^{5} a^{3} x\)	\(810\)

Input:

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

Output:

1/60*x*(160*c^8*x^11+960*b*c^7*x^10+576*a*c^7*x^9+2496*b^2*c^6*x^9+2880*a* 
b*c^6*x^8+3680*b^3*c^5*x^8+720*a^2*c^6*x^7+6120*a*b^2*c^5*x^7+3375*b^4*c^4 
*x^7+2880*a^2*b*c^5*x^6+7200*a*b^3*c^4*x^6+1980*b^5*c^3*x^6+320*a^3*c^5*x^ 
5+4800*a^2*b^2*c^4*x^5+5100*a*b^4*c^3*x^5+730*b^6*c^2*x^5+960*a^3*b*c^4*x^ 
4+4320*a^2*b^3*c^3*x^4+2196*a*b^5*c^2*x^4+156*b^7*c*x^4+1200*a^3*b^2*c^3*x 
^3+2250*a^2*b^4*c^2*x^3+540*a*b^6*c*x^3+15*b^8*x^3+800*a^3*b^3*c^2*x^2+660 
*a^2*b^5*c*x^2+60*a*b^7*x^2+300*a^3*b^4*c*x+90*a^2*b^6*x+60*a^3*b^5)*d^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (93) = 186\).

Time = 0.08 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.39 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^3 \, dx=\frac {8}{3} \, c^{8} d^{5} x^{12} + 16 \, b c^{7} d^{5} x^{11} + \frac {16}{5} \, {\left (13 \, b^{2} c^{6} + 3 \, a c^{7}\right )} d^{5} x^{10} + \frac {8}{3} \, {\left (23 \, b^{3} c^{5} + 18 \, a b c^{6}\right )} d^{5} x^{9} + a^{3} b^{5} d^{5} x + \frac {3}{4} \, {\left (75 \, b^{4} c^{4} + 136 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{5} x^{8} + 3 \, {\left (11 \, b^{5} c^{3} + 40 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{5} x^{7} + \frac {1}{6} \, {\left (73 \, b^{6} c^{2} + 510 \, a b^{4} c^{3} + 480 \, a^{2} b^{2} c^{4} + 32 \, a^{3} c^{5}\right )} d^{5} x^{6} + \frac {1}{5} \, {\left (13 \, b^{7} c + 183 \, a b^{5} c^{2} + 360 \, a^{2} b^{3} c^{3} + 80 \, a^{3} b c^{4}\right )} d^{5} x^{5} + \frac {1}{4} \, {\left (b^{8} + 36 \, a b^{6} c + 150 \, a^{2} b^{4} c^{2} + 80 \, a^{3} b^{2} c^{3}\right )} d^{5} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{7} + 33 \, a^{2} b^{5} c + 40 \, a^{3} b^{3} c^{2}\right )} d^{5} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{6} + 10 \, a^{3} b^{4} c\right )} d^{5} x^{2} \] Input:

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

Output:

8/3*c^8*d^5*x^12 + 16*b*c^7*d^5*x^11 + 16/5*(13*b^2*c^6 + 3*a*c^7)*d^5*x^1 
0 + 8/3*(23*b^3*c^5 + 18*a*b*c^6)*d^5*x^9 + a^3*b^5*d^5*x + 3/4*(75*b^4*c^ 
4 + 136*a*b^2*c^5 + 16*a^2*c^6)*d^5*x^8 + 3*(11*b^5*c^3 + 40*a*b^3*c^4 + 1 
6*a^2*b*c^5)*d^5*x^7 + 1/6*(73*b^6*c^2 + 510*a*b^4*c^3 + 480*a^2*b^2*c^4 + 
 32*a^3*c^5)*d^5*x^6 + 1/5*(13*b^7*c + 183*a*b^5*c^2 + 360*a^2*b^3*c^3 + 8 
0*a^3*b*c^4)*d^5*x^5 + 1/4*(b^8 + 36*a*b^6*c + 150*a^2*b^4*c^2 + 80*a^3*b^ 
2*c^3)*d^5*x^4 + 1/3*(3*a*b^7 + 33*a^2*b^5*c + 40*a^3*b^3*c^2)*d^5*x^3 + 1 
/2*(3*a^2*b^6 + 10*a^3*b^4*c)*d^5*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (100) = 200\).

Time = 0.05 (sec) , antiderivative size = 428, normalized size of antiderivative = 4.24 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^3 \, dx=a^{3} b^{5} d^{5} x + 16 b c^{7} d^{5} x^{11} + \frac {8 c^{8} d^{5} x^{12}}{3} + x^{10} \cdot \left (\frac {48 a c^{7} d^{5}}{5} + \frac {208 b^{2} c^{6} d^{5}}{5}\right ) + x^{9} \cdot \left (48 a b c^{6} d^{5} + \frac {184 b^{3} c^{5} d^{5}}{3}\right ) + x^{8} \cdot \left (12 a^{2} c^{6} d^{5} + 102 a b^{2} c^{5} d^{5} + \frac {225 b^{4} c^{4} d^{5}}{4}\right ) + x^{7} \cdot \left (48 a^{2} b c^{5} d^{5} + 120 a b^{3} c^{4} d^{5} + 33 b^{5} c^{3} d^{5}\right ) + x^{6} \cdot \left (\frac {16 a^{3} c^{5} d^{5}}{3} + 80 a^{2} b^{2} c^{4} d^{5} + 85 a b^{4} c^{3} d^{5} + \frac {73 b^{6} c^{2} d^{5}}{6}\right ) + x^{5} \cdot \left (16 a^{3} b c^{4} d^{5} + 72 a^{2} b^{3} c^{3} d^{5} + \frac {183 a b^{5} c^{2} d^{5}}{5} + \frac {13 b^{7} c d^{5}}{5}\right ) + x^{4} \cdot \left (20 a^{3} b^{2} c^{3} d^{5} + \frac {75 a^{2} b^{4} c^{2} d^{5}}{2} + 9 a b^{6} c d^{5} + \frac {b^{8} d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {40 a^{3} b^{3} c^{2} d^{5}}{3} + 11 a^{2} b^{5} c d^{5} + a b^{7} d^{5}\right ) + x^{2} \cdot \left (5 a^{3} b^{4} c d^{5} + \frac {3 a^{2} b^{6} d^{5}}{2}\right ) \] Input:

integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**3,x)
 

Output:

a**3*b**5*d**5*x + 16*b*c**7*d**5*x**11 + 8*c**8*d**5*x**12/3 + x**10*(48* 
a*c**7*d**5/5 + 208*b**2*c**6*d**5/5) + x**9*(48*a*b*c**6*d**5 + 184*b**3* 
c**5*d**5/3) + x**8*(12*a**2*c**6*d**5 + 102*a*b**2*c**5*d**5 + 225*b**4*c 
**4*d**5/4) + x**7*(48*a**2*b*c**5*d**5 + 120*a*b**3*c**4*d**5 + 33*b**5*c 
**3*d**5) + x**6*(16*a**3*c**5*d**5/3 + 80*a**2*b**2*c**4*d**5 + 85*a*b**4 
*c**3*d**5 + 73*b**6*c**2*d**5/6) + x**5*(16*a**3*b*c**4*d**5 + 72*a**2*b* 
*3*c**3*d**5 + 183*a*b**5*c**2*d**5/5 + 13*b**7*c*d**5/5) + x**4*(20*a**3* 
b**2*c**3*d**5 + 75*a**2*b**4*c**2*d**5/2 + 9*a*b**6*c*d**5 + b**8*d**5/4) 
 + x**3*(40*a**3*b**3*c**2*d**5/3 + 11*a**2*b**5*c*d**5 + a*b**7*d**5) + x 
**2*(5*a**3*b**4*c*d**5 + 3*a**2*b**6*d**5/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (93) = 186\).

Time = 0.04 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.39 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^3 \, dx=\frac {8}{3} \, c^{8} d^{5} x^{12} + 16 \, b c^{7} d^{5} x^{11} + \frac {16}{5} \, {\left (13 \, b^{2} c^{6} + 3 \, a c^{7}\right )} d^{5} x^{10} + \frac {8}{3} \, {\left (23 \, b^{3} c^{5} + 18 \, a b c^{6}\right )} d^{5} x^{9} + a^{3} b^{5} d^{5} x + \frac {3}{4} \, {\left (75 \, b^{4} c^{4} + 136 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{5} x^{8} + 3 \, {\left (11 \, b^{5} c^{3} + 40 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{5} x^{7} + \frac {1}{6} \, {\left (73 \, b^{6} c^{2} + 510 \, a b^{4} c^{3} + 480 \, a^{2} b^{2} c^{4} + 32 \, a^{3} c^{5}\right )} d^{5} x^{6} + \frac {1}{5} \, {\left (13 \, b^{7} c + 183 \, a b^{5} c^{2} + 360 \, a^{2} b^{3} c^{3} + 80 \, a^{3} b c^{4}\right )} d^{5} x^{5} + \frac {1}{4} \, {\left (b^{8} + 36 \, a b^{6} c + 150 \, a^{2} b^{4} c^{2} + 80 \, a^{3} b^{2} c^{3}\right )} d^{5} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{7} + 33 \, a^{2} b^{5} c + 40 \, a^{3} b^{3} c^{2}\right )} d^{5} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{6} + 10 \, a^{3} b^{4} c\right )} d^{5} x^{2} \] Input:

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

Output:

8/3*c^8*d^5*x^12 + 16*b*c^7*d^5*x^11 + 16/5*(13*b^2*c^6 + 3*a*c^7)*d^5*x^1 
0 + 8/3*(23*b^3*c^5 + 18*a*b*c^6)*d^5*x^9 + a^3*b^5*d^5*x + 3/4*(75*b^4*c^ 
4 + 136*a*b^2*c^5 + 16*a^2*c^6)*d^5*x^8 + 3*(11*b^5*c^3 + 40*a*b^3*c^4 + 1 
6*a^2*b*c^5)*d^5*x^7 + 1/6*(73*b^6*c^2 + 510*a*b^4*c^3 + 480*a^2*b^2*c^4 + 
 32*a^3*c^5)*d^5*x^6 + 1/5*(13*b^7*c + 183*a*b^5*c^2 + 360*a^2*b^3*c^3 + 8 
0*a^3*b*c^4)*d^5*x^5 + 1/4*(b^8 + 36*a*b^6*c + 150*a^2*b^4*c^2 + 80*a^3*b^ 
2*c^3)*d^5*x^4 + 1/3*(3*a*b^7 + 33*a^2*b^5*c + 40*a^3*b^3*c^2)*d^5*x^3 + 1 
/2*(3*a^2*b^6 + 10*a^3*b^4*c)*d^5*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (93) = 186\).

Time = 0.12 (sec) , antiderivative size = 424, normalized size of antiderivative = 4.20 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^3 \, dx=\frac {8}{3} \, c^{8} d^{5} x^{12} + 16 \, b c^{7} d^{5} x^{11} + \frac {208}{5} \, b^{2} c^{6} d^{5} x^{10} + \frac {48}{5} \, a c^{7} d^{5} x^{10} + \frac {184}{3} \, b^{3} c^{5} d^{5} x^{9} + 48 \, a b c^{6} d^{5} x^{9} + \frac {225}{4} \, b^{4} c^{4} d^{5} x^{8} + 102 \, a b^{2} c^{5} d^{5} x^{8} + 12 \, a^{2} c^{6} d^{5} x^{8} + 33 \, b^{5} c^{3} d^{5} x^{7} + 120 \, a b^{3} c^{4} d^{5} x^{7} + 48 \, a^{2} b c^{5} d^{5} x^{7} + \frac {73}{6} \, b^{6} c^{2} d^{5} x^{6} + 85 \, a b^{4} c^{3} d^{5} x^{6} + 80 \, a^{2} b^{2} c^{4} d^{5} x^{6} + \frac {16}{3} \, a^{3} c^{5} d^{5} x^{6} + \frac {13}{5} \, b^{7} c d^{5} x^{5} + \frac {183}{5} \, a b^{5} c^{2} d^{5} x^{5} + 72 \, a^{2} b^{3} c^{3} d^{5} x^{5} + 16 \, a^{3} b c^{4} d^{5} x^{5} + \frac {1}{4} \, b^{8} d^{5} x^{4} + 9 \, a b^{6} c d^{5} x^{4} + \frac {75}{2} \, a^{2} b^{4} c^{2} d^{5} x^{4} + 20 \, a^{3} b^{2} c^{3} d^{5} x^{4} + a b^{7} d^{5} x^{3} + 11 \, a^{2} b^{5} c d^{5} x^{3} + \frac {40}{3} \, a^{3} b^{3} c^{2} d^{5} x^{3} + \frac {3}{2} \, a^{2} b^{6} d^{5} x^{2} + 5 \, a^{3} b^{4} c d^{5} x^{2} + a^{3} b^{5} d^{5} x \] Input:

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

Output:

8/3*c^8*d^5*x^12 + 16*b*c^7*d^5*x^11 + 208/5*b^2*c^6*d^5*x^10 + 48/5*a*c^7 
*d^5*x^10 + 184/3*b^3*c^5*d^5*x^9 + 48*a*b*c^6*d^5*x^9 + 225/4*b^4*c^4*d^5 
*x^8 + 102*a*b^2*c^5*d^5*x^8 + 12*a^2*c^6*d^5*x^8 + 33*b^5*c^3*d^5*x^7 + 1 
20*a*b^3*c^4*d^5*x^7 + 48*a^2*b*c^5*d^5*x^7 + 73/6*b^6*c^2*d^5*x^6 + 85*a* 
b^4*c^3*d^5*x^6 + 80*a^2*b^2*c^4*d^5*x^6 + 16/3*a^3*c^5*d^5*x^6 + 13/5*b^7 
*c*d^5*x^5 + 183/5*a*b^5*c^2*d^5*x^5 + 72*a^2*b^3*c^3*d^5*x^5 + 16*a^3*b*c 
^4*d^5*x^5 + 1/4*b^8*d^5*x^4 + 9*a*b^6*c*d^5*x^4 + 75/2*a^2*b^4*c^2*d^5*x^ 
4 + 20*a^3*b^2*c^3*d^5*x^4 + a*b^7*d^5*x^3 + 11*a^2*b^5*c*d^5*x^3 + 40/3*a 
^3*b^3*c^2*d^5*x^3 + 3/2*a^2*b^6*d^5*x^2 + 5*a^3*b^4*c*d^5*x^2 + a^3*b^5*d 
^5*x
 

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.23 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^3 \, dx=\frac {8\,c^8\,d^5\,x^{12}}{3}+\frac {3\,c^4\,d^5\,x^8\,\left (16\,a^2\,c^2+136\,a\,b^2\,c+75\,b^4\right )}{4}+a^3\,b^5\,d^5\,x+16\,b\,c^7\,d^5\,x^{11}+\frac {b^2\,d^5\,x^4\,\left (80\,a^3\,c^3+150\,a^2\,b^2\,c^2+36\,a\,b^4\,c+b^6\right )}{4}+\frac {16\,c^6\,d^5\,x^{10}\,\left (13\,b^2+3\,a\,c\right )}{5}+\frac {c^2\,d^5\,x^6\,\left (32\,a^3\,c^3+480\,a^2\,b^2\,c^2+510\,a\,b^4\,c+73\,b^6\right )}{6}+\frac {a^2\,b^4\,d^5\,x^2\,\left (3\,b^2+10\,a\,c\right )}{2}+\frac {a\,b^3\,d^5\,x^3\,\left (40\,a^2\,c^2+33\,a\,b^2\,c+3\,b^4\right )}{3}+3\,b\,c^3\,d^5\,x^7\,\left (16\,a^2\,c^2+40\,a\,b^2\,c+11\,b^4\right )+\frac {b\,c\,d^5\,x^5\,\left (80\,a^3\,c^3+360\,a^2\,b^2\,c^2+183\,a\,b^4\,c+13\,b^6\right )}{5}+\frac {8\,b\,c^5\,d^5\,x^9\,\left (23\,b^2+18\,a\,c\right )}{3} \] Input:

int((b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^3,x)
 

Output:

(8*c^8*d^5*x^12)/3 + (3*c^4*d^5*x^8*(75*b^4 + 16*a^2*c^2 + 136*a*b^2*c))/4 
 + a^3*b^5*d^5*x + 16*b*c^7*d^5*x^11 + (b^2*d^5*x^4*(b^6 + 80*a^3*c^3 + 15 
0*a^2*b^2*c^2 + 36*a*b^4*c))/4 + (16*c^6*d^5*x^10*(3*a*c + 13*b^2))/5 + (c 
^2*d^5*x^6*(73*b^6 + 32*a^3*c^3 + 480*a^2*b^2*c^2 + 510*a*b^4*c))/6 + (a^2 
*b^4*d^5*x^2*(10*a*c + 3*b^2))/2 + (a*b^3*d^5*x^3*(3*b^4 + 40*a^2*c^2 + 33 
*a*b^2*c))/3 + 3*b*c^3*d^5*x^7*(11*b^4 + 16*a^2*c^2 + 40*a*b^2*c) + (b*c*d 
^5*x^5*(13*b^6 + 80*a^3*c^3 + 360*a^2*b^2*c^2 + 183*a*b^4*c))/5 + (8*b*c^5 
*d^5*x^9*(18*a*c + 23*b^2))/3
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 337, normalized size of antiderivative = 3.34 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^3 \, dx=\frac {d^{5} x \left (160 c^{8} x^{11}+960 b \,c^{7} x^{10}+576 a \,c^{7} x^{9}+2496 b^{2} c^{6} x^{9}+2880 a b \,c^{6} x^{8}+3680 b^{3} c^{5} x^{8}+720 a^{2} c^{6} x^{7}+6120 a \,b^{2} c^{5} x^{7}+3375 b^{4} c^{4} x^{7}+2880 a^{2} b \,c^{5} x^{6}+7200 a \,b^{3} c^{4} x^{6}+1980 b^{5} c^{3} x^{6}+320 a^{3} c^{5} x^{5}+4800 a^{2} b^{2} c^{4} x^{5}+5100 a \,b^{4} c^{3} x^{5}+730 b^{6} c^{2} x^{5}+960 a^{3} b \,c^{4} x^{4}+4320 a^{2} b^{3} c^{3} x^{4}+2196 a \,b^{5} c^{2} x^{4}+156 b^{7} c \,x^{4}+1200 a^{3} b^{2} c^{3} x^{3}+2250 a^{2} b^{4} c^{2} x^{3}+540 a \,b^{6} c \,x^{3}+15 b^{8} x^{3}+800 a^{3} b^{3} c^{2} x^{2}+660 a^{2} b^{5} c \,x^{2}+60 a \,b^{7} x^{2}+300 a^{3} b^{4} c x +90 a^{2} b^{6} x +60 a^{3} b^{5}\right )}{60} \] Input:

int((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^3,x)
 

Output:

(d**5*x*(60*a**3*b**5 + 300*a**3*b**4*c*x + 800*a**3*b**3*c**2*x**2 + 1200 
*a**3*b**2*c**3*x**3 + 960*a**3*b*c**4*x**4 + 320*a**3*c**5*x**5 + 90*a**2 
*b**6*x + 660*a**2*b**5*c*x**2 + 2250*a**2*b**4*c**2*x**3 + 4320*a**2*b**3 
*c**3*x**4 + 4800*a**2*b**2*c**4*x**5 + 2880*a**2*b*c**5*x**6 + 720*a**2*c 
**6*x**7 + 60*a*b**7*x**2 + 540*a*b**6*c*x**3 + 2196*a*b**5*c**2*x**4 + 51 
00*a*b**4*c**3*x**5 + 7200*a*b**3*c**4*x**6 + 6120*a*b**2*c**5*x**7 + 2880 
*a*b*c**6*x**8 + 576*a*c**7*x**9 + 15*b**8*x**3 + 156*b**7*c*x**4 + 730*b* 
*6*c**2*x**5 + 1980*b**5*c**3*x**6 + 3375*b**4*c**4*x**7 + 3680*b**3*c**5* 
x**8 + 2496*b**2*c**6*x**9 + 960*b*c**7*x**10 + 160*c**8*x**11))/60