Integrand size = 24, antiderivative size = 101 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^5}{640 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^7}{896 c^4}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^9}{384 c^4}+\frac {d^4 (b+2 c x)^{11}}{1408 c^4} \] Output:
-1/640*(-4*a*c+b^2)^3*d^4*(2*c*x+b)^5/c^4+3/896*(-4*a*c+b^2)^2*d^4*(2*c*x+ b)^7/c^4-1/384*(-4*a*c+b^2)*d^4*(2*c*x+b)^9/c^4+1/1408*d^4*(2*c*x+b)^11/c^ 4
Leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(101)=202\).
Time = 0.03 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.56 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=d^4 \left (a^3 b^4 x+\frac {1}{2} a^2 b^3 \left (3 b^2+8 a c\right ) x^2+a b^2 \left (b^4+9 a b^2 c+8 a^2 c^2\right ) x^3+\frac {1}{4} b \left (b^6+30 a b^4 c+96 a^2 b^2 c^2+32 a^3 c^3\right ) x^4+\frac {1}{5} c \left (11 b^6+123 a b^4 c+168 a^2 b^2 c^2+16 a^3 c^3\right ) x^5+\frac {1}{2} b c^2 \left (17 b^4+88 a b^2 c+48 a^2 c^2\right ) x^6+\frac {3}{7} c^3 \left (43 b^4+104 a b^2 c+16 a^2 c^2\right ) x^7+24 b c^4 \left (b^2+a c\right ) x^8+\frac {8}{3} c^5 \left (7 b^2+2 a c\right ) x^9+8 b c^6 x^{10}+\frac {16 c^7 x^{11}}{11}\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3,x]
Output:
d^4*(a^3*b^4*x + (a^2*b^3*(3*b^2 + 8*a*c)*x^2)/2 + a*b^2*(b^4 + 9*a*b^2*c + 8*a^2*c^2)*x^3 + (b*(b^6 + 30*a*b^4*c + 96*a^2*b^2*c^2 + 32*a^3*c^3)*x^4 )/4 + (c*(11*b^6 + 123*a*b^4*c + 168*a^2*b^2*c^2 + 16*a^3*c^3)*x^5)/5 + (b *c^2*(17*b^4 + 88*a*b^2*c + 48*a^2*c^2)*x^6)/2 + (3*c^3*(43*b^4 + 104*a*b^ 2*c + 16*a^2*c^2)*x^7)/7 + 24*b*c^4*(b^2 + a*c)*x^8 + (8*c^5*(7*b^2 + 2*a* c)*x^9)/3 + 8*b*c^6*x^10 + (16*c^7*x^11)/11)
Time = 0.39 (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.
| \(\displaystyle \int \left (a+b x+c x^2\right )^3 (b d+2 c d x)^4 \, dx\) |
| \(\Big \downarrow \) 1107 |
| \(\displaystyle \int \left (\frac {3 \left (4 a c-b^2\right ) (b d+2 c d x)^8}{64 c^3 d^4}+\frac {3 \left (4 a c-b^2\right )^2 (b d+2 c d x)^6}{64 c^3 d^2}+\frac {\left (4 a c-b^2\right )^3 (b d+2 c d x)^4}{64 c^3}+\frac {(b d+2 c d x)^{10}}{64 c^3 d^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^9}{384 c^4}+\frac {3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^7}{896 c^4}-\frac {d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^5}{640 c^4}+\frac {d^4 (b+2 c x)^{11}}{1408 c^4}\) |
Input:
Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3,x]
Output:
-1/640*((b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^5)/c^4 + (3*(b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^7)/(896*c^4) - ((b^2 - 4*a*c)*d^4*(b + 2*c*x)^9)/(384*c^4) + (d^ 4*(b + 2*c*x)^11)/(1408*c^4)
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])
Leaf count of result is larger than twice the leaf count of optimal. \(286\) vs. \(2(93)=186\).
Time = 0.76 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.84
| method | result | size |
| gosper | \(\frac {x \left (6720 c^{7} x^{10}+36960 b \,c^{6} x^{9}+24640 x^{8} a \,c^{6}+86240 x^{8} c^{5} b^{2}+110880 a b \,c^{5} x^{7}+110880 b^{3} c^{4} x^{7}+31680 x^{6} a^{2} c^{5}+205920 x^{6} a \,b^{2} c^{4}+85140 b^{4} c^{3} x^{6}+110880 x^{5} a^{2} b \,c^{4}+203280 a \,b^{3} c^{3} x^{5}+39270 b^{5} c^{2} x^{5}+14784 x^{4} a^{3} c^{4}+155232 a^{2} b^{2} c^{3} x^{4}+113652 x^{4} a \,b^{4} c^{2}+10164 b^{6} c \,x^{4}+36960 x^{3} a^{3} b \,c^{3}+110880 x^{3} a^{2} b^{3} c^{2}+34650 a \,b^{5} c \,x^{3}+1155 x^{3} b^{7}+36960 a^{3} b^{2} c^{2} x^{2}+41580 a^{2} b^{4} c \,x^{2}+4620 a \,b^{6} x^{2}+18480 a^{3} b^{3} c x +6930 a^{2} b^{5} x +4620 a^{3} b^{4}\right ) d^{4}}{4620}\) | \(287\) |
| orering | \(\frac {x \left (6720 c^{7} x^{10}+36960 b \,c^{6} x^{9}+24640 x^{8} a \,c^{6}+86240 x^{8} c^{5} b^{2}+110880 a b \,c^{5} x^{7}+110880 b^{3} c^{4} x^{7}+31680 x^{6} a^{2} c^{5}+205920 x^{6} a \,b^{2} c^{4}+85140 b^{4} c^{3} x^{6}+110880 x^{5} a^{2} b \,c^{4}+203280 a \,b^{3} c^{3} x^{5}+39270 b^{5} c^{2} x^{5}+14784 x^{4} a^{3} c^{4}+155232 a^{2} b^{2} c^{3} x^{4}+113652 x^{4} a \,b^{4} c^{2}+10164 b^{6} c \,x^{4}+36960 x^{3} a^{3} b \,c^{3}+110880 x^{3} a^{2} b^{3} c^{2}+34650 a \,b^{5} c \,x^{3}+1155 x^{3} b^{7}+36960 a^{3} b^{2} c^{2} x^{2}+41580 a^{2} b^{4} c \,x^{2}+4620 a \,b^{6} x^{2}+18480 a^{3} b^{3} c x +6930 a^{2} b^{5} x +4620 a^{3} b^{4}\right ) \left (2 c d x +b d \right )^{4}}{4620 \left (2 c x +b \right )^{4}}\) | \(303\) |
| norman | \(\left (\frac {16}{3} a \,c^{6} d^{4}+\frac {56}{3} b^{2} d^{4} c^{5}\right ) x^{9}+\left (4 b^{3} c \,d^{4} a^{3}+\frac {3}{2} a^{2} b^{5} d^{4}\right ) x^{2}+\left (\frac {48}{7} a^{2} c^{5} d^{4}+\frac {312}{7} a \,b^{2} c^{4} d^{4}+\frac {129}{7} b^{4} c^{3} d^{4}\right ) x^{7}+\left (24 a^{2} c^{4} d^{4} b +44 a \,b^{3} c^{3} d^{4}+\frac {17}{2} b^{5} d^{4} c^{2}\right ) x^{6}+\left (\frac {16}{5} d^{4} c^{4} a^{3}+\frac {168}{5} a^{2} c^{3} d^{4} b^{2}+\frac {123}{5} a \,b^{4} c^{2} d^{4}+\frac {11}{5} b^{6} c \,d^{4}\right ) x^{5}+\left (8 a^{3} b \,c^{3} d^{4}+24 b^{3} d^{4} c^{2} a^{2}+\frac {15}{2} a \,b^{5} c \,d^{4}+\frac {1}{4} b^{7} d^{4}\right ) x^{4}+\left (24 a b \,c^{5} d^{4}+24 b^{3} c^{4} d^{4}\right ) x^{8}+\left (8 b^{2} d^{4} c^{2} a^{3}+9 b^{4} c \,d^{4} a^{2}+a \,b^{6} d^{4}\right ) x^{3}+b^{4} d^{4} a^{3} x +\frac {16 d^{4} c^{7} x^{11}}{11}+8 b \,c^{6} d^{4} x^{10}\) | \(333\) |
| risch | \(24 d^{4} x^{6} a^{2} b \,c^{4}+\frac {168}{5} d^{4} x^{5} a^{2} b^{2} c^{3}+\frac {123}{5} d^{4} x^{5} a \,b^{4} c^{2}+8 d^{4} x^{4} a^{3} b \,c^{3}+24 d^{4} x^{4} a^{2} b^{3} c^{2}+24 d^{4} a b \,c^{5} x^{8}+8 d^{4} a^{3} b^{2} c^{2} x^{3}+4 d^{4} a^{3} b^{3} c \,x^{2}+\frac {15}{2} d^{4} a \,b^{5} c \,x^{4}+9 d^{4} a^{2} b^{4} c \,x^{3}+44 d^{4} a \,b^{3} c^{3} x^{6}+8 b \,c^{6} d^{4} x^{10}+b^{4} d^{4} a^{3} x +\frac {312}{7} d^{4} x^{7} a \,b^{2} c^{4}+\frac {1}{4} d^{4} x^{4} b^{7}+\frac {129}{7} d^{4} b^{4} c^{3} x^{7}+\frac {3}{2} d^{4} x^{2} a^{2} b^{5}+24 d^{4} b^{3} c^{4} x^{8}+d^{4} a \,b^{6} x^{3}+\frac {17}{2} d^{4} x^{6} b^{5} c^{2}+\frac {16}{5} d^{4} x^{5} a^{3} c^{4}+\frac {11}{5} d^{4} b^{6} c \,x^{5}+\frac {16}{3} d^{4} x^{9} a \,c^{6}+\frac {56}{3} d^{4} x^{9} c^{5} b^{2}+\frac {48}{7} d^{4} x^{7} a^{2} c^{5}+\frac {16}{11} d^{4} c^{7} x^{11}\) | \(362\) |
| parallelrisch | \(24 d^{4} x^{6} a^{2} b \,c^{4}+\frac {168}{5} d^{4} x^{5} a^{2} b^{2} c^{3}+\frac {123}{5} d^{4} x^{5} a \,b^{4} c^{2}+8 d^{4} x^{4} a^{3} b \,c^{3}+24 d^{4} x^{4} a^{2} b^{3} c^{2}+24 d^{4} a b \,c^{5} x^{8}+8 d^{4} a^{3} b^{2} c^{2} x^{3}+4 d^{4} a^{3} b^{3} c \,x^{2}+\frac {15}{2} d^{4} a \,b^{5} c \,x^{4}+9 d^{4} a^{2} b^{4} c \,x^{3}+44 d^{4} a \,b^{3} c^{3} x^{6}+8 b \,c^{6} d^{4} x^{10}+b^{4} d^{4} a^{3} x +\frac {312}{7} d^{4} x^{7} a \,b^{2} c^{4}+\frac {1}{4} d^{4} x^{4} b^{7}+\frac {129}{7} d^{4} b^{4} c^{3} x^{7}+\frac {3}{2} d^{4} x^{2} a^{2} b^{5}+24 d^{4} b^{3} c^{4} x^{8}+d^{4} a \,b^{6} x^{3}+\frac {17}{2} d^{4} x^{6} b^{5} c^{2}+\frac {16}{5} d^{4} x^{5} a^{3} c^{4}+\frac {11}{5} d^{4} b^{6} c \,x^{5}+\frac {16}{3} d^{4} x^{9} a \,c^{6}+\frac {56}{3} d^{4} x^{9} c^{5} b^{2}+\frac {48}{7} d^{4} x^{7} a^{2} c^{5}+\frac {16}{11} d^{4} c^{7} x^{11}\) | \(362\) |
| default | \(\frac {16 d^{4} c^{7} x^{11}}{11}+8 b \,c^{6} d^{4} x^{10}+\frac {\left (120 b^{2} d^{4} c^{5}+16 d^{4} c^{4} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{9}}{9}+\frac {\left (80 b^{3} c^{4} d^{4}+32 b \,c^{3} d^{4} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+16 d^{4} c^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{8}}{8}+\frac {\left (25 b^{4} c^{3} d^{4}+24 b^{2} d^{4} c^{2} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+32 b \,c^{3} d^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+16 d^{4} c^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{7}}{7}+\frac {\left (3 b^{5} d^{4} c^{2}+8 b^{3} c \,d^{4} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+24 b^{2} d^{4} c^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+32 b \,c^{3} d^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+48 a^{2} c^{4} d^{4} b \right ) x^{6}}{6}+\frac {\left (b^{4} d^{4} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+8 b^{3} c \,d^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+24 b^{2} d^{4} c^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+96 a^{2} c^{3} d^{4} b^{2}+16 d^{4} c^{4} a^{3}\right ) x^{5}}{5}+\frac {\left (b^{4} d^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+8 b^{3} c \,d^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+72 b^{3} d^{4} c^{2} a^{2}+32 a^{3} b \,c^{3} d^{4}\right ) x^{4}}{4}+\frac {\left (b^{4} d^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+24 b^{4} c \,d^{4} a^{2}+24 b^{2} d^{4} c^{2} a^{3}\right ) x^{3}}{3}+\frac {\left (8 b^{3} c \,d^{4} a^{3}+3 a^{2} b^{5} d^{4}\right ) x^{2}}{2}+b^{4} d^{4} a^{3} x\) | \(672\) |
Input:
int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/4620*x*(6720*c^7*x^10+36960*b*c^6*x^9+24640*a*c^6*x^8+86240*b^2*c^5*x^8+ 110880*a*b*c^5*x^7+110880*b^3*c^4*x^7+31680*a^2*c^5*x^6+205920*a*b^2*c^4*x ^6+85140*b^4*c^3*x^6+110880*a^2*b*c^4*x^5+203280*a*b^3*c^3*x^5+39270*b^5*c ^2*x^5+14784*a^3*c^4*x^4+155232*a^2*b^2*c^3*x^4+113652*a*b^4*c^2*x^4+10164 *b^6*c*x^4+36960*a^3*b*c^3*x^3+110880*a^2*b^3*c^2*x^3+34650*a*b^5*c*x^3+11 55*b^7*x^3+36960*a^3*b^2*c^2*x^2+41580*a^2*b^4*c*x^2+4620*a*b^6*x^2+18480* a^3*b^3*c*x+6930*a^2*b^5*x+4620*a^3*b^4)*d^4
Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (93) = 186\).
Time = 0.08 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.87 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac {8}{3} \, {\left (7 \, b^{2} c^{5} + 2 \, a c^{6}\right )} d^{4} x^{9} + 24 \, {\left (b^{3} c^{4} + a b c^{5}\right )} d^{4} x^{8} + a^{3} b^{4} d^{4} x + \frac {3}{7} \, {\left (43 \, b^{4} c^{3} + 104 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{7} + \frac {1}{2} \, {\left (17 \, b^{5} c^{2} + 88 \, a b^{3} c^{3} + 48 \, a^{2} b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (11 \, b^{6} c + 123 \, a b^{4} c^{2} + 168 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (b^{7} + 30 \, a b^{5} c + 96 \, a^{2} b^{3} c^{2} + 32 \, a^{3} b c^{3}\right )} d^{4} x^{4} + {\left (a b^{6} + 9 \, a^{2} b^{4} c + 8 \, a^{3} b^{2} c^{2}\right )} d^{4} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{5} + 8 \, a^{3} b^{3} c\right )} d^{4} x^{2} \] Input:
integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
16/11*c^7*d^4*x^11 + 8*b*c^6*d^4*x^10 + 8/3*(7*b^2*c^5 + 2*a*c^6)*d^4*x^9 + 24*(b^3*c^4 + a*b*c^5)*d^4*x^8 + a^3*b^4*d^4*x + 3/7*(43*b^4*c^3 + 104*a *b^2*c^4 + 16*a^2*c^5)*d^4*x^7 + 1/2*(17*b^5*c^2 + 88*a*b^3*c^3 + 48*a^2*b *c^4)*d^4*x^6 + 1/5*(11*b^6*c + 123*a*b^4*c^2 + 168*a^2*b^2*c^3 + 16*a^3*c ^4)*d^4*x^5 + 1/4*(b^7 + 30*a*b^5*c + 96*a^2*b^3*c^2 + 32*a^3*b*c^3)*d^4*x ^4 + (a*b^6 + 9*a^2*b^4*c + 8*a^3*b^2*c^2)*d^4*x^3 + 1/2*(3*a^2*b^5 + 8*a^ 3*b^3*c)*d^4*x^2
Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (97) = 194\).
Time = 0.05 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.67 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=a^{3} b^{4} d^{4} x + 8 b c^{6} d^{4} x^{10} + \frac {16 c^{7} d^{4} x^{11}}{11} + x^{9} \cdot \left (\frac {16 a c^{6} d^{4}}{3} + \frac {56 b^{2} c^{5} d^{4}}{3}\right ) + x^{8} \cdot \left (24 a b c^{5} d^{4} + 24 b^{3} c^{4} d^{4}\right ) + x^{7} \cdot \left (\frac {48 a^{2} c^{5} d^{4}}{7} + \frac {312 a b^{2} c^{4} d^{4}}{7} + \frac {129 b^{4} c^{3} d^{4}}{7}\right ) + x^{6} \cdot \left (24 a^{2} b c^{4} d^{4} + 44 a b^{3} c^{3} d^{4} + \frac {17 b^{5} c^{2} d^{4}}{2}\right ) + x^{5} \cdot \left (\frac {16 a^{3} c^{4} d^{4}}{5} + \frac {168 a^{2} b^{2} c^{3} d^{4}}{5} + \frac {123 a b^{4} c^{2} d^{4}}{5} + \frac {11 b^{6} c d^{4}}{5}\right ) + x^{4} \cdot \left (8 a^{3} b c^{3} d^{4} + 24 a^{2} b^{3} c^{2} d^{4} + \frac {15 a b^{5} c d^{4}}{2} + \frac {b^{7} d^{4}}{4}\right ) + x^{3} \cdot \left (8 a^{3} b^{2} c^{2} d^{4} + 9 a^{2} b^{4} c d^{4} + a b^{6} d^{4}\right ) + x^{2} \cdot \left (4 a^{3} b^{3} c d^{4} + \frac {3 a^{2} b^{5} d^{4}}{2}\right ) \] Input:
integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**3,x)
Output:
a**3*b**4*d**4*x + 8*b*c**6*d**4*x**10 + 16*c**7*d**4*x**11/11 + x**9*(16* a*c**6*d**4/3 + 56*b**2*c**5*d**4/3) + x**8*(24*a*b*c**5*d**4 + 24*b**3*c* *4*d**4) + x**7*(48*a**2*c**5*d**4/7 + 312*a*b**2*c**4*d**4/7 + 129*b**4*c **3*d**4/7) + x**6*(24*a**2*b*c**4*d**4 + 44*a*b**3*c**3*d**4 + 17*b**5*c* *2*d**4/2) + x**5*(16*a**3*c**4*d**4/5 + 168*a**2*b**2*c**3*d**4/5 + 123*a *b**4*c**2*d**4/5 + 11*b**6*c*d**4/5) + x**4*(8*a**3*b*c**3*d**4 + 24*a**2 *b**3*c**2*d**4 + 15*a*b**5*c*d**4/2 + b**7*d**4/4) + x**3*(8*a**3*b**2*c* *2*d**4 + 9*a**2*b**4*c*d**4 + a*b**6*d**4) + x**2*(4*a**3*b**3*c*d**4 + 3 *a**2*b**5*d**4/2)
Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (93) = 186\).
Time = 0.03 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.87 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac {8}{3} \, {\left (7 \, b^{2} c^{5} + 2 \, a c^{6}\right )} d^{4} x^{9} + 24 \, {\left (b^{3} c^{4} + a b c^{5}\right )} d^{4} x^{8} + a^{3} b^{4} d^{4} x + \frac {3}{7} \, {\left (43 \, b^{4} c^{3} + 104 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{7} + \frac {1}{2} \, {\left (17 \, b^{5} c^{2} + 88 \, a b^{3} c^{3} + 48 \, a^{2} b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (11 \, b^{6} c + 123 \, a b^{4} c^{2} + 168 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (b^{7} + 30 \, a b^{5} c + 96 \, a^{2} b^{3} c^{2} + 32 \, a^{3} b c^{3}\right )} d^{4} x^{4} + {\left (a b^{6} + 9 \, a^{2} b^{4} c + 8 \, a^{3} b^{2} c^{2}\right )} d^{4} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{5} + 8 \, a^{3} b^{3} c\right )} d^{4} x^{2} \] Input:
integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
16/11*c^7*d^4*x^11 + 8*b*c^6*d^4*x^10 + 8/3*(7*b^2*c^5 + 2*a*c^6)*d^4*x^9 + 24*(b^3*c^4 + a*b*c^5)*d^4*x^8 + a^3*b^4*d^4*x + 3/7*(43*b^4*c^3 + 104*a *b^2*c^4 + 16*a^2*c^5)*d^4*x^7 + 1/2*(17*b^5*c^2 + 88*a*b^3*c^3 + 48*a^2*b *c^4)*d^4*x^6 + 1/5*(11*b^6*c + 123*a*b^4*c^2 + 168*a^2*b^2*c^3 + 16*a^3*c ^4)*d^4*x^5 + 1/4*(b^7 + 30*a*b^5*c + 96*a^2*b^3*c^2 + 32*a^3*b*c^3)*d^4*x ^4 + (a*b^6 + 9*a^2*b^4*c + 8*a^3*b^2*c^2)*d^4*x^3 + 1/2*(3*a^2*b^5 + 8*a^ 3*b^3*c)*d^4*x^2
Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (93) = 186\).
Time = 0.13 (sec) , antiderivative size = 361, normalized size of antiderivative = 3.57 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac {56}{3} \, b^{2} c^{5} d^{4} x^{9} + \frac {16}{3} \, a c^{6} d^{4} x^{9} + 24 \, b^{3} c^{4} d^{4} x^{8} + 24 \, a b c^{5} d^{4} x^{8} + \frac {129}{7} \, b^{4} c^{3} d^{4} x^{7} + \frac {312}{7} \, a b^{2} c^{4} d^{4} x^{7} + \frac {48}{7} \, a^{2} c^{5} d^{4} x^{7} + \frac {17}{2} \, b^{5} c^{2} d^{4} x^{6} + 44 \, a b^{3} c^{3} d^{4} x^{6} + 24 \, a^{2} b c^{4} d^{4} x^{6} + \frac {11}{5} \, b^{6} c d^{4} x^{5} + \frac {123}{5} \, a b^{4} c^{2} d^{4} x^{5} + \frac {168}{5} \, a^{2} b^{2} c^{3} d^{4} x^{5} + \frac {16}{5} \, a^{3} c^{4} d^{4} x^{5} + \frac {1}{4} \, b^{7} d^{4} x^{4} + \frac {15}{2} \, a b^{5} c d^{4} x^{4} + 24 \, a^{2} b^{3} c^{2} d^{4} x^{4} + 8 \, a^{3} b c^{3} d^{4} x^{4} + a b^{6} d^{4} x^{3} + 9 \, a^{2} b^{4} c d^{4} x^{3} + 8 \, a^{3} b^{2} c^{2} d^{4} x^{3} + \frac {3}{2} \, a^{2} b^{5} d^{4} x^{2} + 4 \, a^{3} b^{3} c d^{4} x^{2} + a^{3} b^{4} d^{4} x \] Input:
integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
16/11*c^7*d^4*x^11 + 8*b*c^6*d^4*x^10 + 56/3*b^2*c^5*d^4*x^9 + 16/3*a*c^6* d^4*x^9 + 24*b^3*c^4*d^4*x^8 + 24*a*b*c^5*d^4*x^8 + 129/7*b^4*c^3*d^4*x^7 + 312/7*a*b^2*c^4*d^4*x^7 + 48/7*a^2*c^5*d^4*x^7 + 17/2*b^5*c^2*d^4*x^6 + 44*a*b^3*c^3*d^4*x^6 + 24*a^2*b*c^4*d^4*x^6 + 11/5*b^6*c*d^4*x^5 + 123/5*a *b^4*c^2*d^4*x^5 + 168/5*a^2*b^2*c^3*d^4*x^5 + 16/5*a^3*c^4*d^4*x^5 + 1/4* b^7*d^4*x^4 + 15/2*a*b^5*c*d^4*x^4 + 24*a^2*b^3*c^2*d^4*x^4 + 8*a^3*b*c^3* d^4*x^4 + a*b^6*d^4*x^3 + 9*a^2*b^4*c*d^4*x^3 + 8*a^3*b^2*c^2*d^4*x^3 + 3/ 2*a^2*b^5*d^4*x^2 + 4*a^3*b^3*c*d^4*x^2 + a^3*b^4*d^4*x
Time = 5.63 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.71 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {16\,c^7\,d^4\,x^{11}}{11}+\frac {3\,c^3\,d^4\,x^7\,\left (16\,a^2\,c^2+104\,a\,b^2\,c+43\,b^4\right )}{7}+a^3\,b^4\,d^4\,x+8\,b\,c^6\,d^4\,x^{10}+\frac {c\,d^4\,x^5\,\left (16\,a^3\,c^3+168\,a^2\,b^2\,c^2+123\,a\,b^4\,c+11\,b^6\right )}{5}+\frac {8\,c^5\,d^4\,x^9\,\left (7\,b^2+2\,a\,c\right )}{3}+\frac {b\,d^4\,x^4\,\left (32\,a^3\,c^3+96\,a^2\,b^2\,c^2+30\,a\,b^4\,c+b^6\right )}{4}+a\,b^2\,d^4\,x^3\,\left (8\,a^2\,c^2+9\,a\,b^2\,c+b^4\right )+\frac {a^2\,b^3\,d^4\,x^2\,\left (3\,b^2+8\,a\,c\right )}{2}+24\,b\,c^4\,d^4\,x^8\,\left (b^2+a\,c\right )+\frac {b\,c^2\,d^4\,x^6\,\left (48\,a^2\,c^2+88\,a\,b^2\,c+17\,b^4\right )}{2} \] Input:
int((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3,x)
Output:
(16*c^7*d^4*x^11)/11 + (3*c^3*d^4*x^7*(43*b^4 + 16*a^2*c^2 + 104*a*b^2*c)) /7 + a^3*b^4*d^4*x + 8*b*c^6*d^4*x^10 + (c*d^4*x^5*(11*b^6 + 16*a^3*c^3 + 168*a^2*b^2*c^2 + 123*a*b^4*c))/5 + (8*c^5*d^4*x^9*(2*a*c + 7*b^2))/3 + (b *d^4*x^4*(b^6 + 32*a^3*c^3 + 96*a^2*b^2*c^2 + 30*a*b^4*c))/4 + a*b^2*d^4*x ^3*(b^4 + 8*a^2*c^2 + 9*a*b^2*c) + (a^2*b^3*d^4*x^2*(8*a*c + 3*b^2))/2 + 2 4*b*c^4*d^4*x^8*(a*c + b^2) + (b*c^2*d^4*x^6*(17*b^4 + 48*a^2*c^2 + 88*a*b ^2*c))/2
Time = 0.20 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.83 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {d^{4} x \left (6720 c^{7} x^{10}+36960 b \,c^{6} x^{9}+24640 a \,c^{6} x^{8}+86240 b^{2} c^{5} x^{8}+110880 a b \,c^{5} x^{7}+110880 b^{3} c^{4} x^{7}+31680 a^{2} c^{5} x^{6}+205920 a \,b^{2} c^{4} x^{6}+85140 b^{4} c^{3} x^{6}+110880 a^{2} b \,c^{4} x^{5}+203280 a \,b^{3} c^{3} x^{5}+39270 b^{5} c^{2} x^{5}+14784 a^{3} c^{4} x^{4}+155232 a^{2} b^{2} c^{3} x^{4}+113652 a \,b^{4} c^{2} x^{4}+10164 b^{6} c \,x^{4}+36960 a^{3} b \,c^{3} x^{3}+110880 a^{2} b^{3} c^{2} x^{3}+34650 a \,b^{5} c \,x^{3}+1155 b^{7} x^{3}+36960 a^{3} b^{2} c^{2} x^{2}+41580 a^{2} b^{4} c \,x^{2}+4620 a \,b^{6} x^{2}+18480 a^{3} b^{3} c x +6930 a^{2} b^{5} x +4620 a^{3} b^{4}\right )}{4620} \] Input:
int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^3,x)
Output:
(d**4*x*(4620*a**3*b**4 + 18480*a**3*b**3*c*x + 36960*a**3*b**2*c**2*x**2 + 36960*a**3*b*c**3*x**3 + 14784*a**3*c**4*x**4 + 6930*a**2*b**5*x + 41580 *a**2*b**4*c*x**2 + 110880*a**2*b**3*c**2*x**3 + 155232*a**2*b**2*c**3*x** 4 + 110880*a**2*b*c**4*x**5 + 31680*a**2*c**5*x**6 + 4620*a*b**6*x**2 + 34 650*a*b**5*c*x**3 + 113652*a*b**4*c**2*x**4 + 203280*a*b**3*c**3*x**5 + 20 5920*a*b**2*c**4*x**6 + 110880*a*b*c**5*x**7 + 24640*a*c**6*x**8 + 1155*b* *7*x**3 + 10164*b**6*c*x**4 + 39270*b**5*c**2*x**5 + 85140*b**4*c**3*x**6 + 110880*b**3*c**4*x**7 + 86240*b**2*c**5*x**8 + 36960*b*c**6*x**9 + 6720* c**7*x**10))/4620