\(\int (b d+2 c d x)^4 (a+b x+c x^2)^2 \, dx\) [1122]

Optimal result

Integrand size = 24, antiderivative size = 73 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5}{160 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^7}{112 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3} \]

[Out]

1/160*(-4*a*c+b^2)^2*d^4*(2*c*x+b)^5/c^3-1/112*(-4*a*c+b^2)*d^4*(2*c*x+b)^7/c^3+1/288*d^4*(2*c*x+b)^9/c^3

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^7}{112 c^3}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{160 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3} \]

[In]

Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^2,x]

[Out]

((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^5)/(160*c^3) - ((b^2 - 4*a*c)*d^4*(b + 2*c*x)^7)/(112*c^3) + (d^4*(b + 2*c*x)
^9)/(288*c^3)

Rule 697

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] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^2 (b d+2 c d x)^4}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^6}{8 c^2 d^2}+\frac {(b d+2 c d x)^8}{16 c^2 d^4}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5}{160 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^7}{112 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(179\) vs. \(2(73)=146\).

Time = 0.03 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.45 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=d^4 \left (a^2 b^4 x+a b^3 \left (b^2+4 a c\right ) x^2+\frac {1}{3} b^2 \left (b^4+18 a b^2 c+24 a^2 c^2\right ) x^3+\frac {1}{2} b c \left (5 b^4+32 a b^2 c+16 a^2 c^2\right ) x^4+\frac {1}{5} c^2 \left (41 b^4+112 a b^2 c+16 a^2 c^2\right ) x^5+\frac {4}{3} b c^3 \left (11 b^2+12 a c\right ) x^6+\frac {8}{7} c^4 \left (13 b^2+4 a c\right ) x^7+8 b c^5 x^8+\frac {16 c^6 x^9}{9}\right ) \]

[In]

Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^2,x]

[Out]

d^4*(a^2*b^4*x + a*b^3*(b^2 + 4*a*c)*x^2 + (b^2*(b^4 + 18*a*b^2*c + 24*a^2*c^2)*x^3)/3 + (b*c*(5*b^4 + 32*a*b^
2*c + 16*a^2*c^2)*x^4)/2 + (c^2*(41*b^4 + 112*a*b^2*c + 16*a^2*c^2)*x^5)/5 + (4*b*c^3*(11*b^2 + 12*a*c)*x^6)/3
+ (8*c^4*(13*b^2 + 4*a*c)*x^7)/7 + 8*b*c^5*x^8 + (16*c^6*x^9)/9)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(67)=134\).

Time = 2.51 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.60

method	result	size
gosper	\(\frac {x \left (1120 c^{6} x^{8}+5040 b \,c^{5} x^{7}+2880 x^{6} c^{5} a +9360 x^{6} b^{2} c^{4}+10080 x^{5} a b \,c^{4}+9240 x^{5} b^{3} c^{3}+2016 x^{4} a^{2} c^{4}+14112 a \,b^{2} c^{3} x^{4}+5166 c^{2} x^{4} b^{4}+5040 x^{3} a^{2} b \,c^{3}+10080 x^{3} a \,b^{3} c^{2}+1575 x^{3} b^{5} c +5040 x^{2} a^{2} b^{2} c^{2}+3780 x^{2} a \,b^{4} c +210 x^{2} b^{6}+2520 a^{2} b^{3} c x +630 a \,b^{5} x +630 a^{2} b^{4}\right ) d^{4}}{630}\)	\(190\)
norman	\(\left (\frac {32}{7} a \,c^{5} d^{4}+\frac {104}{7} b^{2} d^{4} c^{4}\right ) x^{7}+\left (16 c^{4} d^{4} a b +\frac {44}{3} b^{3} d^{4} c^{3}\right ) x^{6}+\left (\frac {16}{5} c^{4} d^{4} a^{2}+\frac {112}{5} b^{2} c^{3} d^{4} a +\frac {41}{5} b^{4} d^{4} c^{2}\right ) x^{5}+\left (8 b \,c^{3} d^{4} a^{2}+16 b^{3} d^{4} c^{2} a +\frac {5}{2} b^{5} d^{4} c \right ) x^{4}+\left (8 b^{2} d^{4} c^{2} a^{2}+6 b^{4} d^{4} c a +\frac {1}{3} b^{6} d^{4}\right ) x^{3}+\left (4 b^{3} d^{4} c \,a^{2}+b^{5} d^{4} a \right ) x^{2}+b^{4} d^{4} a^{2} x +\frac {16 c^{6} d^{4} x^{9}}{9}+8 b \,c^{5} d^{4} x^{8}\)	\(226\)
risch	\(\frac {16}{9} c^{6} d^{4} x^{9}+8 b \,c^{5} d^{4} x^{8}+\frac {32}{7} d^{4} x^{7} c^{5} a +\frac {104}{7} d^{4} x^{7} b^{2} c^{4}+16 d^{4} x^{6} a b \,c^{4}+\frac {44}{3} d^{4} x^{6} b^{3} c^{3}+\frac {16}{5} d^{4} x^{5} a^{2} c^{4}+\frac {112}{5} d^{4} x^{5} b^{2} c^{3} a +\frac {41}{5} d^{4} b^{4} c^{2} x^{5}+8 d^{4} a^{2} b \,c^{3} x^{4}+16 d^{4} x^{4} a \,b^{3} c^{2}+\frac {5}{2} d^{4} x^{4} c \,b^{5}+8 d^{4} x^{3} a^{2} b^{2} c^{2}+6 d^{4} a \,b^{4} c \,x^{3}+\frac {1}{3} d^{4} x^{3} b^{6}+4 d^{4} a^{2} b^{3} c \,x^{2}+d^{4} a \,b^{5} x^{2}+b^{4} d^{4} a^{2} x\)	\(241\)
parallelrisch	\(\frac {16}{9} c^{6} d^{4} x^{9}+8 b \,c^{5} d^{4} x^{8}+\frac {32}{7} d^{4} x^{7} c^{5} a +\frac {104}{7} d^{4} x^{7} b^{2} c^{4}+16 d^{4} x^{6} a b \,c^{4}+\frac {44}{3} d^{4} x^{6} b^{3} c^{3}+\frac {16}{5} d^{4} x^{5} a^{2} c^{4}+\frac {112}{5} d^{4} x^{5} b^{2} c^{3} a +\frac {41}{5} d^{4} b^{4} c^{2} x^{5}+8 d^{4} a^{2} b \,c^{3} x^{4}+16 d^{4} x^{4} a \,b^{3} c^{2}+\frac {5}{2} d^{4} x^{4} c \,b^{5}+8 d^{4} x^{3} a^{2} b^{2} c^{2}+6 d^{4} a \,b^{4} c \,x^{3}+\frac {1}{3} d^{4} x^{3} b^{6}+4 d^{4} a^{2} b^{3} c \,x^{2}+d^{4} a \,b^{5} x^{2}+b^{4} d^{4} a^{2} x\)	\(241\)
default	\(\frac {16 c^{6} d^{4} x^{9}}{9}+8 b \,c^{5} d^{4} x^{8}+\frac {\left (88 b^{2} d^{4} c^{4}+16 c^{4} d^{4} \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (56 b^{3} d^{4} c^{3}+32 b \,c^{3} d^{4} \left (2 a c +b^{2}\right )+32 c^{4} d^{4} a b \right ) x^{6}}{6}+\frac {\left (17 b^{4} d^{4} c^{2}+24 b^{2} d^{4} c^{2} \left (2 a c +b^{2}\right )+64 b^{2} c^{3} d^{4} a +16 c^{4} d^{4} a^{2}\right ) x^{5}}{5}+\frac {\left (2 b^{5} d^{4} c +8 b^{3} d^{4} c \left (2 a c +b^{2}\right )+48 b^{3} d^{4} c^{2} a +32 b \,c^{3} d^{4} a^{2}\right ) x^{4}}{4}+\frac {\left (b^{4} d^{4} \left (2 a c +b^{2}\right )+16 b^{4} d^{4} c a +24 b^{2} d^{4} c^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (8 b^{3} d^{4} c \,a^{2}+2 b^{5} d^{4} a \right ) x^{2}}{2}+b^{4} d^{4} a^{2} x\)	\(300\)

[In]

int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/630*x*(1120*c^6*x^8+5040*b*c^5*x^7+2880*a*c^5*x^6+9360*b^2*c^4*x^6+10080*a*b*c^4*x^5+9240*b^3*c^3*x^5+2016*a
^2*c^4*x^4+14112*a*b^2*c^3*x^4+5166*b^4*c^2*x^4+5040*a^2*b*c^3*x^3+10080*a*b^3*c^2*x^3+1575*b^5*c*x^3+5040*a^2
*b^2*c^2*x^2+3780*a*b^4*c*x^2+210*b^6*x^2+2520*a^2*b^3*c*x+630*a*b^5*x+630*a^2*b^4)*d^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (67) = 134\).

Time = 0.27 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.75 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {8}{7} \, {\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{7} + a^{2} b^{4} d^{4} x + \frac {4}{3} \, {\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{5} + \frac {1}{2} \, {\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{4} + \frac {1}{3} \, {\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{3} + {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} x^{2} \]

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

16/9*c^6*d^4*x^9 + 8*b*c^5*d^4*x^8 + 8/7*(13*b^2*c^4 + 4*a*c^5)*d^4*x^7 + a^2*b^4*d^4*x + 4/3*(11*b^3*c^3 + 12
*a*b*c^4)*d^4*x^6 + 1/5*(41*b^4*c^2 + 112*a*b^2*c^3 + 16*a^2*c^4)*d^4*x^5 + 1/2*(5*b^5*c + 32*a*b^3*c^2 + 16*a
^2*b*c^3)*d^4*x^4 + 1/3*(b^6 + 18*a*b^4*c + 24*a^2*b^2*c^2)*d^4*x^3 + (a*b^5 + 4*a^2*b^3*c)*d^4*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (68) = 136\).

Time = 0.04 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.40 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^{2} b^{4} d^{4} x + 8 b c^{5} d^{4} x^{8} + \frac {16 c^{6} d^{4} x^{9}}{9} + x^{7} \cdot \left (\frac {32 a c^{5} d^{4}}{7} + \frac {104 b^{2} c^{4} d^{4}}{7}\right ) + x^{6} \cdot \left (16 a b c^{4} d^{4} + \frac {44 b^{3} c^{3} d^{4}}{3}\right ) + x^{5} \cdot \left (\frac {16 a^{2} c^{4} d^{4}}{5} + \frac {112 a b^{2} c^{3} d^{4}}{5} + \frac {41 b^{4} c^{2} d^{4}}{5}\right ) + x^{4} \cdot \left (8 a^{2} b c^{3} d^{4} + 16 a b^{3} c^{2} d^{4} + \frac {5 b^{5} c d^{4}}{2}\right ) + x^{3} \cdot \left (8 a^{2} b^{2} c^{2} d^{4} + 6 a b^{4} c d^{4} + \frac {b^{6} d^{4}}{3}\right ) + x^{2} \cdot \left (4 a^{2} b^{3} c d^{4} + a b^{5} d^{4}\right ) \]

[In]

integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b**4*d**4*x + 8*b*c**5*d**4*x**8 + 16*c**6*d**4*x**9/9 + x**7*(32*a*c**5*d**4/7 + 104*b**2*c**4*d**4/7) +
x**6*(16*a*b*c**4*d**4 + 44*b**3*c**3*d**4/3) + x**5*(16*a**2*c**4*d**4/5 + 112*a*b**2*c**3*d**4/5 + 41*b**4*
c**2*d**4/5) + x**4*(8*a**2*b*c**3*d**4 + 16*a*b**3*c**2*d**4 + 5*b**5*c*d**4/2) + x**3*(8*a**2*b**2*c**2*d**4
+ 6*a*b**4*c*d**4 + b**6*d**4/3) + x**2*(4*a**2*b**3*c*d**4 + a*b**5*d**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (67) = 134\).

Time = 0.19 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.75 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {8}{7} \, {\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{7} + a^{2} b^{4} d^{4} x + \frac {4}{3} \, {\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{5} + \frac {1}{2} \, {\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{4} + \frac {1}{3} \, {\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{3} + {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} x^{2} \]

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

16/9*c^6*d^4*x^9 + 8*b*c^5*d^4*x^8 + 8/7*(13*b^2*c^4 + 4*a*c^5)*d^4*x^7 + a^2*b^4*d^4*x + 4/3*(11*b^3*c^3 + 12
*a*b*c^4)*d^4*x^6 + 1/5*(41*b^4*c^2 + 112*a*b^2*c^3 + 16*a^2*c^4)*d^4*x^5 + 1/2*(5*b^5*c + 32*a*b^3*c^2 + 16*a
^2*b*c^3)*d^4*x^4 + 1/3*(b^6 + 18*a*b^4*c + 24*a^2*b^2*c^2)*d^4*x^3 + (a*b^5 + 4*a^2*b^3*c)*d^4*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (67) = 134\).

Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.29 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {104}{7} \, b^{2} c^{4} d^{4} x^{7} + \frac {32}{7} \, a c^{5} d^{4} x^{7} + \frac {44}{3} \, b^{3} c^{3} d^{4} x^{6} + 16 \, a b c^{4} d^{4} x^{6} + \frac {41}{5} \, b^{4} c^{2} d^{4} x^{5} + \frac {112}{5} \, a b^{2} c^{3} d^{4} x^{5} + \frac {16}{5} \, a^{2} c^{4} d^{4} x^{5} + \frac {5}{2} \, b^{5} c d^{4} x^{4} + 16 \, a b^{3} c^{2} d^{4} x^{4} + 8 \, a^{2} b c^{3} d^{4} x^{4} + \frac {1}{3} \, b^{6} d^{4} x^{3} + 6 \, a b^{4} c d^{4} x^{3} + 8 \, a^{2} b^{2} c^{2} d^{4} x^{3} + a b^{5} d^{4} x^{2} + 4 \, a^{2} b^{3} c d^{4} x^{2} + a^{2} b^{4} d^{4} x \]

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

16/9*c^6*d^4*x^9 + 8*b*c^5*d^4*x^8 + 104/7*b^2*c^4*d^4*x^7 + 32/7*a*c^5*d^4*x^7 + 44/3*b^3*c^3*d^4*x^6 + 16*a*
b*c^4*d^4*x^6 + 41/5*b^4*c^2*d^4*x^5 + 112/5*a*b^2*c^3*d^4*x^5 + 16/5*a^2*c^4*d^4*x^5 + 5/2*b^5*c*d^4*x^4 + 16
*a*b^3*c^2*d^4*x^4 + 8*a^2*b*c^3*d^4*x^4 + 1/3*b^6*d^4*x^3 + 6*a*b^4*c*d^4*x^3 + 8*a^2*b^2*c^2*d^4*x^3 + a*b^5
*d^4*x^2 + 4*a^2*b^3*c*d^4*x^2 + a^2*b^4*d^4*x

Mupad [B] (verification not implemented)

Time = 10.19 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.60 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {16\,c^6\,d^4\,x^9}{9}+\frac {c^2\,d^4\,x^5\,\left (16\,a^2\,c^2+112\,a\,b^2\,c+41\,b^4\right )}{5}+a^2\,b^4\,d^4\,x+8\,b\,c^5\,d^4\,x^8+\frac {8\,c^4\,d^4\,x^7\,\left (13\,b^2+4\,a\,c\right )}{7}+\frac {b^2\,d^4\,x^3\,\left (24\,a^2\,c^2+18\,a\,b^2\,c+b^4\right )}{3}+\frac {b\,c\,d^4\,x^4\,\left (16\,a^2\,c^2+32\,a\,b^2\,c+5\,b^4\right )}{2}+a\,b^3\,d^4\,x^2\,\left (b^2+4\,a\,c\right )+\frac {4\,b\,c^3\,d^4\,x^6\,\left (11\,b^2+12\,a\,c\right )}{3} \]

[In]

int((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^2,x)

[Out]

(16*c^6*d^4*x^9)/9 + (c^2*d^4*x^5*(41*b^4 + 16*a^2*c^2 + 112*a*b^2*c))/5 + a^2*b^4*d^4*x + 8*b*c^5*d^4*x^8 + (
8*c^4*d^4*x^7*(4*a*c + 13*b^2))/7 + (b^2*d^4*x^3*(b^4 + 24*a^2*c^2 + 18*a*b^2*c))/3 + (b*c*d^4*x^4*(5*b^4 + 16
*a^2*c^2 + 32*a*b^2*c))/2 + a*b^3*d^4*x^2*(4*a*c + b^2) + (4*b*c^3*d^4*x^6*(12*a*c + 11*b^2))/3