\(\int (b d+2 c d x)^2 (a+b x+c x^2)^3 \, dx\) [1140]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx=-\frac {\left (b^2-4 a c\right )^3 d^2 (b+2 c x)^3}{384 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^5}{640 c^4}-\frac {3 \left (b^2-4 a c\right ) d^2 (b+2 c x)^7}{896 c^4}+\frac {d^2 (b+2 c x)^9}{1152 c^4} \]

[Out]

-1/384*(-4*a*c+b^2)^3*d^2*(2*c*x+b)^3/c^4+3/640*(-4*a*c+b^2)^2*d^2*(2*c*x+b)^5/c^4-3/896*(-4*a*c+b^2)*d^2*(2*c
*x+b)^7/c^4+1/1152*d^2*(2*c*x+b)^9/c^4

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 101, 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} \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx=-\frac {3 d^2 \left (b^2-4 a c\right ) (b+2 c x)^7}{896 c^4}+\frac {3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{640 c^4}-\frac {d^2 \left (b^2-4 a c\right )^3 (b+2 c x)^3}{384 c^4}+\frac {d^2 (b+2 c x)^9}{1152 c^4} \]

[In]

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

[Out]

-1/384*((b^2 - 4*a*c)^3*d^2*(b + 2*c*x)^3)/c^4 + (3*(b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^5)/(640*c^4) - (3*(b^2 - 4
*a*c)*d^2*(b + 2*c*x)^7)/(896*c^4) + (d^2*(b + 2*c*x)^9)/(1152*c^4)

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 )^3 (b d+2 c d x)^2}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^4}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^6}{64 c^3 d^4}+\frac {(b d+2 c d x)^8}{64 c^3 d^6}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right )^3 d^2 (b+2 c x)^3}{384 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^5}{640 c^4}-\frac {3 \left (b^2-4 a c\right ) d^2 (b+2 c x)^7}{896 c^4}+\frac {d^2 (b+2 c x)^9}{1152 c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.77 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx=d^2 \left (a^3 b^2 x+\frac {1}{2} a^2 b \left (3 b^2+4 a c\right ) x^2+\frac {1}{3} a \left (3 b^4+15 a b^2 c+4 a^2 c^2\right ) x^3+\frac {1}{4} b \left (b^4+18 a b^2 c+24 a^2 c^2\right ) x^4+\frac {1}{5} c \left (7 b^4+39 a b^2 c+12 a^2 c^2\right ) x^5+\frac {1}{6} b c^2 \left (19 b^2+36 a c\right ) x^6+\frac {1}{7} c^3 \left (25 b^2+12 a c\right ) x^7+2 b c^4 x^8+\frac {4 c^5 x^9}{9}\right ) \]

[In]

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

[Out]

d^2*(a^3*b^2*x + (a^2*b*(3*b^2 + 4*a*c)*x^2)/2 + (a*(3*b^4 + 15*a*b^2*c + 4*a^2*c^2)*x^3)/3 + (b*(b^4 + 18*a*b
^2*c + 24*a^2*c^2)*x^4)/4 + (c*(7*b^4 + 39*a*b^2*c + 12*a^2*c^2)*x^5)/5 + (b*c^2*(19*b^2 + 36*a*c)*x^6)/6 + (c
^3*(25*b^2 + 12*a*c)*x^7)/7 + 2*b*c^4*x^8 + (4*c^5*x^9)/9)

Maple [A] (verified)

Time = 2.53 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.83

method	result	size
gosper	\(\frac {x \left (560 c^{5} x^{8}+2520 b \,c^{4} x^{7}+2160 x^{6} a \,c^{4}+4500 x^{6} b^{2} c^{3}+7560 x^{5} a b \,c^{3}+3990 c^{2} x^{5} b^{3}+3024 a^{2} c^{3} x^{4}+9828 x^{4} b^{2} c^{2} a +1764 b^{4} c \,x^{4}+7560 a^{2} b \,c^{2} x^{3}+5670 x^{3} a \,b^{3} c +315 x^{3} b^{5}+1680 x^{2} a^{3} c^{2}+6300 c \,x^{2} a^{2} b^{2}+1260 b^{4} x^{2} a +2520 a^{3} b c x +1890 a^{2} b^{3} x +1260 a^{3} b^{2}\right ) d^{2}}{1260}\)	\(185\)
norman	\(\left (\frac {12}{7} a \,c^{4} d^{2}+\frac {25}{7} b^{2} c^{3} d^{2}\right ) x^{7}+\left (6 a b \,c^{3} d^{2}+\frac {19}{6} b^{3} d^{2} c^{2}\right ) x^{6}+\left (2 a^{3} b c \,d^{2}+\frac {3}{2} a^{2} b^{3} d^{2}\right ) x^{2}+\left (\frac {12}{5} a^{2} c^{3} d^{2}+\frac {39}{5} a \,c^{2} d^{2} b^{2}+\frac {7}{5} b^{4} c \,d^{2}\right ) x^{5}+\left (\frac {4}{3} c^{2} d^{2} a^{3}+5 b^{2} c \,d^{2} a^{2}+a \,b^{4} d^{2}\right ) x^{3}+\left (6 a^{2} b \,c^{2} d^{2}+\frac {9}{2} a \,b^{3} c \,d^{2}+\frac {1}{4} b^{5} d^{2}\right ) x^{4}+b^{2} d^{2} a^{3} x +\frac {4 c^{5} d^{2} x^{9}}{9}+2 c^{4} b \,d^{2} x^{8}\)	\(221\)
risch	\(\frac {4}{9} c^{5} d^{2} x^{9}+2 c^{4} b \,d^{2} x^{8}+\frac {12}{7} d^{2} a \,c^{4} x^{7}+\frac {25}{7} d^{2} x^{7} b^{2} c^{3}+6 d^{2} x^{6} a b \,c^{3}+\frac {19}{6} b^{3} c^{2} d^{2} x^{6}+\frac {12}{5} d^{2} a^{2} c^{3} x^{5}+\frac {39}{5} d^{2} x^{5} b^{2} c^{2} a +\frac {7}{5} d^{2} x^{5} b^{4} c +6 d^{2} a^{2} b \,c^{2} x^{4}+\frac {9}{2} d^{2} x^{4} a \,b^{3} c +\frac {1}{4} d^{2} b^{5} x^{4}+\frac {4}{3} d^{2} x^{3} a^{3} c^{2}+5 a^{2} b^{2} c \,d^{2} x^{3}+d^{2} a \,b^{4} x^{3}+2 d^{2} a^{3} b c \,x^{2}+\frac {3}{2} d^{2} a^{2} b^{3} x^{2}+b^{2} d^{2} a^{3} x\)	\(236\)
parallelrisch	\(\frac {4}{9} c^{5} d^{2} x^{9}+2 c^{4} b \,d^{2} x^{8}+\frac {12}{7} d^{2} a \,c^{4} x^{7}+\frac {25}{7} d^{2} x^{7} b^{2} c^{3}+6 d^{2} x^{6} a b \,c^{3}+\frac {19}{6} b^{3} c^{2} d^{2} x^{6}+\frac {12}{5} d^{2} a^{2} c^{3} x^{5}+\frac {39}{5} d^{2} x^{5} b^{2} c^{2} a +\frac {7}{5} d^{2} x^{5} b^{4} c +6 d^{2} a^{2} b \,c^{2} x^{4}+\frac {9}{2} d^{2} x^{4} a \,b^{3} c +\frac {1}{4} d^{2} b^{5} x^{4}+\frac {4}{3} d^{2} x^{3} a^{3} c^{2}+5 a^{2} b^{2} c \,d^{2} x^{3}+d^{2} a \,b^{4} x^{3}+2 d^{2} a^{3} b c \,x^{2}+\frac {3}{2} d^{2} a^{2} b^{3} x^{2}+b^{2} d^{2} a^{3} x\)	\(236\)
default	\(\frac {4 c^{5} d^{2} x^{9}}{9}+2 c^{4} b \,d^{2} x^{8}+\frac {\left (13 b^{2} c^{3} d^{2}+4 c^{2} d^{2} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 b^{3} d^{2} c^{2}+4 b c \,d^{2} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+4 c^{2} d^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (b^{2} d^{2} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+4 b c \,d^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+4 c^{2} d^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{5}}{5}+\frac {\left (b^{2} d^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+4 b c \,d^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+12 a^{2} b \,c^{2} d^{2}\right ) x^{4}}{4}+\frac {\left (b^{2} d^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+12 b^{2} c \,d^{2} a^{2}+4 c^{2} d^{2} a^{3}\right ) x^{3}}{3}+\frac {\left (4 a^{3} b c \,d^{2}+3 a^{2} b^{3} d^{2}\right ) x^{2}}{2}+b^{2} d^{2} a^{3} x\)	\(396\)

[In]

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

[Out]

1/1260*x*(560*c^5*x^8+2520*b*c^4*x^7+2160*a*c^4*x^6+4500*b^2*c^3*x^6+7560*a*b*c^3*x^5+3990*b^3*c^2*x^5+3024*a^
2*c^3*x^4+9828*a*b^2*c^2*x^4+1764*b^4*c*x^4+7560*a^2*b*c^2*x^3+5670*a*b^3*c*x^3+315*b^5*x^3+1680*a^3*c^2*x^2+6
300*a^2*b^2*c*x^2+1260*a*b^4*x^2+2520*a^3*b*c*x+1890*a^2*b^3*x+1260*a^3*b^2)*d^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (93) = 186\).

Time = 0.39 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.96 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx=\frac {4}{9} \, c^{5} d^{2} x^{9} + 2 \, b c^{4} d^{2} x^{8} + \frac {1}{7} \, {\left (25 \, b^{2} c^{3} + 12 \, a c^{4}\right )} d^{2} x^{7} + \frac {1}{6} \, {\left (19 \, b^{3} c^{2} + 36 \, a b c^{3}\right )} d^{2} x^{6} + a^{3} b^{2} d^{2} x + \frac {1}{5} \, {\left (7 \, b^{4} c + 39 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{2} x^{5} + \frac {1}{4} \, {\left (b^{5} + 18 \, a b^{3} c + 24 \, a^{2} b c^{2}\right )} d^{2} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{4} + 15 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{2} x^{2} \]

[In]

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

[Out]

4/9*c^5*d^2*x^9 + 2*b*c^4*d^2*x^8 + 1/7*(25*b^2*c^3 + 12*a*c^4)*d^2*x^7 + 1/6*(19*b^3*c^2 + 36*a*b*c^3)*d^2*x^
6 + a^3*b^2*d^2*x + 1/5*(7*b^4*c + 39*a*b^2*c^2 + 12*a^2*c^3)*d^2*x^5 + 1/4*(b^5 + 18*a*b^3*c + 24*a^2*b*c^2)*
d^2*x^4 + 1/3*(3*a*b^4 + 15*a^2*b^2*c + 4*a^3*c^2)*d^2*x^3 + 1/2*(3*a^2*b^3 + 4*a^3*b*c)*d^2*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (100) = 200\).

Time = 0.04 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.44 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx=a^{3} b^{2} d^{2} x + 2 b c^{4} d^{2} x^{8} + \frac {4 c^{5} d^{2} x^{9}}{9} + x^{7} \cdot \left (\frac {12 a c^{4} d^{2}}{7} + \frac {25 b^{2} c^{3} d^{2}}{7}\right ) + x^{6} \cdot \left (6 a b c^{3} d^{2} + \frac {19 b^{3} c^{2} d^{2}}{6}\right ) + x^{5} \cdot \left (\frac {12 a^{2} c^{3} d^{2}}{5} + \frac {39 a b^{2} c^{2} d^{2}}{5} + \frac {7 b^{4} c d^{2}}{5}\right ) + x^{4} \cdot \left (6 a^{2} b c^{2} d^{2} + \frac {9 a b^{3} c d^{2}}{2} + \frac {b^{5} d^{2}}{4}\right ) + x^{3} \cdot \left (\frac {4 a^{3} c^{2} d^{2}}{3} + 5 a^{2} b^{2} c d^{2} + a b^{4} d^{2}\right ) + x^{2} \cdot \left (2 a^{3} b c d^{2} + \frac {3 a^{2} b^{3} d^{2}}{2}\right ) \]

[In]

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

[Out]

a**3*b**2*d**2*x + 2*b*c**4*d**2*x**8 + 4*c**5*d**2*x**9/9 + x**7*(12*a*c**4*d**2/7 + 25*b**2*c**3*d**2/7) + x
**6*(6*a*b*c**3*d**2 + 19*b**3*c**2*d**2/6) + x**5*(12*a**2*c**3*d**2/5 + 39*a*b**2*c**2*d**2/5 + 7*b**4*c*d**
2/5) + x**4*(6*a**2*b*c**2*d**2 + 9*a*b**3*c*d**2/2 + b**5*d**2/4) + x**3*(4*a**3*c**2*d**2/3 + 5*a**2*b**2*c*
d**2 + a*b**4*d**2) + x**2*(2*a**3*b*c*d**2 + 3*a**2*b**3*d**2/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (93) = 186\).

Time = 0.20 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.96 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx=\frac {4}{9} \, c^{5} d^{2} x^{9} + 2 \, b c^{4} d^{2} x^{8} + \frac {1}{7} \, {\left (25 \, b^{2} c^{3} + 12 \, a c^{4}\right )} d^{2} x^{7} + \frac {1}{6} \, {\left (19 \, b^{3} c^{2} + 36 \, a b c^{3}\right )} d^{2} x^{6} + a^{3} b^{2} d^{2} x + \frac {1}{5} \, {\left (7 \, b^{4} c + 39 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{2} x^{5} + \frac {1}{4} \, {\left (b^{5} + 18 \, a b^{3} c + 24 \, a^{2} b c^{2}\right )} d^{2} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{4} + 15 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{2} x^{2} \]

[In]

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

[Out]

4/9*c^5*d^2*x^9 + 2*b*c^4*d^2*x^8 + 1/7*(25*b^2*c^3 + 12*a*c^4)*d^2*x^7 + 1/6*(19*b^3*c^2 + 36*a*b*c^3)*d^2*x^
6 + a^3*b^2*d^2*x + 1/5*(7*b^4*c + 39*a*b^2*c^2 + 12*a^2*c^3)*d^2*x^5 + 1/4*(b^5 + 18*a*b^3*c + 24*a^2*b*c^2)*
d^2*x^4 + 1/3*(3*a*b^4 + 15*a^2*b^2*c + 4*a^3*c^2)*d^2*x^3 + 1/2*(3*a^2*b^3 + 4*a^3*b*c)*d^2*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (93) = 186\).

Time = 0.28 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.33 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx=\frac {4}{9} \, c^{5} d^{2} x^{9} + 2 \, b c^{4} d^{2} x^{8} + \frac {25}{7} \, b^{2} c^{3} d^{2} x^{7} + \frac {12}{7} \, a c^{4} d^{2} x^{7} + \frac {19}{6} \, b^{3} c^{2} d^{2} x^{6} + 6 \, a b c^{3} d^{2} x^{6} + \frac {7}{5} \, b^{4} c d^{2} x^{5} + \frac {39}{5} \, a b^{2} c^{2} d^{2} x^{5} + \frac {12}{5} \, a^{2} c^{3} d^{2} x^{5} + \frac {1}{4} \, b^{5} d^{2} x^{4} + \frac {9}{2} \, a b^{3} c d^{2} x^{4} + 6 \, a^{2} b c^{2} d^{2} x^{4} + a b^{4} d^{2} x^{3} + 5 \, a^{2} b^{2} c d^{2} x^{3} + \frac {4}{3} \, a^{3} c^{2} d^{2} x^{3} + \frac {3}{2} \, a^{2} b^{3} d^{2} x^{2} + 2 \, a^{3} b c d^{2} x^{2} + a^{3} b^{2} d^{2} x \]

[In]

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

[Out]

4/9*c^5*d^2*x^9 + 2*b*c^4*d^2*x^8 + 25/7*b^2*c^3*d^2*x^7 + 12/7*a*c^4*d^2*x^7 + 19/6*b^3*c^2*d^2*x^6 + 6*a*b*c
^3*d^2*x^6 + 7/5*b^4*c*d^2*x^5 + 39/5*a*b^2*c^2*d^2*x^5 + 12/5*a^2*c^3*d^2*x^5 + 1/4*b^5*d^2*x^4 + 9/2*a*b^3*c
*d^2*x^4 + 6*a^2*b*c^2*d^2*x^4 + a*b^4*d^2*x^3 + 5*a^2*b^2*c*d^2*x^3 + 4/3*a^3*c^2*d^2*x^3 + 3/2*a^2*b^3*d^2*x
^2 + 2*a^3*b*c*d^2*x^2 + a^3*b^2*d^2*x

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.86 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx=\frac {4\,c^5\,d^2\,x^9}{9}+a^3\,b^2\,d^2\,x+2\,b\,c^4\,d^2\,x^8+\frac {b\,d^2\,x^4\,\left (24\,a^2\,c^2+18\,a\,b^2\,c+b^4\right )}{4}+\frac {c^3\,d^2\,x^7\,\left (25\,b^2+12\,a\,c\right )}{7}+\frac {a\,d^2\,x^3\,\left (4\,a^2\,c^2+15\,a\,b^2\,c+3\,b^4\right )}{3}+\frac {c\,d^2\,x^5\,\left (12\,a^2\,c^2+39\,a\,b^2\,c+7\,b^4\right )}{5}+\frac {a^2\,b\,d^2\,x^2\,\left (3\,b^2+4\,a\,c\right )}{2}+\frac {b\,c^2\,d^2\,x^6\,\left (19\,b^2+36\,a\,c\right )}{6} \]

[In]

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

[Out]

(4*c^5*d^2*x^9)/9 + a^3*b^2*d^2*x + 2*b*c^4*d^2*x^8 + (b*d^2*x^4*(b^4 + 24*a^2*c^2 + 18*a*b^2*c))/4 + (c^3*d^2
*x^7*(12*a*c + 25*b^2))/7 + (a*d^2*x^3*(3*b^4 + 4*a^2*c^2 + 15*a*b^2*c))/3 + (c*d^2*x^5*(7*b^4 + 12*a^2*c^2 +
39*a*b^2*c))/5 + (a^2*b*d^2*x^2*(4*a*c + 3*b^2))/2 + (b*c^2*d^2*x^6*(36*a*c + 19*b^2))/6