### 3.1123 $$\int (b d+2 c d x)^3 (a+b x+c x^2)^2 \, dx$$

Optimal. Leaf size=55 $\frac{1}{12} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3+\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3$

[Out]

((b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^3)/12 + (d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^3)/4

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Rubi [A]  time = 0.0244556, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {692, 629} $\frac{1}{12} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3+\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^3)/12 + (d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^3)/4

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx &=\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3+\frac{1}{4} \left (\left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx\\ &=\frac{1}{12} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^3+\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3\\ \end{align*}

Mathematica [A]  time = 0.0198803, size = 97, normalized size = 1.76 $\frac{1}{3} d^3 x (b+c x) \left (3 a^2 \left (b^2+2 b c x+2 c^2 x^2\right )+a x \left (11 b^2 c x+3 b^3+16 b c^2 x^2+8 c^3 x^3\right )+x^2 (b+c x)^2 \left (b^2+3 b c x+3 c^2 x^2\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d^3*x*(b + c*x)*(3*a^2*(b^2 + 2*b*c*x + 2*c^2*x^2) + x^2*(b + c*x)^2*(b^2 + 3*b*c*x + 3*c^2*x^2) + a*x*(3*b^3
+ 11*b^2*c*x + 16*b*c^2*x^2 + 8*c^3*x^3)))/3

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Maple [B]  time = 0.041, size = 237, normalized size = 4.3 \begin{align*}{c}^{5}{d}^{3}{x}^{8}+4\,b{d}^{3}{c}^{4}{x}^{7}+{\frac{ \left ( 30\,{b}^{2}{d}^{3}{c}^{3}+8\,{c}^{3}{d}^{3} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 13\,{b}^{3}{d}^{3}{c}^{2}+12\,b{d}^{3}{c}^{2} \left ( 2\,ac+{b}^{2} \right ) +16\,{c}^{3}{d}^{3}ab \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{4}{d}^{3}c+6\,{b}^{2}{d}^{3}c \left ( 2\,ac+{b}^{2} \right ) +24\,{b}^{2}{d}^{3}{c}^{2}a+8\,{c}^{3}{d}^{3}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{3}{d}^{3} \left ( 2\,ac+{b}^{2} \right ) +12\,{b}^{3}{d}^{3}ca+12\,b{d}^{3}{c}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{b}^{2}{d}^{3}c{a}^{2}+2\,{b}^{4}{d}^{3}a \right ){x}^{2}}{2}}+{b}^{3}{d}^{3}{a}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^5*d^3*x^8+4*b*d^3*c^4*x^7+1/6*(30*b^2*d^3*c^3+8*c^3*d^3*(2*a*c+b^2))*x^6+1/5*(13*b^3*d^3*c^2+12*b*d^3*c^2*(2
*a*c+b^2)+16*c^3*d^3*a*b)*x^5+1/4*(2*b^4*d^3*c+6*b^2*d^3*c*(2*a*c+b^2)+24*b^2*d^3*c^2*a+8*c^3*d^3*a^2)*x^4+1/3
*(b^3*d^3*(2*a*c+b^2)+12*b^3*d^3*c*a+12*b*d^3*c^2*a^2)*x^3+1/2*(6*a^2*b^2*c*d^3+2*a*b^4*d^3)*x^2+b^3*d^3*a^2*x

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Maxima [B]  time = 1.20458, size = 217, normalized size = 3.95 \begin{align*} c^{5} d^{3} x^{8} + 4 \, b c^{4} d^{3} x^{7} + \frac{1}{3} \,{\left (19 \, b^{2} c^{3} + 8 \, a c^{4}\right )} d^{3} x^{6} + a^{2} b^{3} d^{3} x +{\left (5 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d^{3} x^{5} +{\left (2 \, b^{4} c + 9 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d^{3} x^{4} + \frac{1}{3} \,{\left (b^{5} + 14 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} d^{3} x^{3} +{\left (a b^{4} + 3 \, a^{2} b^{2} c\right )} d^{3} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^5*d^3*x^8 + 4*b*c^4*d^3*x^7 + 1/3*(19*b^2*c^3 + 8*a*c^4)*d^3*x^6 + a^2*b^3*d^3*x + (5*b^3*c^2 + 8*a*b*c^3)*d
^3*x^5 + (2*b^4*c + 9*a*b^2*c^2 + 2*a^2*c^3)*d^3*x^4 + 1/3*(b^5 + 14*a*b^3*c + 12*a^2*b*c^2)*d^3*x^3 + (a*b^4
+ 3*a^2*b^2*c)*d^3*x^2

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Fricas [B]  time = 1.72548, size = 393, normalized size = 7.15 \begin{align*} x^{8} d^{3} c^{5} + 4 x^{7} d^{3} c^{4} b + \frac{19}{3} x^{6} d^{3} c^{3} b^{2} + \frac{8}{3} x^{6} d^{3} c^{4} a + 5 x^{5} d^{3} c^{2} b^{3} + 8 x^{5} d^{3} c^{3} b a + 2 x^{4} d^{3} c b^{4} + 9 x^{4} d^{3} c^{2} b^{2} a + 2 x^{4} d^{3} c^{3} a^{2} + \frac{1}{3} x^{3} d^{3} b^{5} + \frac{14}{3} x^{3} d^{3} c b^{3} a + 4 x^{3} d^{3} c^{2} b a^{2} + x^{2} d^{3} b^{4} a + 3 x^{2} d^{3} c b^{2} a^{2} + x d^{3} b^{3} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^8*d^3*c^5 + 4*x^7*d^3*c^4*b + 19/3*x^6*d^3*c^3*b^2 + 8/3*x^6*d^3*c^4*a + 5*x^5*d^3*c^2*b^3 + 8*x^5*d^3*c^3*b
*a + 2*x^4*d^3*c*b^4 + 9*x^4*d^3*c^2*b^2*a + 2*x^4*d^3*c^3*a^2 + 1/3*x^3*d^3*b^5 + 14/3*x^3*d^3*c*b^3*a + 4*x^
3*d^3*c^2*b*a^2 + x^2*d^3*b^4*a + 3*x^2*d^3*c*b^2*a^2 + x*d^3*b^3*a^2

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Sympy [B]  time = 0.100154, size = 194, normalized size = 3.53 \begin{align*} a^{2} b^{3} d^{3} x + 4 b c^{4} d^{3} x^{7} + c^{5} d^{3} x^{8} + x^{6} \left (\frac{8 a c^{4} d^{3}}{3} + \frac{19 b^{2} c^{3} d^{3}}{3}\right ) + x^{5} \left (8 a b c^{3} d^{3} + 5 b^{3} c^{2} d^{3}\right ) + x^{4} \left (2 a^{2} c^{3} d^{3} + 9 a b^{2} c^{2} d^{3} + 2 b^{4} c d^{3}\right ) + x^{3} \left (4 a^{2} b c^{2} d^{3} + \frac{14 a b^{3} c d^{3}}{3} + \frac{b^{5} d^{3}}{3}\right ) + x^{2} \left (3 a^{2} b^{2} c d^{3} + a b^{4} d^{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*b**3*d**3*x + 4*b*c**4*d**3*x**7 + c**5*d**3*x**8 + x**6*(8*a*c**4*d**3/3 + 19*b**2*c**3*d**3/3) + x**5*(
8*a*b*c**3*d**3 + 5*b**3*c**2*d**3) + x**4*(2*a**2*c**3*d**3 + 9*a*b**2*c**2*d**3 + 2*b**4*c*d**3) + x**3*(4*a
**2*b*c**2*d**3 + 14*a*b**3*c*d**3/3 + b**5*d**3/3) + x**2*(3*a**2*b**2*c*d**3 + a*b**4*d**3)

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Giac [B]  time = 1.21721, size = 261, normalized size = 4.75 \begin{align*} c^{5} d^{3} x^{8} + 4 \, b c^{4} d^{3} x^{7} + \frac{19}{3} \, b^{2} c^{3} d^{3} x^{6} + \frac{8}{3} \, a c^{4} d^{3} x^{6} + 5 \, b^{3} c^{2} d^{3} x^{5} + 8 \, a b c^{3} d^{3} x^{5} + 2 \, b^{4} c d^{3} x^{4} + 9 \, a b^{2} c^{2} d^{3} x^{4} + 2 \, a^{2} c^{3} d^{3} x^{4} + \frac{1}{3} \, b^{5} d^{3} x^{3} + \frac{14}{3} \, a b^{3} c d^{3} x^{3} + 4 \, a^{2} b c^{2} d^{3} x^{3} + a b^{4} d^{3} x^{2} + 3 \, a^{2} b^{2} c d^{3} x^{2} + a^{2} b^{3} d^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^5*d^3*x^8 + 4*b*c^4*d^3*x^7 + 19/3*b^2*c^3*d^3*x^6 + 8/3*a*c^4*d^3*x^6 + 5*b^3*c^2*d^3*x^5 + 8*a*b*c^3*d^3*x
^5 + 2*b^4*c*d^3*x^4 + 9*a*b^2*c^2*d^3*x^4 + 2*a^2*c^3*d^3*x^4 + 1/3*b^5*d^3*x^3 + 14/3*a*b^3*c*d^3*x^3 + 4*a^
2*b*c^2*d^3*x^3 + a*b^4*d^3*x^2 + 3*a^2*b^2*c*d^3*x^2 + a^2*b^3*d^3*x