### 3.1139 $$\int (b d+2 c d x)^3 (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=55 $\frac{1}{20} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4+\frac{1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4$

[Out]

((b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^4)/20 + (d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^4)/5

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Rubi [A]  time = 0.0250731, 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} $\frac{1}{20} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4+\frac{1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^4)/20 + (d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^4)/5

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 )^3 \, dx &=\frac{1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4+\frac{1}{5} \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 )^3 \, dx\\ &=\frac{1}{20} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^4+\frac{1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4\\ \end{align*}

Mathematica [B]  time = 0.0331185, size = 132, normalized size = 2.4 $\frac{1}{20} d^3 x (b+c x) \left (20 a^3 \left (b^2+2 b c x+2 c^2 x^2\right )+10 a^2 x \left (11 b^2 c x+3 b^3+16 b c^2 x^2+8 c^3 x^3\right )+20 a x^2 (b+c x)^2 \left (b^2+3 b c x+3 c^2 x^2\right )+x^3 (b+c x)^3 \left (5 b^2+16 b c x+16 c^2 x^2\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d^3*x*(b + c*x)*(20*a^3*(b^2 + 2*b*c*x + 2*c^2*x^2) + 20*a*x^2*(b + c*x)^2*(b^2 + 3*b*c*x + 3*c^2*x^2) + x^3*
(b + c*x)^3*(5*b^2 + 16*b*c*x + 16*c^2*x^2) + 10*a^2*x*(3*b^3 + 11*b^2*c*x + 16*b*c^2*x^2 + 8*c^3*x^3)))/20

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Maple [B]  time = 0.041, size = 534, normalized size = 9.7 \begin{align*}{\frac{4\,{c}^{6}{d}^{3}{x}^{10}}{5}}+4\,b{d}^{3}{c}^{5}{x}^{9}+{\frac{ \left ( 42\,{b}^{2}{d}^{3}{c}^{4}+8\,{c}^{3}{d}^{3} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 19\,{b}^{3}{d}^{3}{c}^{3}+12\,b{d}^{3}{c}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +8\,{c}^{3}{d}^{3} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{4}{d}^{3}{c}^{2}+6\,{b}^{2}{d}^{3}c \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +12\,b{d}^{3}{c}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +8\,{c}^{3}{d}^{3} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({b}^{3}{d}^{3} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +6\,{b}^{2}{d}^{3}c \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +12\,b{d}^{3}{c}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +24\,{c}^{3}{d}^{3}b{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({b}^{3}{d}^{3} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +6\,{b}^{2}{d}^{3}c \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +36\,{b}^{2}{d}^{3}{c}^{2}{a}^{2}+8\,{c}^{3}{d}^{3}{a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{3}{d}^{3} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +18\,{b}^{3}{d}^{3}c{a}^{2}+12\,b{d}^{3}{c}^{2}{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{b}^{2}{d}^{3}c{a}^{3}+3\,{b}^{4}{d}^{3}{a}^{2} \right ){x}^{2}}{2}}+{b}^{3}{d}^{3}{a}^{3}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)^3,x)

[Out]

4/5*c^6*d^3*x^10+4*b*d^3*c^5*x^9+1/8*(42*b^2*d^3*c^4+8*c^3*d^3*(a*c^2+2*b^2*c+c*(2*a*c+b^2)))*x^8+1/7*(19*b^3*
d^3*c^3+12*b*d^3*c^2*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+8*c^3*d^3*(4*a*b*c+b*(2*a*c+b^2)))*x^7+1/6*(3*b^4*d^3*c^2+6
*b^2*d^3*c*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+12*b*d^3*c^2*(4*a*b*c+b*(2*a*c+b^2))+8*c^3*d^3*(a*(2*a*c+b^2)+2*b^2*a
+a^2*c))*x^6+1/5*(b^3*d^3*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+6*b^2*d^3*c*(4*a*b*c+b*(2*a*c+b^2))+12*b*d^3*c^2*(a*(2
*a*c+b^2)+2*b^2*a+a^2*c)+24*c^3*d^3*b*a^2)*x^5+1/4*(b^3*d^3*(4*a*b*c+b*(2*a*c+b^2))+6*b^2*d^3*c*(a*(2*a*c+b^2)
+2*b^2*a+a^2*c)+36*b^2*d^3*c^2*a^2+8*c^3*d^3*a^3)*x^4+1/3*(b^3*d^3*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+18*b^3*d^3*c*
a^2+12*b*d^3*c^2*a^3)*x^3+1/2*(6*a^3*b^2*c*d^3+3*a^2*b^4*d^3)*x^2+b^3*d^3*a^3*x

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Maxima [B]  time = 1.10006, size = 328, normalized size = 5.96 \begin{align*} \frac{4}{5} \, c^{6} d^{3} x^{10} + 4 \, b c^{5} d^{3} x^{9} + \frac{3}{4} \,{\left (11 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{3} x^{8} + 3 \,{\left (3 \, b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{3} x^{7} + a^{3} b^{3} d^{3} x + \frac{1}{2} \,{\left (11 \, b^{4} c^{2} + 38 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{3} x^{6} + \frac{3}{5} \,{\left (3 \, b^{5} c + 25 \, a b^{3} c^{2} + 20 \, a^{2} b c^{3}\right )} d^{3} x^{5} + \frac{1}{4} \,{\left (b^{6} + 24 \, a b^{4} c + 54 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d^{3} x^{4} +{\left (a b^{5} + 7 \, a^{2} b^{3} c + 4 \, a^{3} b c^{2}\right )} d^{3} x^{3} + \frac{3}{2} \,{\left (a^{2} b^{4} + 2 \, a^{3} 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)^3,x, algorithm="maxima")

[Out]

4/5*c^6*d^3*x^10 + 4*b*c^5*d^3*x^9 + 3/4*(11*b^2*c^4 + 4*a*c^5)*d^3*x^8 + 3*(3*b^3*c^3 + 4*a*b*c^4)*d^3*x^7 +
a^3*b^3*d^3*x + 1/2*(11*b^4*c^2 + 38*a*b^2*c^3 + 8*a^2*c^4)*d^3*x^6 + 3/5*(3*b^5*c + 25*a*b^3*c^2 + 20*a^2*b*c
^3)*d^3*x^5 + 1/4*(b^6 + 24*a*b^4*c + 54*a^2*b^2*c^2 + 8*a^3*c^3)*d^3*x^4 + (a*b^5 + 7*a^2*b^3*c + 4*a^3*b*c^2
)*d^3*x^3 + 3/2*(a^2*b^4 + 2*a^3*b^2*c)*d^3*x^2

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Fricas [B]  time = 1.81675, size = 614, normalized size = 11.16 \begin{align*} \frac{4}{5} x^{10} d^{3} c^{6} + 4 x^{9} d^{3} c^{5} b + \frac{33}{4} x^{8} d^{3} c^{4} b^{2} + 3 x^{8} d^{3} c^{5} a + 9 x^{7} d^{3} c^{3} b^{3} + 12 x^{7} d^{3} c^{4} b a + \frac{11}{2} x^{6} d^{3} c^{2} b^{4} + 19 x^{6} d^{3} c^{3} b^{2} a + 4 x^{6} d^{3} c^{4} a^{2} + \frac{9}{5} x^{5} d^{3} c b^{5} + 15 x^{5} d^{3} c^{2} b^{3} a + 12 x^{5} d^{3} c^{3} b a^{2} + \frac{1}{4} x^{4} d^{3} b^{6} + 6 x^{4} d^{3} c b^{4} a + \frac{27}{2} x^{4} d^{3} c^{2} b^{2} a^{2} + 2 x^{4} d^{3} c^{3} a^{3} + x^{3} d^{3} b^{5} a + 7 x^{3} d^{3} c b^{3} a^{2} + 4 x^{3} d^{3} c^{2} b a^{3} + \frac{3}{2} x^{2} d^{3} b^{4} a^{2} + 3 x^{2} d^{3} c b^{2} a^{3} + x d^{3} b^{3} a^{3} \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)^3,x, algorithm="fricas")

[Out]

4/5*x^10*d^3*c^6 + 4*x^9*d^3*c^5*b + 33/4*x^8*d^3*c^4*b^2 + 3*x^8*d^3*c^5*a + 9*x^7*d^3*c^3*b^3 + 12*x^7*d^3*c
^4*b*a + 11/2*x^6*d^3*c^2*b^4 + 19*x^6*d^3*c^3*b^2*a + 4*x^6*d^3*c^4*a^2 + 9/5*x^5*d^3*c*b^5 + 15*x^5*d^3*c^2*
b^3*a + 12*x^5*d^3*c^3*b*a^2 + 1/4*x^4*d^3*b^6 + 6*x^4*d^3*c*b^4*a + 27/2*x^4*d^3*c^2*b^2*a^2 + 2*x^4*d^3*c^3*
a^3 + x^3*d^3*b^5*a + 7*x^3*d^3*c*b^3*a^2 + 4*x^3*d^3*c^2*b*a^3 + 3/2*x^2*d^3*b^4*a^2 + 3*x^2*d^3*c*b^2*a^3 +
x*d^3*b^3*a^3

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Sympy [B]  time = 0.185013, size = 299, normalized size = 5.44 \begin{align*} a^{3} b^{3} d^{3} x + 4 b c^{5} d^{3} x^{9} + \frac{4 c^{6} d^{3} x^{10}}{5} + x^{8} \left (3 a c^{5} d^{3} + \frac{33 b^{2} c^{4} d^{3}}{4}\right ) + x^{7} \left (12 a b c^{4} d^{3} + 9 b^{3} c^{3} d^{3}\right ) + x^{6} \left (4 a^{2} c^{4} d^{3} + 19 a b^{2} c^{3} d^{3} + \frac{11 b^{4} c^{2} d^{3}}{2}\right ) + x^{5} \left (12 a^{2} b c^{3} d^{3} + 15 a b^{3} c^{2} d^{3} + \frac{9 b^{5} c d^{3}}{5}\right ) + x^{4} \left (2 a^{3} c^{3} d^{3} + \frac{27 a^{2} b^{2} c^{2} d^{3}}{2} + 6 a b^{4} c d^{3} + \frac{b^{6} d^{3}}{4}\right ) + x^{3} \left (4 a^{3} b c^{2} d^{3} + 7 a^{2} b^{3} c d^{3} + a b^{5} d^{3}\right ) + x^{2} \left (3 a^{3} b^{2} c d^{3} + \frac{3 a^{2} b^{4} d^{3}}{2}\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)**3,x)

[Out]

a**3*b**3*d**3*x + 4*b*c**5*d**3*x**9 + 4*c**6*d**3*x**10/5 + x**8*(3*a*c**5*d**3 + 33*b**2*c**4*d**3/4) + x**
7*(12*a*b*c**4*d**3 + 9*b**3*c**3*d**3) + x**6*(4*a**2*c**4*d**3 + 19*a*b**2*c**3*d**3 + 11*b**4*c**2*d**3/2)
+ x**5*(12*a**2*b*c**3*d**3 + 15*a*b**3*c**2*d**3 + 9*b**5*c*d**3/5) + x**4*(2*a**3*c**3*d**3 + 27*a**2*b**2*c
**2*d**3/2 + 6*a*b**4*c*d**3 + b**6*d**3/4) + x**3*(4*a**3*b*c**2*d**3 + 7*a**2*b**3*c*d**3 + a*b**5*d**3) + x
**2*(3*a**3*b**2*c*d**3 + 3*a**2*b**4*d**3/2)

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Giac [B]  time = 1.1588, size = 402, normalized size = 7.31 \begin{align*} \frac{4}{5} \, c^{6} d^{3} x^{10} + 4 \, b c^{5} d^{3} x^{9} + \frac{33}{4} \, b^{2} c^{4} d^{3} x^{8} + 3 \, a c^{5} d^{3} x^{8} + 9 \, b^{3} c^{3} d^{3} x^{7} + 12 \, a b c^{4} d^{3} x^{7} + \frac{11}{2} \, b^{4} c^{2} d^{3} x^{6} + 19 \, a b^{2} c^{3} d^{3} x^{6} + 4 \, a^{2} c^{4} d^{3} x^{6} + \frac{9}{5} \, b^{5} c d^{3} x^{5} + 15 \, a b^{3} c^{2} d^{3} x^{5} + 12 \, a^{2} b c^{3} d^{3} x^{5} + \frac{1}{4} \, b^{6} d^{3} x^{4} + 6 \, a b^{4} c d^{3} x^{4} + \frac{27}{2} \, a^{2} b^{2} c^{2} d^{3} x^{4} + 2 \, a^{3} c^{3} d^{3} x^{4} + a b^{5} d^{3} x^{3} + 7 \, a^{2} b^{3} c d^{3} x^{3} + 4 \, a^{3} b c^{2} d^{3} x^{3} + \frac{3}{2} \, a^{2} b^{4} d^{3} x^{2} + 3 \, a^{3} b^{2} c d^{3} x^{2} + a^{3} 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)^3,x, algorithm="giac")

[Out]

4/5*c^6*d^3*x^10 + 4*b*c^5*d^3*x^9 + 33/4*b^2*c^4*d^3*x^8 + 3*a*c^5*d^3*x^8 + 9*b^3*c^3*d^3*x^7 + 12*a*b*c^4*d
^3*x^7 + 11/2*b^4*c^2*d^3*x^6 + 19*a*b^2*c^3*d^3*x^6 + 4*a^2*c^4*d^3*x^6 + 9/5*b^5*c*d^3*x^5 + 15*a*b^3*c^2*d^
3*x^5 + 12*a^2*b*c^3*d^3*x^5 + 1/4*b^6*d^3*x^4 + 6*a*b^4*c*d^3*x^4 + 27/2*a^2*b^2*c^2*d^3*x^4 + 2*a^3*c^3*d^3*
x^4 + a*b^5*d^3*x^3 + 7*a^2*b^3*c*d^3*x^3 + 4*a^3*b*c^2*d^3*x^3 + 3/2*a^2*b^4*d^3*x^2 + 3*a^3*b^2*c*d^3*x^2 +
a^3*b^3*d^3*x