### 3.1110 $$\int (b d+2 c d x)^3 (a+b x+c x^2) \, dx$$

Optimal. Leaf size=45 $\frac{d^3 (b+2 c x)^6}{48 c^2}-\frac{d^3 \left (b^2-4 a c\right ) (b+2 c x)^4}{32 c^2}$

[Out]

-((b^2 - 4*a*c)*d^3*(b + 2*c*x)^4)/(32*c^2) + (d^3*(b + 2*c*x)^6)/(48*c^2)

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Rubi [A]  time = 0.0535636, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {683} $\frac{d^3 (b+2 c x)^6}{48 c^2}-\frac{d^3 \left (b^2-4 a c\right ) (b+2 c x)^4}{32 c^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((b^2 - 4*a*c)*d^3*(b + 2*c*x)^4)/(32*c^2) + (d^3*(b + 2*c*x)^6)/(48*c^2)

Rule 683

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*} \int (b d+2 c d x)^3 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^3}{4 c}+\frac{(b d+2 c d x)^5}{4 c d^2}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right ) d^3 (b+2 c x)^4}{32 c^2}+\frac{d^3 (b+2 c x)^6}{48 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0145711, size = 66, normalized size = 1.47 $\frac{1}{6} d^3 x (b+c x) \left (6 a \left (b^2+2 b c x+2 c^2 x^2\right )+x \left (11 b^2 c x+3 b^3+16 b c^2 x^2+8 c^3 x^3\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d^3*x*(b + c*x)*(6*a*(b^2 + 2*b*c*x + 2*c^2*x^2) + x*(3*b^3 + 11*b^2*c*x + 16*b*c^2*x^2 + 8*c^3*x^3)))/6

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Maple [B]  time = 0.039, size = 108, normalized size = 2.4 \begin{align*}{\frac{4\,{c}^{4}{d}^{3}{x}^{6}}{3}}+4\,b{d}^{3}{c}^{3}{x}^{5}+{\frac{ \left ( 8\,{c}^{3}{d}^{3}a+18\,{b}^{2}{d}^{3}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 12\,b{d}^{3}{c}^{2}a+7\,{b}^{3}{d}^{3}c \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{b}^{2}{d}^{3}ca+{b}^{4}{d}^{3} \right ){x}^{2}}{2}}+{b}^{3}{d}^{3}ax \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),x)

[Out]

4/3*c^4*d^3*x^6+4*b*d^3*c^3*x^5+1/4*(8*a*c^3*d^3+18*b^2*c^2*d^3)*x^4+1/3*(12*a*b*c^2*d^3+7*b^3*c*d^3)*x^3+1/2*
(6*a*b^2*c*d^3+b^4*d^3)*x^2+b^3*d^3*a*x

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Maxima [B]  time = 1.23227, size = 131, normalized size = 2.91 \begin{align*} \frac{4}{3} \, c^{4} d^{3} x^{6} + 4 \, b c^{3} d^{3} x^{5} + a b^{3} d^{3} x + \frac{1}{2} \,{\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} x^{4} + \frac{1}{3} \,{\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} d^{3} x^{3} + \frac{1}{2} \,{\left (b^{4} + 6 \, a 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),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.02252, size = 225, normalized size = 5. \begin{align*} \frac{4}{3} x^{6} d^{3} c^{4} + 4 x^{5} d^{3} c^{3} b + \frac{9}{2} x^{4} d^{3} c^{2} b^{2} + 2 x^{4} d^{3} c^{3} a + \frac{7}{3} x^{3} d^{3} c b^{3} + 4 x^{3} d^{3} c^{2} b a + \frac{1}{2} x^{2} d^{3} b^{4} + 3 x^{2} d^{3} c b^{2} a + x d^{3} b^{3} a \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),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.120087, size = 114, normalized size = 2.53 \begin{align*} a b^{3} d^{3} x + 4 b c^{3} d^{3} x^{5} + \frac{4 c^{4} d^{3} x^{6}}{3} + x^{4} \left (2 a c^{3} d^{3} + \frac{9 b^{2} c^{2} d^{3}}{2}\right ) + x^{3} \left (4 a b c^{2} d^{3} + \frac{7 b^{3} c d^{3}}{3}\right ) + x^{2} \left (3 a b^{2} c d^{3} + \frac{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),x)

[Out]

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

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Giac [B]  time = 1.22207, size = 146, normalized size = 3.24 \begin{align*} \frac{4}{3} \, c^{4} d^{3} x^{6} + 4 \, b c^{3} d^{3} x^{5} + \frac{9}{2} \, b^{2} c^{2} d^{3} x^{4} + 2 \, a c^{3} d^{3} x^{4} + \frac{7}{3} \, b^{3} c d^{3} x^{3} + 4 \, a b c^{2} d^{3} x^{3} + \frac{1}{2} \, b^{4} d^{3} x^{2} + 3 \, a b^{2} c d^{3} x^{2} + a 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),x, algorithm="giac")

[Out]

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