### 3.1141 $$\int (b d+2 c d x) (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=17 $\frac{1}{4} d \left (a+b x+c x^2\right )^4$

[Out]

(d*(a + b*x + c*x^2)^4)/4

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Rubi [A]  time = 0.0054493, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {629} $\frac{1}{4} d \left (a+b x+c x^2\right )^4$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d*(a + b*x + c*x^2)^4)/4

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

Mathematica [B]  time = 0.0176819, size = 52, normalized size = 3.06 $\frac{1}{4} d x (b+c x) \left (6 a^2 x (b+c x)+4 a^3+4 a x^2 (b+c x)^2+x^3 (b+c x)^3\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.039, size = 231, normalized size = 13.6 \begin{align*}{\frac{{c}^{4}d{x}^{8}}{4}}+bd{c}^{3}{x}^{7}+{\frac{ \left ( 3\,{b}^{2}d{c}^{2}+2\,cd \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( bd \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,cd \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( bd \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,cd \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( bd \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +6\,cdb{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,cd{a}^{3}+3\,{b}^{2}d{a}^{2} \right ){x}^{2}}{2}}+bd{a}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.23692, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{4} \,{\left (c x^{2} + b x + a\right )}^{4} d \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*x^2 + b*x + a)^4*d

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Fricas [B]  time = 1.65394, size = 306, normalized size = 18. \begin{align*} \frac{1}{4} x^{8} d c^{4} + x^{7} d c^{3} b + \frac{3}{2} x^{6} d c^{2} b^{2} + x^{6} d c^{3} a + x^{5} d c b^{3} + 3 x^{5} d c^{2} b a + \frac{1}{4} x^{4} d b^{4} + 3 x^{4} d c b^{2} a + \frac{3}{2} x^{4} d c^{2} a^{2} + x^{3} d b^{3} a + 3 x^{3} d c b a^{2} + \frac{3}{2} x^{2} d b^{2} a^{2} + x^{2} d c a^{3} + x d b a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.128285, size = 144, normalized size = 8.47 \begin{align*} a^{3} b d x + b c^{3} d x^{7} + \frac{c^{4} d x^{8}}{4} + x^{6} \left (a c^{3} d + \frac{3 b^{2} c^{2} d}{2}\right ) + x^{5} \left (3 a b c^{2} d + b^{3} c d\right ) + x^{4} \left (\frac{3 a^{2} c^{2} d}{2} + 3 a b^{2} c d + \frac{b^{4} d}{4}\right ) + x^{3} \left (3 a^{2} b c d + a b^{3} d\right ) + x^{2} \left (a^{3} c d + \frac{3 a^{2} b^{2} d}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.22418, size = 189, normalized size = 11.12 \begin{align*} \frac{1}{4} \, c^{4} d x^{8} + b c^{3} d x^{7} + \frac{3}{2} \, b^{2} c^{2} d x^{6} + a c^{3} d x^{6} + b^{3} c d x^{5} + 3 \, a b c^{2} d x^{5} + \frac{1}{4} \, b^{4} d x^{4} + 3 \, a b^{2} c d x^{4} + \frac{3}{2} \, a^{2} c^{2} d x^{4} + a b^{3} d x^{3} + 3 \, a^{2} b c d x^{3} + \frac{3}{2} \, a^{2} b^{2} d x^{2} + a^{3} c d x^{2} + a^{3} b d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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