### 3.2134 $$\int (d+e x) (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=161 $\frac{1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac{1}{2} a^2 x^2 (a e+3 b d)+a^3 d x+\frac{1}{5} x^5 \left (6 a b c e+3 a c^2 d+3 b^2 c d+b^3 e\right )+\frac{1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8$

[Out]

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

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Rubi [A]  time = 0.157483, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.056, Rules used = {631} $\frac{1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac{1}{2} a^2 x^2 (a e+3 b d)+a^3 d x+\frac{1}{5} x^5 \left (6 a b c e+3 a c^2 d+3 b^2 c d+b^3 e\right )+\frac{1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(a + b*x + c*x^2)^3,x]

[Out]

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

Rule 631

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

Rubi steps

\begin{align*} \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 d+a^2 (3 b d+a e) x+3 a \left (b^2 d+a c d+a b e\right ) x^2+\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^3+\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^4+3 c \left (b c d+b^2 e+a c e\right ) x^5+c^2 (c d+3 b e) x^6+c^3 e x^7\right ) \, dx\\ &=a^3 d x+\frac{1}{2} a^2 (3 b d+a e) x^2+a \left (b^2 d+a c d+a b e\right ) x^3+\frac{1}{4} \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^4+\frac{1}{5} \left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^5+\frac{1}{2} c \left (b c d+b^2 e+a c e\right ) x^6+\frac{1}{7} c^2 (c d+3 b e) x^7+\frac{1}{8} c^3 e x^8\\ \end{align*}

Mathematica [A]  time = 0.037271, size = 161, normalized size = 1. $\frac{1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac{1}{2} a^2 x^2 (a e+3 b d)+a^3 d x+\frac{1}{5} x^5 \left (6 a b c e+3 a c^2 d+3 b^2 c d+b^3 e\right )+\frac{1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 223, normalized size = 1.4 \begin{align*}{\frac{{c}^{3}e{x}^{8}}{8}}+{\frac{ \left ( 3\,eb{c}^{2}+d{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,db{c}^{2}+e \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( d \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +e \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( d \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +e \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +3\,{a}^{2}be \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{3}+3\,db{a}^{2} \right ){x}^{2}}{2}}+{a}^{3}dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00462, size = 220, normalized size = 1.37 \begin{align*} \frac{1}{8} \, c^{3} e x^{8} + \frac{1}{7} \,{\left (c^{3} d + 3 \, b c^{2} e\right )} x^{7} + \frac{1}{2} \,{\left (b c^{2} d +{\left (b^{2} c + a c^{2}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left (3 \,{\left (b^{2} c + a c^{2}\right )} d +{\left (b^{3} + 6 \, a b c\right )} e\right )} x^{5} + a^{3} d x + \frac{1}{4} \,{\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \,{\left (a b^{2} + a^{2} c\right )} e\right )} x^{4} +{\left (a^{2} b e +{\left (a b^{2} + a^{2} c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.78398, size = 444, normalized size = 2.76 \begin{align*} \frac{1}{8} x^{8} e c^{3} + \frac{1}{7} x^{7} d c^{3} + \frac{3}{7} x^{7} e c^{2} b + \frac{1}{2} x^{6} d c^{2} b + \frac{1}{2} x^{6} e c b^{2} + \frac{1}{2} x^{6} e c^{2} a + \frac{3}{5} x^{5} d c b^{2} + \frac{1}{5} x^{5} e b^{3} + \frac{3}{5} x^{5} d c^{2} a + \frac{6}{5} x^{5} e c b a + \frac{1}{4} x^{4} d b^{3} + \frac{3}{2} x^{4} d c b a + \frac{3}{4} x^{4} e b^{2} a + \frac{3}{4} x^{4} e c a^{2} + x^{3} d b^{2} a + x^{3} d c a^{2} + x^{3} e b a^{2} + \frac{3}{2} x^{2} d b a^{2} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.097579, size = 190, normalized size = 1.18 \begin{align*} a^{3} d x + \frac{c^{3} e x^{8}}{8} + x^{7} \left (\frac{3 b c^{2} e}{7} + \frac{c^{3} d}{7}\right ) + x^{6} \left (\frac{a c^{2} e}{2} + \frac{b^{2} c e}{2} + \frac{b c^{2} d}{2}\right ) + x^{5} \left (\frac{6 a b c e}{5} + \frac{3 a c^{2} d}{5} + \frac{b^{3} e}{5} + \frac{3 b^{2} c d}{5}\right ) + x^{4} \left (\frac{3 a^{2} c e}{4} + \frac{3 a b^{2} e}{4} + \frac{3 a b c d}{2} + \frac{b^{3} d}{4}\right ) + x^{3} \left (a^{2} b e + a^{2} c d + a b^{2} d\right ) + x^{2} \left (\frac{a^{3} e}{2} + \frac{3 a^{2} b d}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07771, size = 266, normalized size = 1.65 \begin{align*} \frac{1}{8} \, c^{3} x^{8} e + \frac{1}{7} \, c^{3} d x^{7} + \frac{3}{7} \, b c^{2} x^{7} e + \frac{1}{2} \, b c^{2} d x^{6} + \frac{1}{2} \, b^{2} c x^{6} e + \frac{1}{2} \, a c^{2} x^{6} e + \frac{3}{5} \, b^{2} c d x^{5} + \frac{3}{5} \, a c^{2} d x^{5} + \frac{1}{5} \, b^{3} x^{5} e + \frac{6}{5} \, a b c x^{5} e + \frac{1}{4} \, b^{3} d x^{4} + \frac{3}{2} \, a b c d x^{4} + \frac{3}{4} \, a b^{2} x^{4} e + \frac{3}{4} \, a^{2} c x^{4} e + a b^{2} d x^{3} + a^{2} c d x^{3} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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