### 3.247 $$\int (d+e x) (b x+c x^2)^3 \, dx$$

Optimal. Leaf size=75 $\frac{1}{5} b^2 x^5 (b e+3 c d)+\frac{1}{4} b^3 d x^4+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{2} b c x^6 (b e+c d)+\frac{1}{8} c^3 e x^8$

[Out]

(b^3*d*x^4)/4 + (b^2*(3*c*d + b*e)*x^5)/5 + (b*c*(c*d + b*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 = 0.062641, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {631} $\frac{1}{5} b^2 x^5 (b e+3 c d)+\frac{1}{4} b^3 d x^4+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{2} b c x^6 (b e+c d)+\frac{1}{8} c^3 e x^8$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(b*x + c*x^2)^3,x]

[Out]

(b^3*d*x^4)/4 + (b^2*(3*c*d + b*e)*x^5)/5 + (b*c*(c*d + b*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 (b x+c x^2\right )^3 \, dx &=\int \left (b^3 d x^3+b^2 (3 c d+b e) x^4+3 b c (c d+b e) x^5+c^2 (c d+3 b e) x^6+c^3 e x^7\right ) \, dx\\ &=\frac{1}{4} b^3 d x^4+\frac{1}{5} b^2 (3 c d+b e) x^5+\frac{1}{2} b c (c d+b e) 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 = 0.0102053, size = 75, normalized size = 1. $\frac{1}{5} b^2 x^5 (b e+3 c d)+\frac{1}{4} b^3 d x^4+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{2} b c x^6 (b e+c d)+\frac{1}{8} c^3 e x^8$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b^3*d*x^4)/4 + (b^2*(3*c*d + b*e)*x^5)/5 + (b*c*(c*d + b*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 = 0.044, size = 76, normalized size = 1. \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\,e{b}^{2}c+3\,db{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( e{b}^{3}+3\,d{b}^{2}c \right ){x}^{5}}{5}}+{\frac{{b}^{3}d{x}^{4}}{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.11304, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{8} \, c^{3} e x^{8} + \frac{1}{4} \, b^{3} d x^{4} + \frac{1}{7} \,{\left (c^{3} d + 3 \, b c^{2} e\right )} x^{7} + \frac{1}{2} \,{\left (b c^{2} d + b^{2} c e\right )} x^{6} + \frac{1}{5} \,{\left (3 \, b^{2} c d + b^{3} e\right )} x^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.35286, size = 182, normalized size = 2.43 \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{3}{5} x^{5} d c b^{2} + \frac{1}{5} x^{5} e b^{3} + \frac{1}{4} x^{4} d b^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^2+b*x)^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 + 3/5*x^5*d*c*b^2 + 1/5*x^
5*e*b^3 + 1/4*x^4*d*b^3

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Sympy [A]  time = 0.22159, size = 80, normalized size = 1.07 \begin{align*} \frac{b^{3} d x^{4}}{4} + \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{b^{2} c e}{2} + \frac{b c^{2} d}{2}\right ) + x^{5} \left (\frac{b^{3} e}{5} + \frac{3 b^{2} c d}{5}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22847, size = 109, normalized size = 1.45 \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{3}{5} \, b^{2} c d x^{5} + \frac{1}{5} \, b^{3} x^{5} e + \frac{1}{4} \, b^{3} d x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^2+b*x)^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 + 3/5*b^2*c*d*x^5 + 1/5*b^
3*x^5*e + 1/4*b^3*d*x^4