### 3.222 $$\int (d+e x)^3 (b x+c x^2) \, dx$$

Optimal. Leaf size=62 $-\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{d (d+e x)^4 (c d-b e)}{4 e^3}+\frac{c (d+e x)^6}{6 e^3}$

[Out]

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

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Rubi [A]  time = 0.0488904, antiderivative size = 62, 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 = {698} $-\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{d (d+e x)^4 (c d-b e)}{4 e^3}+\frac{c (d+e x)^6}{6 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(b*x + c*x^2),x]

[Out]

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

Rule 698

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] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0162647, size = 67, normalized size = 1.08 $\frac{1}{60} x^2 \left (20 d^2 x (3 b e+c d)+12 e^2 x^3 (b e+3 c d)+45 d e x^2 (b e+c d)+30 b d^3+10 c e^3 x^4\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x^2*(30*b*d^3 + 20*d^2*(c*d + 3*b*e)*x + 45*d*e*(c*d + b*e)*x^2 + 12*e^2*(3*c*d + b*e)*x^3 + 10*c*e^3*x^4))/6
0

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Maple [A]  time = 0.045, size = 76, normalized size = 1.2 \begin{align*}{\frac{{e}^{3}c{x}^{6}}{6}}+{\frac{ \left ({e}^{3}b+3\,d{e}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,d{e}^{2}b+3\,{d}^{2}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{d}^{2}eb+{d}^{3}c \right ){x}^{3}}{3}}+{\frac{{d}^{3}b{x}^{2}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1356, size = 99, normalized size = 1.6 \begin{align*} \frac{1}{6} \, c e^{3} x^{6} + \frac{1}{2} \, b d^{3} x^{2} + \frac{1}{5} \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + \frac{3}{4} \,{\left (c d^{2} e + b d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{3} + 3 \, b d^{2} e\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42969, size = 177, normalized size = 2.85 \begin{align*} \frac{1}{6} x^{6} e^{3} c + \frac{3}{5} x^{5} e^{2} d c + \frac{1}{5} x^{5} e^{3} b + \frac{3}{4} x^{4} e d^{2} c + \frac{3}{4} x^{4} e^{2} d b + \frac{1}{3} x^{3} d^{3} c + x^{3} e d^{2} b + \frac{1}{2} x^{2} d^{3} b \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.288298, size = 80, normalized size = 1.29 \begin{align*} \frac{b d^{3} x^{2}}{2} + \frac{c e^{3} x^{6}}{6} + x^{5} \left (\frac{b e^{3}}{5} + \frac{3 c d e^{2}}{5}\right ) + x^{4} \left (\frac{3 b d e^{2}}{4} + \frac{3 c d^{2} e}{4}\right ) + x^{3} \left (b d^{2} e + \frac{c d^{3}}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.29579, size = 100, normalized size = 1.61 \begin{align*} \frac{1}{6} \, c x^{6} e^{3} + \frac{3}{5} \, c d x^{5} e^{2} + \frac{3}{4} \, c d^{2} x^{4} e + \frac{1}{3} \, c d^{3} x^{3} + \frac{1}{5} \, b x^{5} e^{3} + \frac{3}{4} \, b d x^{4} e^{2} + b d^{2} x^{3} e + \frac{1}{2} \, b d^{3} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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