### 3.1452 $$\int (d+e x)^3 (a^2+2 a b x+b^2 x^2) \, dx$$

Optimal. Leaf size=65 $-\frac{2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac{(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac{b^2 (d+e x)^6}{6 e^3}$

[Out]

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

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Rubi [A]  time = 0.0679921, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {27, 43} $-\frac{2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac{(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac{b^2 (d+e x)^6}{6 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0177003, size = 122, normalized size = 1.88 $\frac{1}{4} e x^4 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac{1}{3} d x^3 \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+a^2 d^3 x+\frac{1}{2} a d^2 x^2 (3 a e+2 b d)+\frac{1}{5} b e^2 x^5 (2 a e+3 b d)+\frac{1}{6} b^2 e^3 x^6$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.039, size = 125, normalized size = 1.9 \begin{align*}{\frac{{e}^{3}{b}^{2}{x}^{6}}{6}}+{\frac{ \left ( 2\,ab{e}^{3}+3\,d{e}^{2}{b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2}{e}^{3}+6\,d{e}^{2}ab+3\,{d}^{2}e{b}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{2}+6\,{d}^{2}eab+{d}^{3}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{2}+2\,{d}^{3}ab \right ){x}^{2}}{2}}+{d}^{3}{a}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.13521, size = 167, normalized size = 2.57 \begin{align*} \frac{1}{6} \, b^{2} e^{3} x^{6} + a^{2} d^{3} x + \frac{1}{5} \,{\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.76083, size = 285, normalized size = 4.38 \begin{align*} \frac{1}{6} x^{6} e^{3} b^{2} + \frac{3}{5} x^{5} e^{2} d b^{2} + \frac{2}{5} x^{5} e^{3} b a + \frac{3}{4} x^{4} e d^{2} b^{2} + \frac{3}{2} x^{4} e^{2} d b a + \frac{1}{4} x^{4} e^{3} a^{2} + \frac{1}{3} x^{3} d^{3} b^{2} + 2 x^{3} e d^{2} b a + x^{3} e^{2} d a^{2} + x^{2} d^{3} b a + \frac{3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.081116, size = 133, normalized size = 2.05 \begin{align*} a^{2} d^{3} x + \frac{b^{2} e^{3} x^{6}}{6} + x^{5} \left (\frac{2 a b e^{3}}{5} + \frac{3 b^{2} d e^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} e^{3}}{4} + \frac{3 a b d e^{2}}{2} + \frac{3 b^{2} d^{2} e}{4}\right ) + x^{3} \left (a^{2} d e^{2} + 2 a b d^{2} e + \frac{b^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{2} d^{2} e}{2} + a b d^{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.15857, size = 171, normalized size = 2.63 \begin{align*} \frac{1}{6} \, b^{2} x^{6} e^{3} + \frac{3}{5} \, b^{2} d x^{5} e^{2} + \frac{3}{4} \, b^{2} d^{2} x^{4} e + \frac{1}{3} \, b^{2} d^{3} x^{3} + \frac{2}{5} \, a b x^{5} e^{3} + \frac{3}{2} \, a b d x^{4} e^{2} + 2 \, a b d^{2} x^{3} e + a b d^{3} x^{2} + \frac{1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac{3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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