### 3.1771 $$\int (a+b x) (a c+(b c+a d) x+b d x^2)^2 \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0657123, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.074, Rules used = {626, 43} $\frac{2 d (a+b x)^5 (b c-a d)}{5 b^3}+\frac{(a+b x)^4 (b c-a d)^2}{4 b^3}+\frac{d^2 (a+b x)^6}{6 b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 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 (a+b x) \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\int (a+b x)^3 (c+d x)^2 \, dx\\ &=\int \left (\frac{(b c-a d)^2 (a+b x)^3}{b^2}+\frac{2 d (b c-a d) (a+b x)^4}{b^2}+\frac{d^2 (a+b x)^5}{b^2}\right ) \, dx\\ &=\frac{(b c-a d)^2 (a+b x)^4}{4 b^3}+\frac{2 d (b c-a d) (a+b x)^5}{5 b^3}+\frac{d^2 (a+b x)^6}{6 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0144083, size = 122, normalized size = 1.88 $\frac{1}{4} b x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{3} a x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{2} a^2 c x^2 (2 a d+3 b c)+a^3 c^2 x+\frac{1}{5} b^2 d x^5 (3 a d+2 b c)+\frac{1}{6} b^3 d^2 x^6$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.038, size = 147, normalized size = 2.3 \begin{align*}{\frac{{b}^{3}{d}^{2}{x}^{6}}{6}}+{\frac{ \left ( a{b}^{2}{d}^{2}+2\,{b}^{2} \left ( ad+bc \right ) d \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,a \left ( ad+bc \right ) bd+b \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( a \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\,bac \left ( ad+bc \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}c \left ( ad+bc \right ) +{a}^{2}b{c}^{2} \right ){x}^{2}}{2}}+{a}^{3}{c}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.10535, size = 167, normalized size = 2.57 \begin{align*} \frac{1}{6} \, b^{3} d^{2} x^{6} + a^{3} c^{2} x + \frac{1}{5} \,{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.58249, size = 285, normalized size = 4.38 \begin{align*} \frac{1}{6} x^{6} d^{2} b^{3} + \frac{2}{5} x^{5} d c b^{3} + \frac{3}{5} x^{5} d^{2} b^{2} a + \frac{1}{4} x^{4} c^{2} b^{3} + \frac{3}{2} x^{4} d c b^{2} a + \frac{3}{4} x^{4} d^{2} b a^{2} + x^{3} c^{2} b^{2} a + 2 x^{3} d c b a^{2} + \frac{1}{3} x^{3} d^{2} a^{3} + \frac{3}{2} x^{2} c^{2} b a^{2} + x^{2} d c a^{3} + x c^{2} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.102693, size = 133, normalized size = 2.05 \begin{align*} a^{3} c^{2} x + \frac{b^{3} d^{2} x^{6}}{6} + x^{5} \left (\frac{3 a b^{2} d^{2}}{5} + \frac{2 b^{3} c d}{5}\right ) + x^{4} \left (\frac{3 a^{2} b d^{2}}{4} + \frac{3 a b^{2} c d}{2} + \frac{b^{3} c^{2}}{4}\right ) + x^{3} \left (\frac{a^{3} d^{2}}{3} + 2 a^{2} b c d + a b^{2} c^{2}\right ) + x^{2} \left (a^{3} c d + \frac{3 a^{2} b c^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.2061, size = 176, normalized size = 2.71 \begin{align*} \frac{1}{6} \, b^{3} d^{2} x^{6} + \frac{2}{5} \, b^{3} c d x^{5} + \frac{3}{5} \, a b^{2} d^{2} x^{5} + \frac{1}{4} \, b^{3} c^{2} x^{4} + \frac{3}{2} \, a b^{2} c d x^{4} + \frac{3}{4} \, a^{2} b d^{2} x^{4} + a b^{2} c^{2} x^{3} + 2 \, a^{2} b c d x^{3} + \frac{1}{3} \, a^{3} d^{2} x^{3} + \frac{3}{2} \, a^{2} b c^{2} x^{2} + a^{3} c d x^{2} + a^{3} c^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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