### 3.1769 $$\int (a+b x)^3 (a c+(b c+a d) x+b d x^2)^2 \, dx$$

Optimal. Leaf size=65 $\frac{2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac{(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac{d^2 (a+b x)^8}{8 b^3}$

[Out]

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

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Rubi [A]  time = 0.129975, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {626, 43} $\frac{2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac{(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac{d^2 (a+b x)^8}{8 b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [B]  time = 0.0313715, size = 189, normalized size = 2.91 $\frac{1}{6} b^3 x^6 \left (10 a^2 d^2+10 a b c d+b^2 c^2\right )+a b^2 x^5 \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{5}{4} a^2 b x^4 \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+\frac{1}{3} a^3 x^3 \left (a^2 d^2+10 a b c d+10 b^2 c^2\right )+\frac{1}{2} a^4 c x^2 (2 a d+5 b c)+a^5 c^2 x+\frac{1}{7} b^4 d x^7 (5 a d+2 b c)+\frac{1}{8} b^5 d^2 x^8$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a^5*c^2*x + (a^4*c*(5*b*c + 2*a*d)*x^2)/2 + (a^3*(10*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^3)/3 + (5*a^2*b*(2*b^2*
c^2 + 4*a*b*c*d + a^2*d^2)*x^4)/4 + a*b^2*(b^2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^5 + (b^3*(b^2*c^2 + 10*a*b*c*d +
10*a^2*d^2)*x^6)/6 + (b^4*d*(2*b*c + 5*a*d)*x^7)/7 + (b^5*d^2*x^8)/8

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Maple [B]  time = 0.039, size = 315, normalized size = 4.9 \begin{align*}{\frac{{b}^{5}{d}^{2}{x}^{8}}{8}}+{\frac{ \left ( 3\,{b}^{4}a{d}^{2}+2\,{b}^{4} \left ( ad+bc \right ) d \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{3}{a}^{2}{d}^{2}+6\,{b}^{3}a \left ( ad+bc \right ) d+{b}^{3} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{3}{b}^{2}{d}^{2}+6\,{b}^{2}{a}^{2} \left ( ad+bc \right ) d+3\,{b}^{2}a \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\,{b}^{3}ac \left ( ad+bc \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{3} \left ( ad+bc \right ) bd+3\,b{a}^{2} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +6\,{b}^{2}{a}^{2}c \left ( ad+bc \right ) +{a}^{2}{b}^{3}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +6\,b{a}^{3}c \left ( ad+bc \right ) +3\,{b}^{2}{a}^{3}{c}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{4}c \left ( ad+bc \right ) +3\,b{a}^{4}{c}^{2} \right ){x}^{2}}{2}}+{a}^{5}{c}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.0531, size = 266, normalized size = 4.09 \begin{align*} \frac{1}{8} \, b^{5} d^{2} x^{8} + a^{5} c^{2} x + \frac{1}{7} \,{\left (2 \, b^{5} c d + 5 \, a b^{4} d^{2}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} c^{2} + 10 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x^{6} +{\left (a b^{4} c^{2} + 4 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, a^{2} b^{3} c^{2} + 4 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} c^{2} + 10 \, a^{4} b c d + a^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^5*d^2*x^8 + a^5*c^2*x + 1/7*(2*b^5*c*d + 5*a*b^4*d^2)*x^7 + 1/6*(b^5*c^2 + 10*a*b^4*c*d + 10*a^2*b^3*d^2
)*x^6 + (a*b^4*c^2 + 4*a^2*b^3*c*d + 2*a^3*b^2*d^2)*x^5 + 5/4*(2*a^2*b^3*c^2 + 4*a^3*b^2*c*d + a^4*b*d^2)*x^4
+ 1/3*(10*a^3*b^2*c^2 + 10*a^4*b*c*d + a^5*d^2)*x^3 + 1/2*(5*a^4*b*c^2 + 2*a^5*c*d)*x^2

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Fricas [B]  time = 1.57015, size = 460, normalized size = 7.08 \begin{align*} \frac{1}{8} x^{8} d^{2} b^{5} + \frac{2}{7} x^{7} d c b^{5} + \frac{5}{7} x^{7} d^{2} b^{4} a + \frac{1}{6} x^{6} c^{2} b^{5} + \frac{5}{3} x^{6} d c b^{4} a + \frac{5}{3} x^{6} d^{2} b^{3} a^{2} + x^{5} c^{2} b^{4} a + 4 x^{5} d c b^{3} a^{2} + 2 x^{5} d^{2} b^{2} a^{3} + \frac{5}{2} x^{4} c^{2} b^{3} a^{2} + 5 x^{4} d c b^{2} a^{3} + \frac{5}{4} x^{4} d^{2} b a^{4} + \frac{10}{3} x^{3} c^{2} b^{2} a^{3} + \frac{10}{3} x^{3} d c b a^{4} + \frac{1}{3} x^{3} d^{2} a^{5} + \frac{5}{2} x^{2} c^{2} b a^{4} + x^{2} d c a^{5} + x c^{2} a^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*d^2*b^5 + 2/7*x^7*d*c*b^5 + 5/7*x^7*d^2*b^4*a + 1/6*x^6*c^2*b^5 + 5/3*x^6*d*c*b^4*a + 5/3*x^6*d^2*b^3*
a^2 + x^5*c^2*b^4*a + 4*x^5*d*c*b^3*a^2 + 2*x^5*d^2*b^2*a^3 + 5/2*x^4*c^2*b^3*a^2 + 5*x^4*d*c*b^2*a^3 + 5/4*x^
4*d^2*b*a^4 + 10/3*x^3*c^2*b^2*a^3 + 10/3*x^3*d*c*b*a^4 + 1/3*x^3*d^2*a^5 + 5/2*x^2*c^2*b*a^4 + x^2*d*c*a^5 +
x*c^2*a^5

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Sympy [B]  time = 0.219368, size = 218, normalized size = 3.35 \begin{align*} a^{5} c^{2} x + \frac{b^{5} d^{2} x^{8}}{8} + x^{7} \left (\frac{5 a b^{4} d^{2}}{7} + \frac{2 b^{5} c d}{7}\right ) + x^{6} \left (\frac{5 a^{2} b^{3} d^{2}}{3} + \frac{5 a b^{4} c d}{3} + \frac{b^{5} c^{2}}{6}\right ) + x^{5} \left (2 a^{3} b^{2} d^{2} + 4 a^{2} b^{3} c d + a b^{4} c^{2}\right ) + x^{4} \left (\frac{5 a^{4} b d^{2}}{4} + 5 a^{3} b^{2} c d + \frac{5 a^{2} b^{3} c^{2}}{2}\right ) + x^{3} \left (\frac{a^{5} d^{2}}{3} + \frac{10 a^{4} b c d}{3} + \frac{10 a^{3} b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{5} c d + \frac{5 a^{4} b c^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*c**2*x + b**5*d**2*x**8/8 + x**7*(5*a*b**4*d**2/7 + 2*b**5*c*d/7) + x**6*(5*a**2*b**3*d**2/3 + 5*a*b**4*c
*d/3 + b**5*c**2/6) + x**5*(2*a**3*b**2*d**2 + 4*a**2*b**3*c*d + a*b**4*c**2) + x**4*(5*a**4*b*d**2/4 + 5*a**3
*b**2*c*d + 5*a**2*b**3*c**2/2) + x**3*(a**5*d**2/3 + 10*a**4*b*c*d/3 + 10*a**3*b**2*c**2/3) + x**2*(a**5*c*d
+ 5*a**4*b*c**2/2)

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Giac [B]  time = 1.15711, size = 286, normalized size = 4.4 \begin{align*} \frac{1}{8} \, b^{5} d^{2} x^{8} + \frac{2}{7} \, b^{5} c d x^{7} + \frac{5}{7} \, a b^{4} d^{2} x^{7} + \frac{1}{6} \, b^{5} c^{2} x^{6} + \frac{5}{3} \, a b^{4} c d x^{6} + \frac{5}{3} \, a^{2} b^{3} d^{2} x^{6} + a b^{4} c^{2} x^{5} + 4 \, a^{2} b^{3} c d x^{5} + 2 \, a^{3} b^{2} d^{2} x^{5} + \frac{5}{2} \, a^{2} b^{3} c^{2} x^{4} + 5 \, a^{3} b^{2} c d x^{4} + \frac{5}{4} \, a^{4} b d^{2} x^{4} + \frac{10}{3} \, a^{3} b^{2} c^{2} x^{3} + \frac{10}{3} \, a^{4} b c d x^{3} + \frac{1}{3} \, a^{5} d^{2} x^{3} + \frac{5}{2} \, a^{4} b c^{2} x^{2} + a^{5} c d x^{2} + a^{5} c^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^5*d^2*x^8 + 2/7*b^5*c*d*x^7 + 5/7*a*b^4*d^2*x^7 + 1/6*b^5*c^2*x^6 + 5/3*a*b^4*c*d*x^6 + 5/3*a^2*b^3*d^2*
x^6 + a*b^4*c^2*x^5 + 4*a^2*b^3*c*d*x^5 + 2*a^3*b^2*d^2*x^5 + 5/2*a^2*b^3*c^2*x^4 + 5*a^3*b^2*c*d*x^4 + 5/4*a^
4*b*d^2*x^4 + 10/3*a^3*b^2*c^2*x^3 + 10/3*a^4*b*c*d*x^3 + 1/3*a^5*d^2*x^3 + 5/2*a^4*b*c^2*x^2 + a^5*c*d*x^2 +
a^5*c^2*x