### 3.1770 $$\int (a+b x)^2 (a c+(b c+a d) x+b d x^2)^2 \, dx$$

Optimal. Leaf size=65 $\frac{d (a+b x)^6 (b c-a d)}{3 b^3}+\frac{(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac{d^2 (a+b x)^7}{7 b^3}$

[Out]

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

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Rubi [A]  time = 0.0897019, 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} $\frac{d (a+b x)^6 (b c-a d)}{3 b^3}+\frac{(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac{d^2 (a+b x)^7}{7 b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((b*c - a*d)^2*(a + b*x)^5)/(5*b^3) + (d*(b*c - a*d)*(a + b*x)^6)/(3*b^3) + (d^2*(a + b*x)^7)/(7*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)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\int (a+b x)^4 (c+d x)^2 \, dx\\ &=\int \left (\frac{(b c-a d)^2 (a+b x)^4}{b^2}+\frac{2 d (b c-a d) (a+b x)^5}{b^2}+\frac{d^2 (a+b x)^6}{b^2}\right ) \, dx\\ &=\frac{(b c-a d)^2 (a+b x)^5}{5 b^3}+\frac{d (b c-a d) (a+b x)^6}{3 b^3}+\frac{d^2 (a+b x)^7}{7 b^3}\\ \end{align*}

Mathematica [B]  time = 0.0200078, size = 148, normalized size = 2.28 $\frac{1}{5} b^2 x^5 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a b x^4 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 x^3 \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+a^3 c x^2 (a d+2 b c)+a^4 c^2 x+\frac{1}{3} b^3 d x^6 (2 a d+b c)+\frac{1}{7} b^4 d^2 x^7$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.039, size = 231, normalized size = 3.6 \begin{align*}{\frac{{b}^{4}{d}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{d}^{2}a{b}^{3}+2\,{b}^{3} \left ( ad+bc \right ) d \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}{b}^{2}{d}^{2}+4\,a{b}^{2} \left ( ad+bc \right ) d+{b}^{2} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{2} \left ( ad+bc \right ) bd+2\,ab \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\,{b}^{2}ac \left ( ad+bc \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{2} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +4\,{a}^{2}bc \left ( ad+bc \right ) +{a}^{2}{b}^{2}{c}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{3}c \left ( ad+bc \right ) +2\,{a}^{3}b{c}^{2} \right ){x}^{2}}{2}}+{a}^{4}{c}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.04526, size = 211, normalized size = 3.25 \begin{align*} \frac{1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac{1}{3} \,{\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} +{\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} +{\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.68828, size = 363, normalized size = 5.58 \begin{align*} \frac{1}{7} x^{7} d^{2} b^{4} + \frac{1}{3} x^{6} d c b^{4} + \frac{2}{3} x^{6} d^{2} b^{3} a + \frac{1}{5} x^{5} c^{2} b^{4} + \frac{8}{5} x^{5} d c b^{3} a + \frac{6}{5} x^{5} d^{2} b^{2} a^{2} + x^{4} c^{2} b^{3} a + 3 x^{4} d c b^{2} a^{2} + x^{4} d^{2} b a^{3} + 2 x^{3} c^{2} b^{2} a^{2} + \frac{8}{3} x^{3} d c b a^{3} + \frac{1}{3} x^{3} d^{2} a^{4} + 2 x^{2} c^{2} b a^{3} + x^{2} d c a^{4} + x c^{2} a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.126889, size = 168, normalized size = 2.58 \begin{align*} a^{4} c^{2} x + \frac{b^{4} d^{2} x^{7}}{7} + x^{6} \left (\frac{2 a b^{3} d^{2}}{3} + \frac{b^{4} c d}{3}\right ) + x^{5} \left (\frac{6 a^{2} b^{2} d^{2}}{5} + \frac{8 a b^{3} c d}{5} + \frac{b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac{a^{4} d^{2}}{3} + \frac{8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.191, size = 230, normalized size = 3.54 \begin{align*} \frac{1}{7} \, b^{4} d^{2} x^{7} + \frac{1}{3} \, b^{4} c d x^{6} + \frac{2}{3} \, a b^{3} d^{2} x^{6} + \frac{1}{5} \, b^{4} c^{2} x^{5} + \frac{8}{5} \, a b^{3} c d x^{5} + \frac{6}{5} \, a^{2} b^{2} d^{2} x^{5} + a b^{3} c^{2} x^{4} + 3 \, a^{2} b^{2} c d x^{4} + a^{3} b d^{2} x^{4} + 2 \, a^{2} b^{2} c^{2} x^{3} + \frac{8}{3} \, a^{3} b c d x^{3} + \frac{1}{3} \, a^{4} d^{2} x^{3} + 2 \, a^{3} b c^{2} x^{2} + a^{4} c d x^{2} + a^{4} c^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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