### 3.1140 $$\int (b d+2 c d x)^2 (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=101 $-\frac{3 d^2 \left (b^2-4 a c\right ) (b+2 c x)^7}{896 c^4}+\frac{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{640 c^4}-\frac{d^2 \left (b^2-4 a c\right )^3 (b+2 c x)^3}{384 c^4}+\frac{d^2 (b+2 c x)^9}{1152 c^4}$

[Out]

-((b^2 - 4*a*c)^3*d^2*(b + 2*c*x)^3)/(384*c^4) + (3*(b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^5)/(640*c^4) - (3*(b^2 - 4
*a*c)*d^2*(b + 2*c*x)^7)/(896*c^4) + (d^2*(b + 2*c*x)^9)/(1152*c^4)

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Rubi [A]  time = 0.12897, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {683} $-\frac{3 d^2 \left (b^2-4 a c\right ) (b+2 c x)^7}{896 c^4}+\frac{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{640 c^4}-\frac{d^2 \left (b^2-4 a c\right )^3 (b+2 c x)^3}{384 c^4}+\frac{d^2 (b+2 c x)^9}{1152 c^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((b^2 - 4*a*c)^3*d^2*(b + 2*c*x)^3)/(384*c^4) + (3*(b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^5)/(640*c^4) - (3*(b^2 - 4
*a*c)*d^2*(b + 2*c*x)^7)/(896*c^4) + (d^2*(b + 2*c*x)^9)/(1152*c^4)

Rule 683

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] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin{align*} \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3 (b d+2 c d x)^2}{64 c^3}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^4}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^6}{64 c^3 d^4}+\frac{(b d+2 c d x)^8}{64 c^3 d^6}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)^3}{384 c^4}+\frac{3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^5}{640 c^4}-\frac{3 \left (b^2-4 a c\right ) d^2 (b+2 c x)^7}{896 c^4}+\frac{d^2 (b+2 c x)^9}{1152 c^4}\\ \end{align*}

Mathematica [A]  time = 0.0302483, size = 179, normalized size = 1.77 $d^2 \left (\frac{1}{5} c x^5 \left (12 a^2 c^2+39 a b^2 c+7 b^4\right )+\frac{1}{4} b x^4 \left (24 a^2 c^2+18 a b^2 c+b^4\right )+\frac{1}{3} a x^3 \left (4 a^2 c^2+15 a b^2 c+3 b^4\right )+\frac{1}{2} a^2 b x^2 \left (4 a c+3 b^2\right )+a^3 b^2 x+\frac{1}{7} c^3 x^7 \left (12 a c+25 b^2\right )+\frac{1}{6} b c^2 x^6 \left (36 a c+19 b^2\right )+2 b c^4 x^8+\frac{4 c^5 x^9}{9}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

d^2*(a^3*b^2*x + (a^2*b*(3*b^2 + 4*a*c)*x^2)/2 + (a*(3*b^4 + 15*a*b^2*c + 4*a^2*c^2)*x^3)/3 + (b*(b^4 + 18*a*b
^2*c + 24*a^2*c^2)*x^4)/4 + (c*(7*b^4 + 39*a*b^2*c + 12*a^2*c^2)*x^5)/5 + (b*c^2*(19*b^2 + 36*a*c)*x^6)/6 + (c
^3*(25*b^2 + 12*a*c)*x^7)/7 + 2*b*c^4*x^8 + (4*c^5*x^9)/9)

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Maple [B]  time = 0.04, size = 396, normalized size = 3.9 \begin{align*}{\frac{4\,{c}^{5}{d}^{2}{x}^{9}}{9}}+2\,b{d}^{2}{c}^{4}{x}^{8}+{\frac{ \left ( 13\,{b}^{2}{d}^{2}{c}^{3}+4\,{c}^{2}{d}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{3}{d}^{2}{c}^{2}+4\,b{d}^{2}c \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,{c}^{2}{d}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({b}^{2}{d}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,b{d}^{2}c \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,{c}^{2}{d}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{5}}{5}}+{\frac{ \left ({b}^{2}{d}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,b{d}^{2}c \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +12\,{a}^{2}b{c}^{2}{d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}{d}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +12\,{b}^{2}{d}^{2}c{a}^{2}+4\,{c}^{2}{d}^{2}{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,b{d}^{2}c{a}^{3}+3\,{b}^{3}{d}^{2}{a}^{2} \right ){x}^{2}}{2}}+{b}^{2}{d}^{2}{a}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9*c^5*d^2*x^9+2*b*d^2*c^4*x^8+1/7*(13*b^2*d^2*c^3+4*c^2*d^2*(a*c^2+2*b^2*c+c*(2*a*c+b^2)))*x^7+1/6*(3*b^3*d^
2*c^2+4*b*d^2*c*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+4*c^2*d^2*(4*a*b*c+b*(2*a*c+b^2)))*x^6+1/5*(b^2*d^2*(a*c^2+2*b^2
*c+c*(2*a*c+b^2))+4*b*d^2*c*(4*a*b*c+b*(2*a*c+b^2))+4*c^2*d^2*(a*(2*a*c+b^2)+2*b^2*a+a^2*c))*x^5+1/4*(b^2*d^2*
(4*a*b*c+b*(2*a*c+b^2))+4*b*d^2*c*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+12*a^2*b*c^2*d^2)*x^4+1/3*(b^2*d^2*(a*(2*a*c+b
^2)+2*b^2*a+a^2*c)+12*b^2*d^2*c*a^2+4*c^2*d^2*a^3)*x^3+1/2*(4*a^3*b*c*d^2+3*a^2*b^3*d^2)*x^2+b^2*d^2*a^3*x

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Maxima [B]  time = 1.14134, size = 267, normalized size = 2.64 \begin{align*} \frac{4}{9} \, c^{5} d^{2} x^{9} + 2 \, b c^{4} d^{2} x^{8} + \frac{1}{7} \,{\left (25 \, b^{2} c^{3} + 12 \, a c^{4}\right )} d^{2} x^{7} + \frac{1}{6} \,{\left (19 \, b^{3} c^{2} + 36 \, a b c^{3}\right )} d^{2} x^{6} + a^{3} b^{2} d^{2} x + \frac{1}{5} \,{\left (7 \, b^{4} c + 39 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{2} x^{5} + \frac{1}{4} \,{\left (b^{5} + 18 \, a b^{3} c + 24 \, a^{2} b c^{2}\right )} d^{2} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{4} + 15 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{2} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{2} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9*c^5*d^2*x^9 + 2*b*c^4*d^2*x^8 + 1/7*(25*b^2*c^3 + 12*a*c^4)*d^2*x^7 + 1/6*(19*b^3*c^2 + 36*a*b*c^3)*d^2*x^
6 + a^3*b^2*d^2*x + 1/5*(7*b^4*c + 39*a*b^2*c^2 + 12*a^2*c^3)*d^2*x^5 + 1/4*(b^5 + 18*a*b^3*c + 24*a^2*b*c^2)*
d^2*x^4 + 1/3*(3*a*b^4 + 15*a^2*b^2*c + 4*a^3*c^2)*d^2*x^3 + 1/2*(3*a^2*b^3 + 4*a^3*b*c)*d^2*x^2

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Fricas [B]  time = 1.70073, size = 500, normalized size = 4.95 \begin{align*} \frac{4}{9} x^{9} d^{2} c^{5} + 2 x^{8} d^{2} c^{4} b + \frac{25}{7} x^{7} d^{2} c^{3} b^{2} + \frac{12}{7} x^{7} d^{2} c^{4} a + \frac{19}{6} x^{6} d^{2} c^{2} b^{3} + 6 x^{6} d^{2} c^{3} b a + \frac{7}{5} x^{5} d^{2} c b^{4} + \frac{39}{5} x^{5} d^{2} c^{2} b^{2} a + \frac{12}{5} x^{5} d^{2} c^{3} a^{2} + \frac{1}{4} x^{4} d^{2} b^{5} + \frac{9}{2} x^{4} d^{2} c b^{3} a + 6 x^{4} d^{2} c^{2} b a^{2} + x^{3} d^{2} b^{4} a + 5 x^{3} d^{2} c b^{2} a^{2} + \frac{4}{3} x^{3} d^{2} c^{2} a^{3} + \frac{3}{2} x^{2} d^{2} b^{3} a^{2} + 2 x^{2} d^{2} c b a^{3} + x d^{2} b^{2} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9*x^9*d^2*c^5 + 2*x^8*d^2*c^4*b + 25/7*x^7*d^2*c^3*b^2 + 12/7*x^7*d^2*c^4*a + 19/6*x^6*d^2*c^2*b^3 + 6*x^6*d
^2*c^3*b*a + 7/5*x^5*d^2*c*b^4 + 39/5*x^5*d^2*c^2*b^2*a + 12/5*x^5*d^2*c^3*a^2 + 1/4*x^4*d^2*b^5 + 9/2*x^4*d^2
*c*b^3*a + 6*x^4*d^2*c^2*b*a^2 + x^3*d^2*b^4*a + 5*x^3*d^2*c*b^2*a^2 + 4/3*x^3*d^2*c^2*a^3 + 3/2*x^2*d^2*b^3*a
^2 + 2*x^2*d^2*c*b*a^3 + x*d^2*b^2*a^3

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Sympy [B]  time = 0.108754, size = 246, normalized size = 2.44 \begin{align*} a^{3} b^{2} d^{2} x + 2 b c^{4} d^{2} x^{8} + \frac{4 c^{5} d^{2} x^{9}}{9} + x^{7} \left (\frac{12 a c^{4} d^{2}}{7} + \frac{25 b^{2} c^{3} d^{2}}{7}\right ) + x^{6} \left (6 a b c^{3} d^{2} + \frac{19 b^{3} c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac{12 a^{2} c^{3} d^{2}}{5} + \frac{39 a b^{2} c^{2} d^{2}}{5} + \frac{7 b^{4} c d^{2}}{5}\right ) + x^{4} \left (6 a^{2} b c^{2} d^{2} + \frac{9 a b^{3} c d^{2}}{2} + \frac{b^{5} d^{2}}{4}\right ) + x^{3} \left (\frac{4 a^{3} c^{2} d^{2}}{3} + 5 a^{2} b^{2} c d^{2} + a b^{4} d^{2}\right ) + x^{2} \left (2 a^{3} b c d^{2} + \frac{3 a^{2} b^{3} d^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*b**2*d**2*x + 2*b*c**4*d**2*x**8 + 4*c**5*d**2*x**9/9 + x**7*(12*a*c**4*d**2/7 + 25*b**2*c**3*d**2/7) + x
**6*(6*a*b*c**3*d**2 + 19*b**3*c**2*d**2/6) + x**5*(12*a**2*c**3*d**2/5 + 39*a*b**2*c**2*d**2/5 + 7*b**4*c*d**
2/5) + x**4*(6*a**2*b*c**2*d**2 + 9*a*b**3*c*d**2/2 + b**5*d**2/4) + x**3*(4*a**3*c**2*d**2/3 + 5*a**2*b**2*c*
d**2 + a*b**4*d**2) + x**2*(2*a**3*b*c*d**2 + 3*a**2*b**3*d**2/2)

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Giac [B]  time = 1.17085, size = 317, normalized size = 3.14 \begin{align*} \frac{4}{9} \, c^{5} d^{2} x^{9} + 2 \, b c^{4} d^{2} x^{8} + \frac{25}{7} \, b^{2} c^{3} d^{2} x^{7} + \frac{12}{7} \, a c^{4} d^{2} x^{7} + \frac{19}{6} \, b^{3} c^{2} d^{2} x^{6} + 6 \, a b c^{3} d^{2} x^{6} + \frac{7}{5} \, b^{4} c d^{2} x^{5} + \frac{39}{5} \, a b^{2} c^{2} d^{2} x^{5} + \frac{12}{5} \, a^{2} c^{3} d^{2} x^{5} + \frac{1}{4} \, b^{5} d^{2} x^{4} + \frac{9}{2} \, a b^{3} c d^{2} x^{4} + 6 \, a^{2} b c^{2} d^{2} x^{4} + a b^{4} d^{2} x^{3} + 5 \, a^{2} b^{2} c d^{2} x^{3} + \frac{4}{3} \, a^{3} c^{2} d^{2} x^{3} + \frac{3}{2} \, a^{2} b^{3} d^{2} x^{2} + 2 \, a^{3} b c d^{2} x^{2} + a^{3} b^{2} d^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9*c^5*d^2*x^9 + 2*b*c^4*d^2*x^8 + 25/7*b^2*c^3*d^2*x^7 + 12/7*a*c^4*d^2*x^7 + 19/6*b^3*c^2*d^2*x^6 + 6*a*b*c
^3*d^2*x^6 + 7/5*b^4*c*d^2*x^5 + 39/5*a*b^2*c^2*d^2*x^5 + 12/5*a^2*c^3*d^2*x^5 + 1/4*b^5*d^2*x^4 + 9/2*a*b^3*c
*d^2*x^4 + 6*a^2*b*c^2*d^2*x^4 + a*b^4*d^2*x^3 + 5*a^2*b^2*c*d^2*x^3 + 4/3*a^3*c^2*d^2*x^3 + 3/2*a^2*b^3*d^2*x
^2 + 2*a^3*b*c*d^2*x^2 + a^3*b^2*d^2*x