### 3.1122 $$\int (b d+2 c d x)^4 (a+b x+c x^2)^2 \, dx$$

Optimal. Leaf size=73 $-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^7}{112 c^3}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{160 c^3}+\frac{d^4 (b+2 c x)^9}{288 c^3}$

[Out]

((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^5)/(160*c^3) - ((b^2 - 4*a*c)*d^4*(b + 2*c*x)^7)/(112*c^3) + (d^4*(b + 2*c*x)
^9)/(288*c^3)

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Rubi [A]  time = 0.124237, antiderivative size = 73, 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} $-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^7}{112 c^3}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{160 c^3}+\frac{d^4 (b+2 c x)^9}{288 c^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^2,x]

[Out]

((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^5)/(160*c^3) - ((b^2 - 4*a*c)*d^4*(b + 2*c*x)^7)/(112*c^3) + (d^4*(b + 2*c*x)
^9)/(288*c^3)

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)^4 \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2 (b d+2 c d x)^4}{16 c^2}+\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^6}{8 c^2 d^2}+\frac{(b d+2 c d x)^8}{16 c^2 d^4}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5}{160 c^3}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^7}{112 c^3}+\frac{d^4 (b+2 c x)^9}{288 c^3}\\ \end{align*}

Mathematica [B]  time = 0.026301, size = 179, normalized size = 2.45 $d^4 \left (\frac{1}{5} c^2 x^5 \left (16 a^2 c^2+112 a b^2 c+41 b^4\right )+\frac{1}{2} b c x^4 \left (16 a^2 c^2+32 a b^2 c+5 b^4\right )+\frac{1}{3} b^2 x^3 \left (24 a^2 c^2+18 a b^2 c+b^4\right )+a^2 b^4 x+\frac{8}{7} c^4 x^7 \left (4 a c+13 b^2\right )+\frac{4}{3} b c^3 x^6 \left (12 a c+11 b^2\right )+a b^3 x^2 \left (4 a c+b^2\right )+8 b c^5 x^8+\frac{16 c^6 x^9}{9}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^2,x]

[Out]

d^4*(a^2*b^4*x + a*b^3*(b^2 + 4*a*c)*x^2 + (b^2*(b^4 + 18*a*b^2*c + 24*a^2*c^2)*x^3)/3 + (b*c*(5*b^4 + 32*a*b^
2*c + 16*a^2*c^2)*x^4)/2 + (c^2*(41*b^4 + 112*a*b^2*c + 16*a^2*c^2)*x^5)/5 + (4*b*c^3*(11*b^2 + 12*a*c)*x^6)/3
+ (8*c^4*(13*b^2 + 4*a*c)*x^7)/7 + 8*b*c^5*x^8 + (16*c^6*x^9)/9)

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Maple [B]  time = 0.039, size = 300, normalized size = 4.1 \begin{align*}{\frac{16\,{c}^{6}{d}^{4}{x}^{9}}{9}}+8\,b{d}^{4}{c}^{5}{x}^{8}+{\frac{ \left ( 88\,{b}^{2}{d}^{4}{c}^{4}+16\,{c}^{4}{d}^{4} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 56\,{b}^{3}{d}^{4}{c}^{3}+32\,b{d}^{4}{c}^{3} \left ( 2\,ac+{b}^{2} \right ) +32\,{c}^{4}{d}^{4}ab \right ){x}^{6}}{6}}+{\frac{ \left ( 17\,{b}^{4}{d}^{4}{c}^{2}+24\,{b}^{2}{d}^{4}{c}^{2} \left ( 2\,ac+{b}^{2} \right ) +64\,{b}^{2}{d}^{4}{c}^{3}a+16\,{c}^{4}{d}^{4}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{5}{d}^{4}c+8\,{b}^{3}{d}^{4}c \left ( 2\,ac+{b}^{2} \right ) +48\,{b}^{3}{d}^{4}{c}^{2}a+32\,b{d}^{4}{c}^{3}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{4}{d}^{4} \left ( 2\,ac+{b}^{2} \right ) +16\,{b}^{4}{d}^{4}ca+24\,{b}^{2}{d}^{4}{c}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 8\,{b}^{3}{d}^{4}c{a}^{2}+2\,{b}^{5}{d}^{4}a \right ){x}^{2}}{2}}+{b}^{4}{d}^{4}{a}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x)

[Out]

16/9*c^6*d^4*x^9+8*b*d^4*c^5*x^8+1/7*(88*b^2*d^4*c^4+16*c^4*d^4*(2*a*c+b^2))*x^7+1/6*(56*b^3*d^4*c^3+32*b*d^4*
c^3*(2*a*c+b^2)+32*c^4*d^4*a*b)*x^6+1/5*(17*b^4*d^4*c^2+24*b^2*d^4*c^2*(2*a*c+b^2)+64*b^2*d^4*c^3*a+16*c^4*d^4
*a^2)*x^5+1/4*(2*b^5*d^4*c+8*b^3*d^4*c*(2*a*c+b^2)+48*b^3*d^4*c^2*a+32*b*d^4*c^3*a^2)*x^4+1/3*(b^4*d^4*(2*a*c+
b^2)+16*b^4*d^4*c*a+24*b^2*d^4*c^2*a^2)*x^3+1/2*(8*a^2*b^3*c*d^4+2*a*b^5*d^4)*x^2+b^4*d^4*a^2*x

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Maxima [B]  time = 1.20309, size = 271, normalized size = 3.71 \begin{align*} \frac{16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac{8}{7} \,{\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{7} + a^{2} b^{4} d^{4} x + \frac{4}{3} \,{\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{6} + \frac{1}{5} \,{\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{5} + \frac{1}{2} \,{\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{4} + \frac{1}{3} \,{\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{3} +{\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

16/9*c^6*d^4*x^9 + 8*b*c^5*d^4*x^8 + 8/7*(13*b^2*c^4 + 4*a*c^5)*d^4*x^7 + a^2*b^4*d^4*x + 4/3*(11*b^3*c^3 + 12
*a*b*c^4)*d^4*x^6 + 1/5*(41*b^4*c^2 + 112*a*b^2*c^3 + 16*a^2*c^4)*d^4*x^5 + 1/2*(5*b^5*c + 32*a*b^3*c^2 + 16*a
^2*b*c^3)*d^4*x^4 + 1/3*(b^6 + 18*a*b^4*c + 24*a^2*b^2*c^2)*d^4*x^3 + (a*b^5 + 4*a^2*b^3*c)*d^4*x^2

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Fricas [B]  time = 1.93283, size = 510, normalized size = 6.99 \begin{align*} \frac{16}{9} x^{9} d^{4} c^{6} + 8 x^{8} d^{4} c^{5} b + \frac{104}{7} x^{7} d^{4} c^{4} b^{2} + \frac{32}{7} x^{7} d^{4} c^{5} a + \frac{44}{3} x^{6} d^{4} c^{3} b^{3} + 16 x^{6} d^{4} c^{4} b a + \frac{41}{5} x^{5} d^{4} c^{2} b^{4} + \frac{112}{5} x^{5} d^{4} c^{3} b^{2} a + \frac{16}{5} x^{5} d^{4} c^{4} a^{2} + \frac{5}{2} x^{4} d^{4} c b^{5} + 16 x^{4} d^{4} c^{2} b^{3} a + 8 x^{4} d^{4} c^{3} b a^{2} + \frac{1}{3} x^{3} d^{4} b^{6} + 6 x^{3} d^{4} c b^{4} a + 8 x^{3} d^{4} c^{2} b^{2} a^{2} + x^{2} d^{4} b^{5} a + 4 x^{2} d^{4} c b^{3} a^{2} + x d^{4} b^{4} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

16/9*x^9*d^4*c^6 + 8*x^8*d^4*c^5*b + 104/7*x^7*d^4*c^4*b^2 + 32/7*x^7*d^4*c^5*a + 44/3*x^6*d^4*c^3*b^3 + 16*x^
6*d^4*c^4*b*a + 41/5*x^5*d^4*c^2*b^4 + 112/5*x^5*d^4*c^3*b^2*a + 16/5*x^5*d^4*c^4*a^2 + 5/2*x^4*d^4*c*b^5 + 16
*x^4*d^4*c^2*b^3*a + 8*x^4*d^4*c^3*b*a^2 + 1/3*x^3*d^4*b^6 + 6*x^3*d^4*c*b^4*a + 8*x^3*d^4*c^2*b^2*a^2 + x^2*d
^4*b^5*a + 4*x^2*d^4*c*b^3*a^2 + x*d^4*b^4*a^2

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Sympy [B]  time = 0.105083, size = 248, normalized size = 3.4 \begin{align*} a^{2} b^{4} d^{4} x + 8 b c^{5} d^{4} x^{8} + \frac{16 c^{6} d^{4} x^{9}}{9} + x^{7} \left (\frac{32 a c^{5} d^{4}}{7} + \frac{104 b^{2} c^{4} d^{4}}{7}\right ) + x^{6} \left (16 a b c^{4} d^{4} + \frac{44 b^{3} c^{3} d^{4}}{3}\right ) + x^{5} \left (\frac{16 a^{2} c^{4} d^{4}}{5} + \frac{112 a b^{2} c^{3} d^{4}}{5} + \frac{41 b^{4} c^{2} d^{4}}{5}\right ) + x^{4} \left (8 a^{2} b c^{3} d^{4} + 16 a b^{3} c^{2} d^{4} + \frac{5 b^{5} c d^{4}}{2}\right ) + x^{3} \left (8 a^{2} b^{2} c^{2} d^{4} + 6 a b^{4} c d^{4} + \frac{b^{6} d^{4}}{3}\right ) + x^{2} \left (4 a^{2} b^{3} c d^{4} + a b^{5} d^{4}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b**4*d**4*x + 8*b*c**5*d**4*x**8 + 16*c**6*d**4*x**9/9 + x**7*(32*a*c**5*d**4/7 + 104*b**2*c**4*d**4/7) +
x**6*(16*a*b*c**4*d**4 + 44*b**3*c**3*d**4/3) + x**5*(16*a**2*c**4*d**4/5 + 112*a*b**2*c**3*d**4/5 + 41*b**4*
c**2*d**4/5) + x**4*(8*a**2*b*c**3*d**4 + 16*a*b**3*c**2*d**4 + 5*b**5*c*d**4/2) + x**3*(8*a**2*b**2*c**2*d**4
+ 6*a*b**4*c*d**4 + b**6*d**4/3) + x**2*(4*a**2*b**3*c*d**4 + a*b**5*d**4)

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Giac [B]  time = 1.27151, size = 324, normalized size = 4.44 \begin{align*} \frac{16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac{104}{7} \, b^{2} c^{4} d^{4} x^{7} + \frac{32}{7} \, a c^{5} d^{4} x^{7} + \frac{44}{3} \, b^{3} c^{3} d^{4} x^{6} + 16 \, a b c^{4} d^{4} x^{6} + \frac{41}{5} \, b^{4} c^{2} d^{4} x^{5} + \frac{112}{5} \, a b^{2} c^{3} d^{4} x^{5} + \frac{16}{5} \, a^{2} c^{4} d^{4} x^{5} + \frac{5}{2} \, b^{5} c d^{4} x^{4} + 16 \, a b^{3} c^{2} d^{4} x^{4} + 8 \, a^{2} b c^{3} d^{4} x^{4} + \frac{1}{3} \, b^{6} d^{4} x^{3} + 6 \, a b^{4} c d^{4} x^{3} + 8 \, a^{2} b^{2} c^{2} d^{4} x^{3} + a b^{5} d^{4} x^{2} + 4 \, a^{2} b^{3} c d^{4} x^{2} + a^{2} b^{4} d^{4} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

16/9*c^6*d^4*x^9 + 8*b*c^5*d^4*x^8 + 104/7*b^2*c^4*d^4*x^7 + 32/7*a*c^5*d^4*x^7 + 44/3*b^3*c^3*d^4*x^6 + 16*a*
b*c^4*d^4*x^6 + 41/5*b^4*c^2*d^4*x^5 + 112/5*a*b^2*c^3*d^4*x^5 + 16/5*a^2*c^4*d^4*x^5 + 5/2*b^5*c*d^4*x^4 + 16
*a*b^3*c^2*d^4*x^4 + 8*a^2*b*c^3*d^4*x^4 + 1/3*b^6*d^4*x^3 + 6*a*b^4*c*d^4*x^3 + 8*a^2*b^2*c^2*d^4*x^3 + a*b^5
*d^4*x^2 + 4*a^2*b^3*c*d^4*x^2 + a^2*b^4*d^4*x