### 3.1124 $$\int (b d+2 c d x)^2 (a+b x+c x^2)^2 \, dx$$

Optimal. Leaf size=73 $-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac{d^2 (b+2 c x)^7}{224 c^3}$

[Out]

((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^3)/(96*c^3) - ((b^2 - 4*a*c)*d^2*(b + 2*c*x)^5)/(80*c^3) + (d^2*(b + 2*c*x)^7
)/(224*c^3)

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Rubi [A]  time = 0.0843479, 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} $-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac{d^2 (b+2 c x)^7}{224 c^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^3)/(96*c^3) - ((b^2 - 4*a*c)*d^2*(b + 2*c*x)^5)/(80*c^3) + (d^2*(b + 2*c*x)^7
)/(224*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)^2 \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)^2}{16 c^2}+\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^4}{8 c^2 d^2}+\frac{(b d+2 c d x)^6}{16 c^2 d^4}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3}{96 c^3}-\frac{\left (b^2-4 a c\right ) d^2 (b+2 c x)^5}{80 c^3}+\frac{d^2 (b+2 c x)^7}{224 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0183955, size = 111, normalized size = 1.52 $d^2 \left (\frac{1}{3} x^3 \left (4 a^2 c^2+10 a b^2 c+b^4\right )+a^2 b^2 x+\frac{1}{5} c^2 x^5 \left (8 a c+13 b^2\right )+\frac{1}{2} b c x^4 \left (8 a c+3 b^2\right )+a b x^2 \left (2 a c+b^2\right )+2 b c^3 x^6+\frac{4 c^4 x^7}{7}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

d^2*(a^2*b^2*x + a*b*(b^2 + 2*a*c)*x^2 + ((b^4 + 10*a*b^2*c + 4*a^2*c^2)*x^3)/3 + (b*c*(3*b^2 + 8*a*c)*x^4)/2
+ (c^2*(13*b^2 + 8*a*c)*x^5)/5 + 2*b*c^3*x^6 + (4*c^4*x^7)/7)

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Maple [B]  time = 0.039, size = 176, normalized size = 2.4 \begin{align*}{\frac{4\,{c}^{4}{d}^{2}{x}^{7}}{7}}+2\,b{d}^{2}{c}^{3}{x}^{6}+{\frac{ \left ( 9\,{b}^{2}{d}^{2}{c}^{2}+4\,{c}^{2}{d}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{3}{d}^{2}c+4\,b{d}^{2}c \left ( 2\,ac+{b}^{2} \right ) +8\,{c}^{2}{d}^{2}ab \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}{d}^{2} \left ( 2\,ac+{b}^{2} \right ) +8\,{b}^{2}{d}^{2}ca+4\,{c}^{2}{d}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,cb{a}^{2}{d}^{2}+2\,{b}^{3}{d}^{2}a \right ){x}^{2}}{2}}+{b}^{2}{d}^{2}{a}^{2}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)^2,x)

[Out]

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

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Maxima [A]  time = 1.14253, size = 171, normalized size = 2.34 \begin{align*} \frac{4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac{1}{5} \,{\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{5} + a^{2} b^{2} d^{2} x + \frac{1}{2} \,{\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{4} + \frac{1}{3} \,{\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{3} +{\left (a b^{3} + 2 \, a^{2} 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)^2,x, algorithm="maxima")

[Out]

4/7*c^4*d^2*x^7 + 2*b*c^3*d^2*x^6 + 1/5*(13*b^2*c^2 + 8*a*c^3)*d^2*x^5 + a^2*b^2*d^2*x + 1/2*(3*b^3*c + 8*a*b*
c^2)*d^2*x^4 + 1/3*(b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*x^3 + (a*b^3 + 2*a^2*b*c)*d^2*x^2

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Fricas [B]  time = 1.67429, size = 315, normalized size = 4.32 \begin{align*} \frac{4}{7} x^{7} d^{2} c^{4} + 2 x^{6} d^{2} c^{3} b + \frac{13}{5} x^{5} d^{2} c^{2} b^{2} + \frac{8}{5} x^{5} d^{2} c^{3} a + \frac{3}{2} x^{4} d^{2} c b^{3} + 4 x^{4} d^{2} c^{2} b a + \frac{1}{3} x^{3} d^{2} b^{4} + \frac{10}{3} x^{3} d^{2} c b^{2} a + \frac{4}{3} x^{3} d^{2} c^{2} a^{2} + x^{2} d^{2} b^{3} a + 2 x^{2} d^{2} c b a^{2} + x d^{2} b^{2} a^{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)^2,x, algorithm="fricas")

[Out]

4/7*x^7*d^2*c^4 + 2*x^6*d^2*c^3*b + 13/5*x^5*d^2*c^2*b^2 + 8/5*x^5*d^2*c^3*a + 3/2*x^4*d^2*c*b^3 + 4*x^4*d^2*c
^2*b*a + 1/3*x^3*d^2*b^4 + 10/3*x^3*d^2*c*b^2*a + 4/3*x^3*d^2*c^2*a^2 + x^2*d^2*b^3*a + 2*x^2*d^2*c*b*a^2 + x*
d^2*b^2*a^2

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Sympy [B]  time = 0.099276, size = 156, normalized size = 2.14 \begin{align*} a^{2} b^{2} d^{2} x + 2 b c^{3} d^{2} x^{6} + \frac{4 c^{4} d^{2} x^{7}}{7} + x^{5} \left (\frac{8 a c^{3} d^{2}}{5} + \frac{13 b^{2} c^{2} d^{2}}{5}\right ) + x^{4} \left (4 a b c^{2} d^{2} + \frac{3 b^{3} c d^{2}}{2}\right ) + x^{3} \left (\frac{4 a^{2} c^{2} d^{2}}{3} + \frac{10 a b^{2} c d^{2}}{3} + \frac{b^{4} d^{2}}{3}\right ) + x^{2} \left (2 a^{2} b c d^{2} + a b^{3} d^{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)**2,x)

[Out]

a**2*b**2*d**2*x + 2*b*c**3*d**2*x**6 + 4*c**4*d**2*x**7/7 + x**5*(8*a*c**3*d**2/5 + 13*b**2*c**2*d**2/5) + x*
*4*(4*a*b*c**2*d**2 + 3*b**3*c*d**2/2) + x**3*(4*a**2*c**2*d**2/3 + 10*a*b**2*c*d**2/3 + b**4*d**2/3) + x**2*(
2*a**2*b*c*d**2 + a*b**3*d**2)

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Giac [B]  time = 1.18166, size = 200, normalized size = 2.74 \begin{align*} \frac{4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac{13}{5} \, b^{2} c^{2} d^{2} x^{5} + \frac{8}{5} \, a c^{3} d^{2} x^{5} + \frac{3}{2} \, b^{3} c d^{2} x^{4} + 4 \, a b c^{2} d^{2} x^{4} + \frac{1}{3} \, b^{4} d^{2} x^{3} + \frac{10}{3} \, a b^{2} c d^{2} x^{3} + \frac{4}{3} \, a^{2} c^{2} d^{2} x^{3} + a b^{3} d^{2} x^{2} + 2 \, a^{2} b c d^{2} x^{2} + a^{2} 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)^2,x, algorithm="giac")

[Out]

4/7*c^4*d^2*x^7 + 2*b*c^3*d^2*x^6 + 13/5*b^2*c^2*d^2*x^5 + 8/5*a*c^3*d^2*x^5 + 3/2*b^3*c*d^2*x^4 + 4*a*b*c^2*d
^2*x^4 + 1/3*b^4*d^2*x^3 + 10/3*a*b^2*c*d^2*x^3 + 4/3*a^2*c^2*d^2*x^3 + a*b^3*d^2*x^2 + 2*a^2*b*c*d^2*x^2 + a^
2*b^2*d^2*x