### 3.1121 $$\int (b d+2 c d x)^5 (a+b x+c x^2)^2 \, dx$$

Optimal. Leaf size=73 $-\frac{d^5 \left (b^2-4 a c\right ) (b+2 c x)^8}{128 c^3}+\frac{d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^6}{192 c^3}+\frac{d^5 (b+2 c x)^{10}}{320 c^3}$

[Out]

((b^2 - 4*a*c)^2*d^5*(b + 2*c*x)^6)/(192*c^3) - ((b^2 - 4*a*c)*d^5*(b + 2*c*x)^8)/(128*c^3) + (d^5*(b + 2*c*x)
^10)/(320*c^3)

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Rubi [A]  time = 0.169635, 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} $-\frac{d^5 \left (b^2-4 a c\right ) (b+2 c x)^8}{128 c^3}+\frac{d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^6}{192 c^3}+\frac{d^5 (b+2 c x)^{10}}{320 c^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^2,x]

[Out]

((b^2 - 4*a*c)^2*d^5*(b + 2*c*x)^6)/(192*c^3) - ((b^2 - 4*a*c)*d^5*(b + 2*c*x)^8)/(128*c^3) + (d^5*(b + 2*c*x)
^10)/(320*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)^5 \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)^5}{16 c^2}+\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^7}{8 c^2 d^2}+\frac{(b d+2 c d x)^9}{16 c^2 d^4}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^2 d^5 (b+2 c x)^6}{192 c^3}-\frac{\left (b^2-4 a c\right ) d^5 (b+2 c x)^8}{128 c^3}+\frac{d^5 (b+2 c x)^{10}}{320 c^3}\\ \end{align*}

Mathematica [B]  time = 0.0598018, size = 168, normalized size = 2.3 $\frac{1}{15} d^5 x (b+c x) \left (5 a^2 \left (28 b^2 c^2 x^2+12 b^3 c x+3 b^4+32 b c^3 x^3+16 c^4 x^4\right )+5 a x \left (56 b^3 c^2 x^2+88 b^2 c^3 x^3+19 b^4 c x+3 b^5+72 b c^4 x^4+24 c^5 x^5\right )+x^2 (b+c x)^2 \left (78 b^2 c^2 x^2+30 b^3 c x+5 b^4+96 b c^3 x^3+48 c^4 x^4\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^2,x]

[Out]

(d^5*x*(b + c*x)*(5*a^2*(3*b^4 + 12*b^3*c*x + 28*b^2*c^2*x^2 + 32*b*c^3*x^3 + 16*c^4*x^4) + x^2*(b + c*x)^2*(5
*b^4 + 30*b^3*c*x + 78*b^2*c^2*x^2 + 96*b*c^3*x^3 + 48*c^4*x^4) + 5*a*x*(3*b^5 + 19*b^4*c*x + 56*b^3*c^2*x^2 +
88*b^2*c^3*x^3 + 72*b*c^4*x^4 + 24*c^5*x^5)))/15

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Maple [B]  time = 0.059, size = 362, normalized size = 5. \begin{align*}{\frac{16\,{c}^{7}{d}^{5}{x}^{10}}{5}}+16\,b{d}^{5}{c}^{6}{x}^{9}+{\frac{ \left ( 240\,{b}^{2}{d}^{5}{c}^{5}+32\,{c}^{5}{d}^{5} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 200\,{b}^{3}{d}^{5}{c}^{4}+80\,b{d}^{5}{c}^{4} \left ( 2\,ac+{b}^{2} \right ) +64\,{c}^{5}{d}^{5}ab \right ){x}^{7}}{7}}+{\frac{ \left ( 90\,{b}^{4}{d}^{5}{c}^{3}+80\,{b}^{2}{d}^{5}{c}^{3} \left ( 2\,ac+{b}^{2} \right ) +160\,{b}^{2}{d}^{5}{c}^{4}a+32\,{c}^{5}{d}^{5}{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 21\,{b}^{5}{d}^{5}{c}^{2}+40\,{b}^{3}{d}^{5}{c}^{2} \left ( 2\,ac+{b}^{2} \right ) +160\,{b}^{3}{d}^{5}{c}^{3}a+80\,b{d}^{5}{c}^{4}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{6}{d}^{5}c+10\,{b}^{4}{d}^{5}c \left ( 2\,ac+{b}^{2} \right ) +80\,{b}^{4}{d}^{5}{c}^{2}a+80\,{b}^{2}{d}^{5}{c}^{3}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{5}{d}^{5} \left ( 2\,ac+{b}^{2} \right ) +20\,{b}^{5}{d}^{5}ca+40\,{b}^{3}{d}^{5}{c}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 10\,{b}^{4}{d}^{5}c{a}^{2}+2\,{b}^{6}{d}^{5}a \right ){x}^{2}}{2}}+{b}^{5}{d}^{5}{a}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^2,x)

[Out]

16/5*c^7*d^5*x^10+16*b*d^5*c^6*x^9+1/8*(240*b^2*d^5*c^5+32*c^5*d^5*(2*a*c+b^2))*x^8+1/7*(200*b^3*d^5*c^4+80*b*
d^5*c^4*(2*a*c+b^2)+64*c^5*d^5*a*b)*x^7+1/6*(90*b^4*d^5*c^3+80*b^2*d^5*c^3*(2*a*c+b^2)+160*b^2*d^5*c^4*a+32*c^
5*d^5*a^2)*x^6+1/5*(21*b^5*d^5*c^2+40*b^3*d^5*c^2*(2*a*c+b^2)+160*b^3*d^5*c^3*a+80*b*d^5*c^4*a^2)*x^5+1/4*(2*b
^6*d^5*c+10*b^4*d^5*c*(2*a*c+b^2)+80*b^4*d^5*c^2*a+80*b^2*d^5*c^3*a^2)*x^4+1/3*(b^5*d^5*(2*a*c+b^2)+20*b^5*d^5
*c*a+40*b^3*d^5*c^2*a^2)*x^3+1/2*(10*a^2*b^4*c*d^5+2*a*b^6*d^5)*x^2+b^5*d^5*a^2*x

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Maxima [B]  time = 1.20255, size = 320, normalized size = 4.38 \begin{align*} \frac{16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 2 \,{\left (17 \, b^{2} c^{5} + 4 \, a c^{6}\right )} d^{5} x^{8} + a^{2} b^{5} d^{5} x + 8 \,{\left (5 \, b^{3} c^{4} + 4 \, a b c^{5}\right )} d^{5} x^{7} + \frac{1}{3} \,{\left (85 \, b^{4} c^{3} + 160 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{5} x^{6} + \frac{1}{5} \,{\left (61 \, b^{5} c^{2} + 240 \, a b^{3} c^{3} + 80 \, a^{2} b c^{4}\right )} d^{5} x^{5} +{\left (3 \, b^{6} c + 25 \, a b^{4} c^{2} + 20 \, a^{2} b^{2} c^{3}\right )} d^{5} x^{4} + \frac{1}{3} \,{\left (b^{7} + 22 \, a b^{5} c + 40 \, a^{2} b^{3} c^{2}\right )} d^{5} x^{3} +{\left (a b^{6} + 5 \, a^{2} b^{4} c\right )} d^{5} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

16/5*c^7*d^5*x^10 + 16*b*c^6*d^5*x^9 + 2*(17*b^2*c^5 + 4*a*c^6)*d^5*x^8 + a^2*b^5*d^5*x + 8*(5*b^3*c^4 + 4*a*b
*c^5)*d^5*x^7 + 1/3*(85*b^4*c^3 + 160*a*b^2*c^4 + 16*a^2*c^5)*d^5*x^6 + 1/5*(61*b^5*c^2 + 240*a*b^3*c^3 + 80*a
^2*b*c^4)*d^5*x^5 + (3*b^6*c + 25*a*b^4*c^2 + 20*a^2*b^2*c^3)*d^5*x^4 + 1/3*(b^7 + 22*a*b^5*c + 40*a^2*b^3*c^2
)*d^5*x^3 + (a*b^6 + 5*a^2*b^4*c)*d^5*x^2

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Fricas [B]  time = 2.08502, size = 605, normalized size = 8.29 \begin{align*} \frac{16}{5} x^{10} d^{5} c^{7} + 16 x^{9} d^{5} c^{6} b + 34 x^{8} d^{5} c^{5} b^{2} + 8 x^{8} d^{5} c^{6} a + 40 x^{7} d^{5} c^{4} b^{3} + 32 x^{7} d^{5} c^{5} b a + \frac{85}{3} x^{6} d^{5} c^{3} b^{4} + \frac{160}{3} x^{6} d^{5} c^{4} b^{2} a + \frac{16}{3} x^{6} d^{5} c^{5} a^{2} + \frac{61}{5} x^{5} d^{5} c^{2} b^{5} + 48 x^{5} d^{5} c^{3} b^{3} a + 16 x^{5} d^{5} c^{4} b a^{2} + 3 x^{4} d^{5} c b^{6} + 25 x^{4} d^{5} c^{2} b^{4} a + 20 x^{4} d^{5} c^{3} b^{2} a^{2} + \frac{1}{3} x^{3} d^{5} b^{7} + \frac{22}{3} x^{3} d^{5} c b^{5} a + \frac{40}{3} x^{3} d^{5} c^{2} b^{3} a^{2} + x^{2} d^{5} b^{6} a + 5 x^{2} d^{5} c b^{4} a^{2} + x d^{5} b^{5} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

16/5*x^10*d^5*c^7 + 16*x^9*d^5*c^6*b + 34*x^8*d^5*c^5*b^2 + 8*x^8*d^5*c^6*a + 40*x^7*d^5*c^4*b^3 + 32*x^7*d^5*
c^5*b*a + 85/3*x^6*d^5*c^3*b^4 + 160/3*x^6*d^5*c^4*b^2*a + 16/3*x^6*d^5*c^5*a^2 + 61/5*x^5*d^5*c^2*b^5 + 48*x^
5*d^5*c^3*b^3*a + 16*x^5*d^5*c^4*b*a^2 + 3*x^4*d^5*c*b^6 + 25*x^4*d^5*c^2*b^4*a + 20*x^4*d^5*c^3*b^2*a^2 + 1/3
*x^3*d^5*b^7 + 22/3*x^3*d^5*c*b^5*a + 40/3*x^3*d^5*c^2*b^3*a^2 + x^2*d^5*b^6*a + 5*x^2*d^5*c*b^4*a^2 + x*d^5*b
^5*a^2

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Sympy [B]  time = 0.136747, size = 291, normalized size = 3.99 \begin{align*} a^{2} b^{5} d^{5} x + 16 b c^{6} d^{5} x^{9} + \frac{16 c^{7} d^{5} x^{10}}{5} + x^{8} \left (8 a c^{6} d^{5} + 34 b^{2} c^{5} d^{5}\right ) + x^{7} \left (32 a b c^{5} d^{5} + 40 b^{3} c^{4} d^{5}\right ) + x^{6} \left (\frac{16 a^{2} c^{5} d^{5}}{3} + \frac{160 a b^{2} c^{4} d^{5}}{3} + \frac{85 b^{4} c^{3} d^{5}}{3}\right ) + x^{5} \left (16 a^{2} b c^{4} d^{5} + 48 a b^{3} c^{3} d^{5} + \frac{61 b^{5} c^{2} d^{5}}{5}\right ) + x^{4} \left (20 a^{2} b^{2} c^{3} d^{5} + 25 a b^{4} c^{2} d^{5} + 3 b^{6} c d^{5}\right ) + x^{3} \left (\frac{40 a^{2} b^{3} c^{2} d^{5}}{3} + \frac{22 a b^{5} c d^{5}}{3} + \frac{b^{7} d^{5}}{3}\right ) + x^{2} \left (5 a^{2} b^{4} c d^{5} + a b^{6} d^{5}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b**5*d**5*x + 16*b*c**6*d**5*x**9 + 16*c**7*d**5*x**10/5 + x**8*(8*a*c**6*d**5 + 34*b**2*c**5*d**5) + x**
7*(32*a*b*c**5*d**5 + 40*b**3*c**4*d**5) + x**6*(16*a**2*c**5*d**5/3 + 160*a*b**2*c**4*d**5/3 + 85*b**4*c**3*d
**5/3) + x**5*(16*a**2*b*c**4*d**5 + 48*a*b**3*c**3*d**5 + 61*b**5*c**2*d**5/5) + x**4*(20*a**2*b**2*c**3*d**5
+ 25*a*b**4*c**2*d**5 + 3*b**6*c*d**5) + x**3*(40*a**2*b**3*c**2*d**5/3 + 22*a*b**5*c*d**5/3 + b**7*d**5/3) +
x**2*(5*a**2*b**4*c*d**5 + a*b**6*d**5)

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Giac [B]  time = 1.18512, size = 386, normalized size = 5.29 \begin{align*} \frac{16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 34 \, b^{2} c^{5} d^{5} x^{8} + 8 \, a c^{6} d^{5} x^{8} + 40 \, b^{3} c^{4} d^{5} x^{7} + 32 \, a b c^{5} d^{5} x^{7} + \frac{85}{3} \, b^{4} c^{3} d^{5} x^{6} + \frac{160}{3} \, a b^{2} c^{4} d^{5} x^{6} + \frac{16}{3} \, a^{2} c^{5} d^{5} x^{6} + \frac{61}{5} \, b^{5} c^{2} d^{5} x^{5} + 48 \, a b^{3} c^{3} d^{5} x^{5} + 16 \, a^{2} b c^{4} d^{5} x^{5} + 3 \, b^{6} c d^{5} x^{4} + 25 \, a b^{4} c^{2} d^{5} x^{4} + 20 \, a^{2} b^{2} c^{3} d^{5} x^{4} + \frac{1}{3} \, b^{7} d^{5} x^{3} + \frac{22}{3} \, a b^{5} c d^{5} x^{3} + \frac{40}{3} \, a^{2} b^{3} c^{2} d^{5} x^{3} + a b^{6} d^{5} x^{2} + 5 \, a^{2} b^{4} c d^{5} x^{2} + a^{2} b^{5} d^{5} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

16/5*c^7*d^5*x^10 + 16*b*c^6*d^5*x^9 + 34*b^2*c^5*d^5*x^8 + 8*a*c^6*d^5*x^8 + 40*b^3*c^4*d^5*x^7 + 32*a*b*c^5*
d^5*x^7 + 85/3*b^4*c^3*d^5*x^6 + 160/3*a*b^2*c^4*d^5*x^6 + 16/3*a^2*c^5*d^5*x^6 + 61/5*b^5*c^2*d^5*x^5 + 48*a*
b^3*c^3*d^5*x^5 + 16*a^2*b*c^4*d^5*x^5 + 3*b^6*c*d^5*x^4 + 25*a*b^4*c^2*d^5*x^4 + 20*a^2*b^2*c^3*d^5*x^4 + 1/3
*b^7*d^5*x^3 + 22/3*a*b^5*c*d^5*x^3 + 40/3*a^2*b^3*c^2*d^5*x^3 + a*b^6*d^5*x^2 + 5*a^2*b^4*c*d^5*x^2 + a^2*b^5
*d^5*x