3.1123 \(\int (b d+2 c d x)^3 (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=55 \[ \frac {1}{12} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3+\frac {1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3 \]

[Out]

1/12*(-4*a*c+b^2)*d^3*(c*x^2+b*x+a)^3+1/4*d^3*(2*c*x+b)^2*(c*x^2+b*x+a)^3

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Rubi [A]  time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {692, 629} \[ \frac {1}{12} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3+\frac {1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^3)/12 + (d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^3)/4

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rubi steps

\begin {align*} \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx &=\frac {1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3+\frac {1}{4} \left (\left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx\\ &=\frac {1}{12} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^3+\frac {1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 97, normalized size = 1.76 \[ \frac {1}{3} d^3 x (b+c x) \left (3 a^2 \left (b^2+2 b c x+2 c^2 x^2\right )+a x \left (3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right )+x^2 (b+c x)^2 \left (b^2+3 b c x+3 c^2 x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*x*(b + c*x)*(3*a^2*(b^2 + 2*b*c*x + 2*c^2*x^2) + x^2*(b + c*x)^2*(b^2 + 3*b*c*x + 3*c^2*x^2) + a*x*(3*b^3
+ 11*b^2*c*x + 16*b*c^2*x^2 + 8*c^3*x^3)))/3

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fricas [B]  time = 1.83, size = 193, normalized size = 3.51 \[ x^{8} d^{3} c^{5} + 4 x^{7} d^{3} c^{4} b + \frac {19}{3} x^{6} d^{3} c^{3} b^{2} + \frac {8}{3} x^{6} d^{3} c^{4} a + 5 x^{5} d^{3} c^{2} b^{3} + 8 x^{5} d^{3} c^{3} b a + 2 x^{4} d^{3} c b^{4} + 9 x^{4} d^{3} c^{2} b^{2} a + 2 x^{4} d^{3} c^{3} a^{2} + \frac {1}{3} x^{3} d^{3} b^{5} + \frac {14}{3} x^{3} d^{3} c b^{3} a + 4 x^{3} d^{3} c^{2} b a^{2} + x^{2} d^{3} b^{4} a + 3 x^{2} d^{3} c b^{2} a^{2} + x d^{3} b^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^8*d^3*c^5 + 4*x^7*d^3*c^4*b + 19/3*x^6*d^3*c^3*b^2 + 8/3*x^6*d^3*c^4*a + 5*x^5*d^3*c^2*b^3 + 8*x^5*d^3*c^3*b
*a + 2*x^4*d^3*c*b^4 + 9*x^4*d^3*c^2*b^2*a + 2*x^4*d^3*c^3*a^2 + 1/3*x^3*d^3*b^5 + 14/3*x^3*d^3*c*b^3*a + 4*x^
3*d^3*c^2*b*a^2 + x^2*d^3*b^4*a + 3*x^2*d^3*c*b^2*a^2 + x*d^3*b^3*a^2

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giac [B]  time = 0.19, size = 124, normalized size = 2.25 \[ {\left (c d x^{2} + b d x\right )} a^{2} b^{2} d^{2} + \frac {3 \, {\left (c d x^{2} + b d x\right )}^{2} a b^{2} d^{2} + 6 \, {\left (c d x^{2} + b d x\right )}^{2} a^{2} c d^{2} + {\left (c d x^{2} + b d x\right )}^{3} b^{2} d + 8 \, {\left (c d x^{2} + b d x\right )}^{3} a c d + 3 \, {\left (c d x^{2} + b d x\right )}^{4} c}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 237, normalized size = 4.31 \[ c^{5} d^{3} x^{8}+4 b \,c^{4} d^{3} x^{7}+a^{2} b^{3} d^{3} x +\frac {\left (30 b^{2} c^{3} d^{3}+8 \left (2 a c +b^{2}\right ) c^{3} d^{3}\right ) x^{6}}{6}+\frac {\left (16 a b \,c^{3} d^{3}+13 b^{3} c^{2} d^{3}+12 \left (2 a c +b^{2}\right ) b \,c^{2} d^{3}\right ) x^{5}}{5}+\frac {\left (8 a^{2} c^{3} d^{3}+24 a \,b^{2} c^{2} d^{3}+2 b^{4} c \,d^{3}+6 \left (2 a c +b^{2}\right ) b^{2} c \,d^{3}\right ) x^{4}}{4}+\frac {\left (12 a^{2} b \,c^{2} d^{3}+12 a \,b^{3} c \,d^{3}+\left (2 a c +b^{2}\right ) b^{3} d^{3}\right ) x^{3}}{3}+\frac {\left (6 b^{2} d^{3} c \,a^{2}+2 b^{4} d^{3} a \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^5*d^3*x^8+4*b*d^3*c^4*x^7+1/6*(30*b^2*c^3*d^3+8*c^3*d^3*(2*a*c+b^2))*x^6+1/5*(13*b^3*d^3*c^2+12*b*d^3*c^2*(2
*a*c+b^2)+16*c^3*d^3*a*b)*x^5+1/4*(2*b^4*d^3*c+6*b^2*d^3*c*(2*a*c+b^2)+24*b^2*d^3*c^2*a+8*c^3*d^3*a^2)*x^4+1/3
*(b^3*d^3*(2*a*c+b^2)+12*b^3*d^3*c*a+12*b*d^3*c^2*a^2)*x^3+1/2*(6*a^2*b^2*c*d^3+2*a*b^4*d^3)*x^2+b^3*d^3*a^2*x

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maxima [B]  time = 1.36, size = 161, normalized size = 2.93 \[ c^{5} d^{3} x^{8} + 4 \, b c^{4} d^{3} x^{7} + \frac {1}{3} \, {\left (19 \, b^{2} c^{3} + 8 \, a c^{4}\right )} d^{3} x^{6} + a^{2} b^{3} d^{3} x + {\left (5 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d^{3} x^{5} + {\left (2 \, b^{4} c + 9 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d^{3} x^{4} + \frac {1}{3} \, {\left (b^{5} + 14 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} d^{3} x^{3} + {\left (a b^{4} + 3 \, a^{2} b^{2} c\right )} d^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^5*d^3*x^8 + 4*b*c^4*d^3*x^7 + 1/3*(19*b^2*c^3 + 8*a*c^4)*d^3*x^6 + a^2*b^3*d^3*x + (5*b^3*c^2 + 8*a*b*c^3)*d
^3*x^5 + (2*b^4*c + 9*a*b^2*c^2 + 2*a^2*c^3)*d^3*x^4 + 1/3*(b^5 + 14*a*b^3*c + 12*a^2*b*c^2)*d^3*x^3 + (a*b^4
+ 3*a^2*b^2*c)*d^3*x^2

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mupad [B]  time = 0.07, size = 152, normalized size = 2.76 \[ c^5\,d^3\,x^8+a^2\,b^3\,d^3\,x+4\,b\,c^4\,d^3\,x^7+\frac {b\,d^3\,x^3\,\left (12\,a^2\,c^2+14\,a\,b^2\,c+b^4\right )}{3}+\frac {c^3\,d^3\,x^6\,\left (19\,b^2+8\,a\,c\right )}{3}+c\,d^3\,x^4\,\left (2\,a^2\,c^2+9\,a\,b^2\,c+2\,b^4\right )+a\,b^2\,d^3\,x^2\,\left (b^2+3\,a\,c\right )+b\,c^2\,d^3\,x^5\,\left (5\,b^2+8\,a\,c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^5*d^3*x^8 + a^2*b^3*d^3*x + 4*b*c^4*d^3*x^7 + (b*d^3*x^3*(b^4 + 12*a^2*c^2 + 14*a*b^2*c))/3 + (c^3*d^3*x^6*(
8*a*c + 19*b^2))/3 + c*d^3*x^4*(2*b^4 + 2*a^2*c^2 + 9*a*b^2*c) + a*b^2*d^3*x^2*(3*a*c + b^2) + b*c^2*d^3*x^5*(
8*a*c + 5*b^2)

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sympy [B]  time = 0.11, size = 194, normalized size = 3.53 \[ a^{2} b^{3} d^{3} x + 4 b c^{4} d^{3} x^{7} + c^{5} d^{3} x^{8} + x^{6} \left (\frac {8 a c^{4} d^{3}}{3} + \frac {19 b^{2} c^{3} d^{3}}{3}\right ) + x^{5} \left (8 a b c^{3} d^{3} + 5 b^{3} c^{2} d^{3}\right ) + x^{4} \left (2 a^{2} c^{3} d^{3} + 9 a b^{2} c^{2} d^{3} + 2 b^{4} c d^{3}\right ) + x^{3} \left (4 a^{2} b c^{2} d^{3} + \frac {14 a b^{3} c d^{3}}{3} + \frac {b^{5} d^{3}}{3}\right ) + x^{2} \left (3 a^{2} b^{2} c d^{3} + a b^{4} d^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*b**3*d**3*x + 4*b*c**4*d**3*x**7 + c**5*d**3*x**8 + x**6*(8*a*c**4*d**3/3 + 19*b**2*c**3*d**3/3) + x**5*(
8*a*b*c**3*d**3 + 5*b**3*c**2*d**3) + x**4*(2*a**2*c**3*d**3 + 9*a*b**2*c**2*d**3 + 2*b**4*c*d**3) + x**3*(4*a
**2*b*c**2*d**3 + 14*a*b**3*c*d**3/3 + b**5*d**3/3) + x**2*(3*a**2*b**2*c*d**3 + a*b**4*d**3)

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