3.1141 \(\int (b d+2 c d x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{4} d \left (a+b x+c x^2\right )^4 \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {629} \[ \frac {1}{4} d \left (a+b x+c x^2\right )^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(a + b*x + c*x^2)^4)/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]

Rubi steps

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

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Mathematica [B]  time = 0.02, size = 52, normalized size = 3.06 \[ \frac {1}{4} d x (b+c x) \left (4 a^3+6 a^2 x (b+c x)+4 a x^2 (b+c x)^2+x^3 (b+c x)^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.74, size = 140, normalized size = 8.24 \[ \frac {1}{4} x^{8} d c^{4} + x^{7} d c^{3} b + \frac {3}{2} x^{6} d c^{2} b^{2} + x^{6} d c^{3} a + x^{5} d c b^{3} + 3 x^{5} d c^{2} b a + \frac {1}{4} x^{4} d b^{4} + 3 x^{4} d c b^{2} a + \frac {3}{2} x^{4} d c^{2} a^{2} + x^{3} d b^{3} a + 3 x^{3} d c b a^{2} + \frac {3}{2} x^{2} d b^{2} a^{2} + x^{2} d c a^{3} + x d b a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.16, size = 73, normalized size = 4.29 \[ {\left (c d x^{2} + b d x\right )} a^{3} + \frac {6 \, {\left (c d x^{2} + b d x\right )}^{2} a^{2} d^{2} + 4 \, {\left (c d x^{2} + b d x\right )}^{3} a d + {\left (c d x^{2} + b d x\right )}^{4}}{4 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 231, normalized size = 13.59 \[ \frac {c^{4} d \,x^{8}}{4}+b \,c^{3} d \,x^{7}+a^{3} b d x +\frac {\left (3 b^{2} c^{2} d +2 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c d \right ) x^{6}}{6}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b d +2 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c d \right ) x^{5}}{5}+\frac {\left (\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b d +2 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c d \right ) x^{4}}{4}+\frac {\left (6 a^{2} b c d +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b d \right ) x^{3}}{3}+\frac {\left (2 c d \,a^{3}+3 b^{2} d \,a^{2}\right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 15, normalized size = 0.88 \[ \frac {1}{4} \, {\left (c x^{2} + b x + a\right )}^{4} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*x^2 + b*x + a)^4*d

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mupad [B]  time = 0.44, size = 119, normalized size = 7.00 \[ \frac {c^4\,d\,x^8}{4}+\frac {d\,x^4\,\left (6\,a^2\,c^2+12\,a\,b^2\,c+b^4\right )}{4}+\frac {c^2\,d\,x^6\,\left (3\,b^2+2\,a\,c\right )}{2}+a^3\,b\,d\,x+b\,c^3\,d\,x^7+\frac {a^2\,d\,x^2\,\left (3\,b^2+2\,a\,c\right )}{2}+a\,b\,d\,x^3\,\left (b^2+3\,a\,c\right )+b\,c\,d\,x^5\,\left (b^2+3\,a\,c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.10, size = 144, normalized size = 8.47 \[ a^{3} b d x + b c^{3} d x^{7} + \frac {c^{4} d x^{8}}{4} + x^{6} \left (a c^{3} d + \frac {3 b^{2} c^{2} d}{2}\right ) + x^{5} \left (3 a b c^{2} d + b^{3} c d\right ) + x^{4} \left (\frac {3 a^{2} c^{2} d}{2} + 3 a b^{2} c d + \frac {b^{4} d}{4}\right ) + x^{3} \left (3 a^{2} b c d + a b^{3} d\right ) + x^{2} \left (a^{3} c d + \frac {3 a^{2} b^{2} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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