\(\int (b d+2 c d x) (a+b x+c x^2)^2 \, dx\) [1125]

Optimal result

Integrand size = 22, antiderivative size = 17 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{3} d \left (a+b x+c x^2\right )^3 \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {643} \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{3} d \left (a+b x+c x^2\right )^3 \]

[In]

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

[Out]

(d*(a + b*x + c*x^2)^3)/3

Rule 643

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*} \text {integral}& = \frac {1}{3} d \left (a+b x+c x^2\right )^3 \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(17)=34\).

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.18 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{3} d x (b+c x) \left (3 a^2+3 a x (b+c x)+x^2 (b+c x)^2\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {d \left (c \,x^{2}+b x +a \right )^{3}}{3}\)	\(16\)
gosper	\(\frac {d x \left (c^{3} x^{5}+3 b \,c^{2} x^{4}+3 a \,c^{2} x^{3}+3 b^{2} c \,x^{3}+6 a b c \,x^{2}+b^{3} x^{2}+3 a^{2} c x +3 a \,b^{2} x +3 a^{2} b \right )}{3}\)	\(75\)
norman	\(\left (2 a b c d +\frac {1}{3} b^{3} d \right ) x^{3}+\left (a \,c^{2} d +b^{2} c d \right ) x^{4}+\left (a^{2} c d +a \,b^{2} d \right ) x^{2}+a^{2} b d x +b \,c^{2} d \,x^{5}+\frac {c^{3} d \,x^{6}}{3}\)	\(78\)
parallelrisch	\(\frac {1}{3} c^{3} d \,x^{6}+b \,c^{2} d \,x^{5}+a \,c^{2} d \,x^{4}+b^{2} c d \,x^{4}+2 x^{3} b d a c +\frac {1}{3} b^{3} d \,x^{3}+a^{2} c d \,x^{2}+a \,b^{2} d \,x^{2}+a^{2} b d x\)	\(81\)
risch	\(\frac {1}{3} c^{3} d \,x^{6}+b \,c^{2} d \,x^{5}+a \,c^{2} d \,x^{4}+b^{2} c d \,x^{4}+2 x^{3} b d a c +\frac {1}{3} b^{3} d \,x^{3}+a^{2} c d \,x^{2}+a \,b^{2} d \,x^{2}+a^{2} b d x +\frac {1}{3} a^{3} d\)	\(87\)

[In]

int((2*c*d*x+b*d)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (15) = 30\).

Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 4.29 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{3} \, c^{3} d x^{6} + b c^{2} d x^{5} + {\left (b^{2} c + a c^{2}\right )} d x^{4} + a^{2} b d x + \frac {1}{3} \, {\left (b^{3} + 6 \, a b c\right )} d x^{3} + {\left (a b^{2} + a^{2} c\right )} d x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (14) = 28\).

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.71 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx=a^{2} b d x + b c^{2} d x^{5} + \frac {c^{3} d x^{6}}{3} + x^{4} \left (a c^{2} d + b^{2} c d\right ) + x^{3} \cdot \left (2 a b c d + \frac {b^{3} d}{3}\right ) + x^{2} \left (a^{2} c d + a b^{2} d\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{3} \, {\left (c x^{2} + b x + a\right )}^{3} d \]

[In]

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

[Out]

1/3*(c*x^2 + b*x + a)^3*d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (15) = 30\).

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 3.06 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx={\left (c d x^{2} + b d x\right )} a^{2} + \frac {3 \, {\left (c d x^{2} + b d x\right )}^{2} a d + {\left (c d x^{2} + b d x\right )}^{3}}{3 \, d^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.89 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.94 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx=\frac {c^3\,d\,x^6}{3}+a\,d\,x^2\,\left (b^2+a\,c\right )+\frac {b\,d\,x^3\,\left (b^2+6\,a\,c\right )}{3}+c\,d\,x^4\,\left (b^2+a\,c\right )+a^2\,b\,d\,x+b\,c^2\,d\,x^5 \]

[In]

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

[Out]

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