3.12.12 \(\int (b d+2 c d x) (a+b x+c x^2) \, dx\) [1112]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {643} \begin {gather*} \frac {1}{2} d \left (a+b x+c x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(a + b*x + c*x^2)^2)/2

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

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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.29 \begin {gather*} \frac {1}{2} d x (b+c x) (2 a+x (b+c x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x*(b + c*x)*(2*a + x*(b + c*x)))/2

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Maple [A]
time = 0.09, size = 16, normalized size = 0.94

method	result	size
default	\(\frac {d \left (c \,x^{2}+b x +a \right )^{2}}{2}\)	\(16\)
gosper	\(\frac {d x \left (c^{2} x^{3}+2 c \,x^{2} b +2 a c x +b^{2} x +2 a b \right )}{2}\)	\(34\)
norman	\(\left (a c d +\frac {1}{2} b^{2} d \right ) x^{2}+a b d x +b c d \,x^{3}+\frac {c^{2} d \,x^{4}}{2}\)	\(38\)
risch	\(\frac {1}{2} c^{2} d \,x^{4}+b c d \,x^{3}+a c d \,x^{2}+\frac {1}{2} x^{2} b^{2} d +a b d x +\frac {1}{2} a^{2} d\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, {\left (c x^{2} + b x + a\right )}^{2} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2 + b*x + a)^2*d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (15) = 30\).
time = 2.81, size = 36, normalized size = 2.12 \begin {gather*} \frac {1}{2} \, c^{2} d x^{4} + b c d x^{3} + a b d x + \frac {1}{2} \, {\left (b^{2} + 2 \, a c\right )} d x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (14) = 28\).
time = 0.01, size = 39, normalized size = 2.29 \begin {gather*} a b d x + b c d x^{3} + \frac {c^{2} d x^{4}}{2} + x^{2} \left (a c d + \frac {b^{2} d}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
time = 1.16, size = 32, normalized size = 1.88 \begin {gather*} {\left (c d x^{2} + b d x\right )} a + \frac {{\left (c d x^{2} + b d x\right )}^{2}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.05, size = 36, normalized size = 2.12 \begin {gather*} \frac {c^2\,d\,x^4}{2}+\frac {d\,x^2\,\left (b^2+2\,a\,c\right )}{2}+b\,c\,d\,x^3+a\,b\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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