3.1101 \(\int (b d+2 c d x) \left (a+b x+c x^2\right ) \, dx\)

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

[Out]

(d*(a + b*x + c*x^2)^2)/2

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Rubi [A]  time = 0.0144629, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{1}{2} d \left (a+b x+c x^2\right )^2 \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d \int ^{a + b x + c x^{2}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)*(c*x**2+b*x+a),x)

[Out]

d*Integral(x, (x, a + b*x + c*x**2))

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Mathematica [A]  time = 0.00966029, size = 22, normalized size = 1.29 \[ \frac{1}{2} d x (b+c x) (2 a+x (b+c x)) \]

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 [B]  time = 0., size = 39, normalized size = 2.3 \[{\frac{{c}^{2}d{x}^{4}}{2}}+bcd{x}^{3}+{\frac{ \left ( 2\,acd+{b}^{2}d \right ){x}^{2}}{2}}+bdax \]

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)

[Out]

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

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Maxima [A]  time = 0.671946, size = 20, normalized size = 1.18 \[ \frac{1}{2} \,{\left (c x^{2} + b x + a\right )}^{2} d \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(c*x^2 + b*x + a)^2*d

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Fricas [A]  time = 0.190148, size = 1, normalized size = 0.06 \[ \frac{1}{2} x^{4} d c^{2} + x^{3} d c b + \frac{1}{2} x^{2} d b^{2} + x^{2} d c a + x d b a \]

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

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Sympy [A]  time = 0.094575, size = 39, normalized size = 2.29 \[ a b d x + b c d x^{3} + \frac{c^{2} d x^{4}}{2} + x^{2} \left (a c d + \frac{b^{2} d}{2}\right ) \]

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/XCAS [A]  time = 0.212538, size = 51, normalized size = 3. \[ \frac{1}{2} \, c^{2} d x^{4} + b c d x^{3} + \frac{1}{2} \, b^{2} d x^{2} + a c d x^{2} + a b d x \]

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]

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