3.430 \(\int (d x+c (a+b x)) \, dx\)

Optimal. Leaf size=24 \[ \frac{c (a+b x)^2}{2 b}+\frac{d x^2}{2} \]

[Out]

(d*x^2)/2 + (c*(a + b*x)^2)/(2*b)

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Rubi [A]  time = 0.0166772, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \frac{c (a+b x)^2}{2 b}+\frac{d x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[d*x + c*(a + b*x),x]

[Out]

(d*x^2)/2 + (c*(a + b*x)^2)/(2*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(d*x+(b*x+a)*c,x)

[Out]

d*Integral(x, x) + c*(a + b*x)**2/(2*b)

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Mathematica [A]  time = 0.00166935, size = 22, normalized size = 0.92 \[ a c x+\frac{1}{2} b c x^2+\frac{d x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[d*x + c*(a + b*x),x]

[Out]

a*c*x + (b*c*x^2)/2 + (d*x^2)/2

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Maple [A]  time = 0.001, size = 20, normalized size = 0.8 \[{\frac{d{x}^{2}}{2}}+c \left ( ax+{\frac{b{x}^{2}}{2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(d*x+(b*x+a)*c,x)

[Out]

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

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Maxima [A]  time = 0.791181, size = 27, normalized size = 1.12 \[ \frac{1}{2} \, d x^{2} + \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*x^2 + 1/2*(b*x^2 + 2*a*x)*c

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Fricas [A]  time = 0.248856, size = 1, normalized size = 0.04 \[ \frac{1}{2} x^{2} c b + \frac{1}{2} x^{2} d + x c a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2*c*b + 1/2*x^2*d + x*c*a

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Sympy [A]  time = 0.074944, size = 15, normalized size = 0.62 \[ a c x + x^{2} \left (\frac{b c}{2} + \frac{d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d*x+(b*x+a)*c,x)

[Out]

a*c*x + x**2*(b*c/2 + d/2)

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GIAC/XCAS [A]  time = 0.256977, size = 27, normalized size = 1.12 \[ \frac{1}{2} \, d x^{2} + \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*x^2 + 1/2*(b*x^2 + 2*a*x)*c