3.155 \(\int c (a+b x) \, dx\)

Optimal. Leaf size=15 \[ \frac{c (a+b x)^2}{2 b} \]

[Out]

(c*(a + b*x)^2)/(2*b)

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Rubi [A]  time = 0.00743257, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{c (a+b x)^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[c*(a + b*x),x]

[Out]

(c*(a + b*x)^2)/(2*b)

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Rubi in Sympy [A]  time = 1.70861, size = 10, normalized size = 0.67 \[ \frac{c \left (a + b x\right )^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(c*(b*x+a),x)

[Out]

c*(a + b*x)**2/(2*b)

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Mathematica [A]  time = 0.000647646, size = 14, normalized size = 0.93 \[ c \left (a x+\frac{b x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[c*(a + b*x),x]

[Out]

c*(a*x + (b*x^2)/2)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \[ c \left ( ax+{\frac{b{x}^{2}}{2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(c*(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.34169, size = 18, normalized size = 1.2 \[ \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 + 2*a*x)*c

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Fricas [A]  time = 0.170187, size = 1, normalized size = 0.07 \[ \frac{1}{2} x^{2} c b + x c a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2*c*b + x*c*a

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Sympy [A]  time = 0.058057, size = 12, normalized size = 0.8 \[ a c x + \frac{b c x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(c*(b*x+a),x)

[Out]

a*c*x + b*c*x**2/2

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GIAC/XCAS [A]  time = 0.203124, size = 18, normalized size = 1.2 \[ \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 + 2*a*x)*c