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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ a c x+\frac {1}{2} x^2 (b c+d) \]

Antiderivative was successfully verified.

[In]

Int[a*c + (b*c + d)*x,x]

[Out]

a*c*x + ((b*c + d)*x^2)/2

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 22, normalized size = 1.29 \[ a c x+\frac {1}{2} b c x^2+\frac {d x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[a*c + (b*c + d)*x,x]

[Out]

a*c*x + (b*c*x^2)/2 + (d*x^2)/2

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 18, normalized size = 1.06 \[ \frac {1}{2} x^{2} c b + \frac {1}{2} x^{2} d + x c a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*c*b + 1/2*x^2*d + x*c*a

________________________________________________________________________________________

giac [A]  time = 0.39, size = 15, normalized size = 0.88 \[ a c x + \frac {1}{2} \, {\left (b c + d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x + 1/2*(b*c + d)*x^2

________________________________________________________________________________________

maple [A]  time = 0.00, size = 16, normalized size = 0.94 \[ a c x +\frac {\left (b c +d \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.66, size = 15, normalized size = 0.88 \[ a c x + \frac {1}{2} \, {\left (b c + d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x + 1/2*(b*c + d)*x^2

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 17, normalized size = 1.00 \[ \left (\frac {d}{2}+\frac {b\,c}{2}\right )\,x^2+a\,c\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a*c + x*(d + b*c),x)

[Out]

x^2*(d/2 + (b*c)/2) + a*c*x

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 15, normalized size = 0.88 \[ a c x + x^{2} \left (\frac {b c}{2} + \frac {d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x + x**2*(b*c/2 + d/2)

________________________________________________________________________________________