\(\int (a c+(b c+a d) x+b d x^2) \, dx\) [1762]

Optimal result

Integrand size = 19, antiderivative size = 28 \[ \int \left (a c+(b c+a d) x+b d x^2\right ) \, dx=a c x+\frac {1}{2} (b c+a d) x^2+\frac {1}{3} b d x^3 \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {1}{2} x^2 (a d+b c)+a c x+\frac {1}{3} b d x^3 \]

[In]

Int[a*c + (b*c + a*d)*x + b*d*x^2,x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = a c x+\frac {1}{2} (b c+a d) x^2+\frac {1}{3} b d x^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \left (a c+(b c+a d) x+b d x^2\right ) \, dx=a c x+\frac {1}{2} b c x^2+\frac {1}{2} a d x^2+\frac {1}{3} b d x^3 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
default	\(a c x +\frac {\left (a d +b c \right ) x^{2}}{2}+\frac {b d \,x^{3}}{3}\)	\(25\)
gosper	\(\frac {x \left (2 b d \,x^{2}+3 a d x +3 b c x +6 a c \right )}{6}\)	\(26\)
norman	\(\frac {b d \,x^{3}}{3}+\left (\frac {a d}{2}+\frac {b c}{2}\right ) x^{2}+a c x\)	\(26\)
risch	\(a c x +\frac {1}{2} a d \,x^{2}+\frac {1}{2} c b \,x^{2}+\frac {1}{3} b d \,x^{3}\)	\(27\)
parallelrisch	\(a c x +\frac {1}{2} a d \,x^{2}+\frac {1}{2} c b \,x^{2}+\frac {1}{3} b d \,x^{3}\)	\(27\)
parts	\(a c x +\frac {1}{2} a d \,x^{2}+\frac {1}{2} c b \,x^{2}+\frac {1}{3} b d \,x^{3}\)	\(27\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {1}{3} \, b d x^{3} + a c x + \frac {1}{2} \, {\left (b c + a d\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \left (a c+(b c+a d) x+b d x^2\right ) \, dx=a c x + \frac {b d x^{3}}{3} + x^{2} \left (\frac {a d}{2} + \frac {b c}{2}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {1}{3} \, b d x^{3} + a c x + \frac {1}{2} \, {\left (b c + a d\right )} x^{2} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {1}{3} \, b d x^{3} + a c x + \frac {1}{2} \, {\left (b c + a d\right )} x^{2} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {b\,d\,x^3}{3}+\left (\frac {a\,d}{2}+\frac {b\,c}{2}\right )\,x^2+a\,c\,x \]

[In]

int(a*c + x*(a*d + b*c) + b*d*x^2,x)

[Out]

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