Optimal. Leaf size=28 \[ a c x+\frac {1}{2} (b c+a d) x^2+\frac {1}{3} b d x^3 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {1}{2} x^2 (a d+b c)+a c x+\frac {1}{3} b d x^3 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int (a+b x) (c+d x) \, dx &=\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\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} a c x+\frac {1}{2} (b c+a d) x^2+\frac {1}{3} b d x^3 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [A]
time = 1.70, size = 23, normalized size = 0.82 \begin {gather*} x \left (a c+\frac {x \left (a d+b c\right )}{2}+\frac {b d x^2}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 25, normalized size = 0.89
method | result | size |
default | \(a c x +\frac {\left (a d +b c \right ) x^{2}}{2}+\frac {b d \,x^{3}}{3}\) | \(25\) |
norman | \(\frac {b d \,x^{3}}{3}+\left (\frac {a d}{2}+\frac {b c}{2}\right ) x^{2}+a c x\) | \(26\) |
gosper | \(\frac {1}{3} b d \,x^{3}+\frac {1}{2} x^{2} a d +\frac {1}{2} b c \,x^{2}+a c x\) | \(27\) |
risch | \(\frac {1}{3} b d \,x^{3}+\frac {1}{2} x^{2} a d +\frac {1}{2} b c \,x^{2}+a c x\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, b d x^{3} + a c x + \frac {1}{2} \, {\left (b c + a d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.26, size = 26, normalized size = 0.93 \begin {gather*} \frac {1}{3} x^{3} d b + \frac {1}{2} x^{2} c b + \frac {1}{2} x^{2} d a + x c a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.03, size = 26, normalized size = 0.93 \begin {gather*} a c x + \frac {b d x^{3}}{3} + x^{2} \left (\frac {a d}{2} + \frac {b c}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 32, normalized size = 1.14 \begin {gather*} \frac {1}{3} x^{3} b d+\frac {1}{2} x^{2} b c+\frac {1}{2} x^{2} a d+x a c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 25, normalized size = 0.89 \begin {gather*} \frac {b\,d\,x^3}{3}+\left (\frac {a\,d}{2}+\frac {b\,c}{2}\right )\,x^2+a\,c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________