Optimal. Leaf size=24 \[ \frac {d x^2}{2}+\frac {c (a+b x)^2}{2 b} \]
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Rubi [A]
time = 0.00, antiderivative size = 24, 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 = {}
\begin {gather*} \frac {c (a+b x)^2}{2 b}+\frac {d x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rubi steps
\begin {align*} \int (d x+c (a+b x)) \, dx &=\frac {d x^2}{2}+\frac {c (a+b x)^2}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.92 \begin {gather*} a c x+\frac {1}{2} b c x^2+\frac {d x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 19, normalized size = 0.79
method | result | size |
gosper | \(\frac {x \left (b c x +2 a c +d x \right )}{2}\) | \(16\) |
norman | \(\left (\frac {b c}{2}+\frac {d}{2}\right ) x^{2}+a c x\) | \(18\) |
default | \(\frac {1}{2} b c \,x^{2}+a c x +\frac {1}{2} d \,x^{2}\) | \(19\) |
risch | \(\frac {1}{2} b c \,x^{2}+a c x +\frac {1}{2} d \,x^{2}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 20, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, d x^{2} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 18, normalized size = 0.75 \begin {gather*} \frac {1}{2} x^{2} c b + \frac {1}{2} x^{2} d + x c a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 15, normalized size = 0.62 \begin {gather*} a c x + x^{2} \left (\frac {b c}{2} + \frac {d}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.61, size = 20, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, d x^{2} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 17, normalized size = 0.71 \begin {gather*} \left (\frac {d}{2}+\frac {b\,c}{2}\right )\,x^2+a\,c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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