Optimal. Leaf size=12 \[ c x+\frac {d x^2}{2} \]
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Rubi [A]
time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 1, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {1600}
\begin {gather*} c x+\frac {d x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1600
Rubi steps
\begin {align*} \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^2} \, dx &=\int \frac {a c+a d x+b c x^2+b d x^3}{a+b x^2} \, dx\\ &=\int (c+d x) \, dx\\ &=c x+\frac {d x^2}{2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} c x+\frac {d x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 11, normalized size = 0.92
method | result | size |
gosper | \(\frac {x \left (d x +2 c \right )}{2}\) | \(11\) |
default | \(c x +\frac {1}{2} d \,x^{2}\) | \(11\) |
risch | \(c x +\frac {1}{2} d \,x^{2}\) | \(11\) |
norman | \(\frac {a c x +b c \,x^{3}-\frac {a^{2} d}{2 b}+\frac {b d \,x^{4}}{2}}{b \,x^{2}+a}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, d x^{2} + c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, d x^{2} + c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 8, normalized size = 0.67 \begin {gather*} c x + \frac {d x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.35, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, d x^{2} + c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 10, normalized size = 0.83 \begin {gather*} \frac {d\,x^2}{2}+c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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