Optimal. Leaf size=17 \[ 2 d \sqrt {a+b x+c x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {643}
\begin {gather*} 2 d \sqrt {a+b x+c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rubi steps
\begin {align*} \int \frac {b d+2 c d x}{\sqrt {a+b x+c x^2}} \, dx &=2 d \sqrt {a+b x+c x^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} 2 d \sqrt {a+x (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.65, size = 16, normalized size = 0.94
method | result | size |
gosper | \(2 d \sqrt {c \,x^{2}+b x +a}\) | \(16\) |
default | \(2 d \sqrt {c \,x^{2}+b x +a}\) | \(16\) |
trager | \(2 d \sqrt {c \,x^{2}+b x +a}\) | \(16\) |
risch | \(2 d \sqrt {c \,x^{2}+b x +a}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 15, normalized size = 0.88 \begin {gather*} 2 \, \sqrt {c x^{2} + b x + a} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.30, size = 15, normalized size = 0.88 \begin {gather*} 2 \, \sqrt {c x^{2} + b x + a} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} 2 d \sqrt {a + b x + c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.76, size = 15, normalized size = 0.88 \begin {gather*} 2 \, \sqrt {c x^{2} + b x + a} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 15, normalized size = 0.88 \begin {gather*} 2\,d\,\sqrt {c\,x^2+b\,x+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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