Optimal. Leaf size=37 \[ \frac {2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) d^2 (b+2 c x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {696}
\begin {gather*} \frac {2 \sqrt {a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 696
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^2 \sqrt {a+b x+c x^2}} \, dx &=\frac {2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) d^2 (b+2 c x)}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 36, normalized size = 0.97 \begin {gather*} \frac {2 \sqrt {a+x (b+c x)}}{\left (b^2-4 a c\right ) d^2 (b+2 c x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.81, size = 61, normalized size = 1.65
method | result | size |
gosper | \(-\frac {2 \sqrt {c \,x^{2}+b x +a}}{\left (2 c x +b \right ) d^{2} \left (4 a c -b^{2}\right )}\) | \(38\) |
trager | \(-\frac {2 \sqrt {c \,x^{2}+b x +a}}{\left (2 c x +b \right ) d^{2} \left (4 a c -b^{2}\right )}\) | \(38\) |
default | \(-\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{d^{2} c \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.06, size = 48, normalized size = 1.30 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x + a}}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x + {\left (b^{3} - 4 \, a b c\right )} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{b^{2} \sqrt {a + b x + c x^{2}} + 4 b c x \sqrt {a + b x + c x^{2}} + 4 c^{2} x^{2} \sqrt {a + b x + c x^{2}}}\, dx}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 139 vs.
\(2 (35) = 70\).
time = 4.42, size = 139, normalized size = 3.76 \begin {gather*} -\frac {\sqrt {c} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{b^{2} c d^{2} - 4 \, a c^{2} d^{2}} + \frac {\sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}}{b^{2} c d^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 4 \, a c^{2} d^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 37, normalized size = 1.00 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x+a}}{d^2\,\left (4\,a\,c-b^2\right )\,\left (b+2\,c\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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