Optimal. Leaf size=37 \[ \frac {2 \sqrt {a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]
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Rubi [A] time = 0.02, 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 = {682} \[ \frac {2 \sqrt {a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]
Antiderivative was successfully verified.
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Rule 682
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 = 0.02, size = 36, normalized size = 0.97 \[ \frac {2 \sqrt {a+x (b+c x)}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 48, normalized size = 1.30 \[ \frac {2 \, \sqrt {c x^{2} + b x + a}}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x + {\left (b^{3} - 4 \, a b c\right )} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 139, normalized size = 3.76 \[ -\frac {\sqrt {c} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d)}{b^{2} c d^{2} - 4 \, a c^{2} d^{2}} + \frac {\sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}}{b^{2} c d^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d) - 4 \, a c^{2} d^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 38, normalized size = 1.03 \[ -\frac {2 \sqrt {c \,x^{2}+b x +a}}{\left (2 c x +b \right ) \left (4 a c -b^{2}\right ) d^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ -\frac {2\,\sqrt {c\,x^2+b\,x+a}}{d^2\,\left (4\,a\,c-b^2\right )\,\left (b+2\,c\,x\right )} \]
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 \[ \frac {\int \frac {1}{b^{2} \sqrt {a + b x + c x^{2}} + 4 b c x \sqrt {a + b x + c x^{2}} + 4 c^{2} x^{2} \sqrt {a + b x + c x^{2}}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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