Optimal. Leaf size=32 \[ -\frac {2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {627}
\begin {gather*} -\frac {2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 627
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 32, normalized size = 1.00 \begin {gather*} -\frac {2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 33, normalized size = 1.03
method | result | size |
gosper | \(\frac {4 c x +2 b}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\) | \(33\) |
default | \(\frac {4 c x +2 b}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\) | \(33\) |
trager | \(\frac {4 c x +2 b}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\) | \(33\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (30) = 60\).
time = 3.41, size = 61, normalized size = 1.91 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )}}{a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x} \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*} \int \frac {1}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.19, size = 41, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left (\frac {2 \, c x}{b^{2} - 4 \, a c} + \frac {b}{b^{2} - 4 \, a c}\right )}}{\sqrt {c x^{2} + b x + a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.01, size = 31, normalized size = 0.97 \begin {gather*} \frac {\frac {b}{2}+c\,x}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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