Optimal. Leaf size=45 \[ -\frac {2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {650}
\begin {gather*} -\frac {2 (-2 a e+x (2 c d-b e)+b d)}{\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 650
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 43, normalized size = 0.96 \begin {gather*} \frac {-2 b d+4 a e-4 c d x+2 b e x}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs.
\(2(43)=86\).
time = 0.73, size = 91, normalized size = 2.02
method | result | size |
gosper | \(-\frac {2 \left (b e x -2 c d x +2 a e -b d \right )}{\sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )}\) | \(45\) |
trager | \(-\frac {2 \left (b e x -2 c d x +2 a e -b d \right )}{\sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )}\) | \(45\) |
default | \(e \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\frac {2 d \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\) | \(91\) |
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 = 2.95, size = 75, normalized size = 1.67 \begin {gather*} -\frac {2 \, {\left (2 \, c d x + b d - {\left (b x + 2 \, a\right )} e\right )} \sqrt {c x^{2} + b x + a}}{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 {d + e x}{\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 = 1.43, size = 57, normalized size = 1.27 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (2 \, c d - b e\right )} x}{b^{2} - 4 \, a c} + \frac {b d - 2 \, a e}{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 = 1.17, size = 45, normalized size = 1.00 \begin {gather*} -\frac {4\,a\,e-2\,b\,d+2\,b\,e\,x-4\,c\,d\,x}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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