Optimal. Leaf size=38 \[ \frac {x}{4 c d^2}+\frac {b^2-4 a c}{8 c^2 d^2 (b+2 c x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {697}
\begin {gather*} \frac {b^2-4 a c}{8 c^2 d^2 (b+2 c x)}+\frac {x}{4 c d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(b d+2 c d x)^2} \, dx &=\int \left (\frac {1}{4 c d^2}+\frac {-b^2+4 a c}{4 c d^2 (b+2 c x)^2}\right ) \, dx\\ &=\frac {x}{4 c d^2}+\frac {b^2-4 a c}{8 c^2 d^2 (b+2 c x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 1.08 \begin {gather*} \frac {\frac {b^2-4 a c}{8 c^2 (b+2 c x)}+\frac {b+2 c x}{8 c^2}}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 35, normalized size = 0.92
method | result | size |
gosper | \(-\frac {-c \,x^{2}+a}{2 \left (2 c x +b \right ) c \,d^{2}}\) | \(25\) |
norman | \(\frac {\frac {a x}{b d}+\frac {x^{2}}{2 d}}{d \left (2 c x +b \right )}\) | \(31\) |
default | \(\frac {\frac {x}{4 c}-\frac {4 a c -b^{2}}{8 c^{2} \left (2 c x +b \right )}}{d^{2}}\) | \(35\) |
risch | \(\frac {x}{4 c \,d^{2}}-\frac {a}{2 c \,d^{2} \left (2 c x +b \right )}+\frac {b^{2}}{8 c^{2} d^{2} \left (2 c x +b \right )}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 40, normalized size = 1.05 \begin {gather*} \frac {b^{2} - 4 \, a c}{8 \, {\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )}} + \frac {x}{4 \, c d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.58, size = 43, normalized size = 1.13 \begin {gather*} \frac {4 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 4 \, a c}{8 \, {\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 36, normalized size = 0.95 \begin {gather*} \frac {- 4 a c + b^{2}}{8 b c^{2} d^{2} + 16 c^{3} d^{2} x} + \frac {x}{4 c 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. 170 vs.
\(2 (34) = 68\).
time = 0.83, size = 170, normalized size = 4.47 \begin {gather*} -\frac {1}{8} \, c {\left (\frac {b^{2}}{{\left (2 \, c d x + b d\right )} c^{3} d} - \frac {2 \, b \log \left (\frac {{\left | 2 \, c d x + b d \right |}}{2 \, {\left (2 \, c d x + b d\right )}^{2} {\left | c \right |} {\left | d \right |}}\right )}{c^{3} d^{2}} - \frac {2 \, c d x + b d}{c^{3} d^{3}}\right )} + \frac {b {\left (\frac {b}{{\left (2 \, c d x + b d\right )} c} - \frac {\log \left (\frac {{\left | 2 \, c d x + b d \right |}}{2 \, {\left (2 \, c d x + b d\right )}^{2} {\left | c \right |} {\left | d \right |}}\right )}{c d}\right )}}{4 \, c d} - \frac {a}{2 \, {\left (2 \, c d x + b d\right )} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 36, normalized size = 0.95 \begin {gather*} \frac {x}{4\,c\,d^2}-\frac {\frac {a\,c}{2}-\frac {b^2}{8}}{c^2\,d^2\,\left (b+2\,c\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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