Optimal. Leaf size=14 \[ -\frac {1}{2 d (c+d x)^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 32}
\begin {gather*} -\frac {1}{2 d (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 640
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac {1}{(c+d x)^3} \, dx\\ &=-\frac {1}{2 d (c+d x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{2 d (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 13, normalized size = 0.93
method | result | size |
gosper | \(-\frac {1}{2 d \left (d x +c \right )^{2}}\) | \(13\) |
default | \(-\frac {1}{2 d \left (d x +c \right )^{2}}\) | \(13\) |
risch | \(-\frac {1}{2 d \left (d x +c \right )^{2}}\) | \(13\) |
norman | \(\frac {-\frac {b a x}{d}-\frac {a^{2}}{2 d}-\frac {b^{2} x^{2}}{2 d}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 24, normalized size = 1.71 \begin {gather*} -\frac {1}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.12, size = 24, normalized size = 1.71 \begin {gather*} -\frac {1}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (12) = 24\).
time = 0.10, size = 26, normalized size = 1.86 \begin {gather*} - \frac {1}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{2 \, {\left (d x + c\right )}^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 26, normalized size = 1.86 \begin {gather*} -\frac {1}{2\,c^2\,d+4\,c\,d^2\,x+2\,d^3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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