Optimal. Leaf size=13 \[ -\frac {b}{d^2 (c+d x)} \]
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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32}
\begin {gather*} -\frac {b}{d^2 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {\frac {b c}{d}+b x}{(c+d x)^3} \, dx &=\frac {b \int \frac {1}{(c+d x)^2} \, dx}{d}\\ &=-\frac {b}{d^2 (c+d x)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {b}{d^2 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 14, normalized size = 1.08
method | result | size |
gosper | \(-\frac {b}{d^{2} \left (d x +c \right )}\) | \(14\) |
default | \(-\frac {b}{d^{2} \left (d x +c \right )}\) | \(14\) |
risch | \(-\frac {b}{d^{2} \left (d x +c \right )}\) | \(14\) |
norman | \(\frac {-\frac {c b}{d^{2}}-\frac {b x}{d}}{\left (d x +c \right )^{2}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 16, normalized size = 1.23 \begin {gather*} -\frac {b}{d^{3} x + c d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 16, normalized size = 1.23 \begin {gather*} -\frac {b}{d^{3} x + c d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 12, normalized size = 0.92 \begin {gather*} - \frac {b}{c d^{2} + d^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.60, size = 13, normalized size = 1.00 \begin {gather*} -\frac {b}{{\left (d x + c\right )} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 13, normalized size = 1.00 \begin {gather*} -\frac {b}{d^2\,\left (c+d\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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