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} \[ -\frac {b}{d^2 (c+d x)} \]
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 \[ -\frac {b}{d^2 (c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 16, normalized size = 1.23 \[ -\frac {b}{d^{3} x + c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 13, normalized size = 1.00 \[ -\frac {b}{{\left (d x + c\right )} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 1.08 \[ -\frac {b}{\left (d x +c \right ) d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 16, normalized size = 1.23 \[ -\frac {b}{d^{3} x + c d^{2}} \]
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 \[ -\frac {b}{d^2\,\left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 12, normalized size = 0.92 \[ - \frac {b}{c d^{2} + d^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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