Optimal. Leaf size=15 \[ -\frac {b^2}{d^3 (a+b x)} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, 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^2}{d^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx &=\frac {b^3 \int \frac {1}{(a+b x)^2} \, dx}{d^3}\\ &=-\frac {b^2}{d^3 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {b^2}{d^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 16, normalized size = 1.07
method | result | size |
gosper | \(-\frac {b^{2}}{d^{3} \left (b x +a \right )}\) | \(16\) |
default | \(-\frac {b^{2}}{d^{3} \left (b x +a \right )}\) | \(16\) |
risch | \(-\frac {b^{2}}{d^{3} \left (b x +a \right )}\) | \(16\) |
norman | \(\frac {-\frac {a \,b^{2}}{d}-\frac {b^{3} x}{d}}{d^{2} \left (b x +a \right )^{2}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 19, normalized size = 1.27 \begin {gather*} -\frac {b^{2}}{b d^{3} x + a d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 19, normalized size = 1.27 \begin {gather*} -\frac {b^{2}}{b d^{3} x + a d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 1.27 \begin {gather*} - \frac {b^{3}}{a b d^{3} + b^{2} d^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 15, normalized size = 1.00 \begin {gather*} -\frac {b^{2}}{{\left (b x + a\right )} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 15, normalized size = 1.00 \begin {gather*} -\frac {b^2}{d^3\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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