Optimal. Leaf size=12 \[ -\frac {1}{b (a+b x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 32}
\begin {gather*} -\frac {1}{b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac {1}{(a+b x)^2} \, dx\\ &=-\frac {1}{b (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 13, normalized size = 1.08
method | result | size |
gosper | \(-\frac {1}{b \left (b x +a \right )}\) | \(13\) |
default | \(-\frac {1}{b \left (b x +a \right )}\) | \(13\) |
norman | \(\frac {x}{a \left (b x +a \right )}\) | \(13\) |
risch | \(-\frac {1}{b \left (b x +a \right )}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 1.08 \begin {gather*} -\frac {1}{b^{2} x + a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.43, size = 13, normalized size = 1.08 \begin {gather*} -\frac {1}{b^{2} x + a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 10, normalized size = 0.83 \begin {gather*} - \frac {1}{a b + b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.67, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (b x + a\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.17, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{b\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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