Optimal. Leaf size=18 \[ \frac {x}{b}-\frac {a \log (a+b x)}{b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 45}
\begin {gather*} \frac {x}{b}-\frac {a \log (a+b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^3}{a x^2+b x^3} \, dx &=\int \frac {x}{a+b x} \, dx\\ &=\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx\\ &=\frac {x}{b}-\frac {a \log (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {x}{b}-\frac {a \log (a+b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 19, normalized size = 1.06
method | result | size |
default | \(\frac {x}{b}-\frac {a \ln \left (b x +a \right )}{b^{2}}\) | \(19\) |
norman | \(\frac {x}{b}-\frac {a \ln \left (b x +a \right )}{b^{2}}\) | \(19\) |
risch | \(\frac {x}{b}-\frac {a \ln \left (b x +a \right )}{b^{2}}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 18, normalized size = 1.00 \begin {gather*} \frac {x}{b} - \frac {a \log \left (b x + a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.35, size = 17, normalized size = 0.94 \begin {gather*} \frac {b x - a \log \left (b x + a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 14, normalized size = 0.78 \begin {gather*} - \frac {a \log {\left (a + b x \right )}}{b^{2}} + \frac {x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.03, size = 19, normalized size = 1.06 \begin {gather*} \frac {x}{b} - \frac {a \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} -\frac {a\,\ln \left (a+b\,x\right )-b\,x}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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