Optimal. Leaf size=13 \[ \frac {b^2 \log (a+b x)}{d^3} \]
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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 31}
\begin {gather*} \frac {b^2 \log (a+b x)}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 31
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{\left (\frac {a d}{b}+d x\right )^3} \, dx &=\frac {b^3 \int \frac {1}{a+b x} \, dx}{d^3}\\ &=\frac {b^2 \log (a+b x)}{d^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^2 \log (a+b x)}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 14, normalized size = 1.08
method | result | size |
default | \(\frac {b^{2} \ln \left (b x +a \right )}{d^{3}}\) | \(14\) |
norman | \(\frac {b^{2} \ln \left (b x +a \right )}{d^{3}}\) | \(14\) |
risch | \(\frac {b^{2} \ln \left (b x +a \right )}{d^{3}}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^{2} \log \left (b x + a\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^{2} \log \left (b x + a\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 19, normalized size = 1.46 \begin {gather*} \frac {b^{2} \log {\left (a d^{3} + b d^{3} x \right )}}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.07, size = 14, normalized size = 1.08 \begin {gather*} \frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^2\,\ln \left (a+b\,x\right )}{d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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