Optimal. Leaf size=13 \[ \frac {b^2 \log (c+d x)}{d^3} \]
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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 (c+d x)}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 31
Rubi steps
\begin {align*} \int \frac {\left (\frac {b c}{d}+b x\right )^2}{(c+d x)^3} \, dx &=\frac {b^2 \int \frac {1}{c+d x} \, dx}{d^2}\\ &=\frac {b^2 \log (c+d 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 (c+d x)}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 14, normalized size = 1.08
method | result | size |
default | \(\frac {b^{2} \ln \left (d x +c \right )}{d^{3}}\) | \(14\) |
norman | \(\frac {b^{2} \ln \left (d x +c \right )}{d^{3}}\) | \(14\) |
risch | \(\frac {b^{2} \ln \left (d x +c \right )}{d^{3}}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^{2} \log \left (d x + c\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^{2} \log \left (d x + c\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 17, normalized size = 1.31 \begin {gather*} \frac {b^{2} \log {\left (c d^{2} + 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 = 2.14, size = 14, normalized size = 1.08 \begin {gather*} \frac {b^{2} \log \left ({\left | d x + c \right |}\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^2\,\ln \left (c+d\,x\right )}{d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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