Optimal. Leaf size=26 \[ \frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2} \]
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Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 43} \begin {gather*} \frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx &=\int \frac {a+b x}{c+d x} \, dx\\ &=\int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx\\ &=\frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.96 \begin {gather*} \frac {(a d-b c) \log (c+d x)}{d^2}+\frac {b x}{d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 25, normalized size = 0.96 \begin {gather*} \frac {b d x - {\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 27, normalized size = 1.04 \begin {gather*} \frac {b x}{d} - \frac {{\left (b c - a d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 32, normalized size = 1.23 \begin {gather*} \frac {a \ln \left (d x +c \right )}{d}-\frac {b c \ln \left (d x +c \right )}{d^{2}}+\frac {b x}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 26, normalized size = 1.00 \begin {gather*} \frac {b x}{d} - \frac {{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 25, normalized size = 0.96 \begin {gather*} \frac {\ln \left (c+d\,x\right )\,\left (a\,d-b\,c\right )}{d^2}+\frac {b\,x}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 20, normalized size = 0.77 \begin {gather*} \frac {b x}{d} + \frac {\left (a d - b c\right ) \log {\left (c + d x \right )}}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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