Optimal. Leaf size=62 \[ -\frac {(a d+b c)^2 \log (a-b x)}{2 a b^3}+\frac {(b c-a d)^2 \log (a+b x)}{2 a b^3}-\frac {d^2 x}{b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {72} \[ -\frac {(a d+b c)^2 \log (a-b x)}{2 a b^3}+\frac {(b c-a d)^2 \log (a+b x)}{2 a b^3}-\frac {d^2 x}{b^2} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{(a-b x) (a+b x)} \, dx &=\int \left (-\frac {d^2}{b^2}+\frac {(b c+a d)^2}{2 a b^2 (a-b x)}+\frac {(-b c+a d)^2}{2 a b^2 (a+b x)}\right ) \, dx\\ &=-\frac {d^2 x}{b^2}-\frac {(b c+a d)^2 \log (a-b x)}{2 a b^3}+\frac {(b c-a d)^2 \log (a+b x)}{2 a b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 54, normalized size = 0.87 \[ \frac {-(a d+b c)^2 \log (a-b x)+(b c-a d)^2 \log (a+b x)-2 a b d^2 x}{2 a b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 76, normalized size = 1.23 \[ -\frac {2 \, a b d^{2} x - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right ) + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x - a\right )}{2 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 84, normalized size = 1.35 \[ -\frac {d^{2} x}{b^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{3}} - \frac {{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 107, normalized size = 1.73 \[ -\frac {a \,d^{2} \ln \left (b x -a \right )}{2 b^{3}}+\frac {a \,d^{2} \ln \left (b x +a \right )}{2 b^{3}}-\frac {c^{2} \ln \left (b x -a \right )}{2 a b}+\frac {c^{2} \ln \left (b x +a \right )}{2 a b}-\frac {c d \ln \left (b x -a \right )}{b^{2}}-\frac {c d \ln \left (b x +a \right )}{b^{2}}-\frac {d^{2} x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 82, normalized size = 1.32 \[ -\frac {d^{2} x}{b^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \, a b^{3}} - \frac {{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x - a\right )}{2 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 81, normalized size = 1.31 \[ \frac {\ln \left (a+b\,x\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a\,b^3}-\frac {d^2\,x}{b^2}-\frac {\ln \left (a-b\,x\right )\,\left (a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.59, size = 112, normalized size = 1.81 \[ - \frac {d^{2} x}{b^{2}} + \frac {\left (a d - b c\right )^{2} \log {\left (x + \frac {2 a^{2} c d + \frac {a \left (a d - b c\right )^{2}}{b}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 a b^{3}} - \frac {\left (a d + b c\right )^{2} \log {\left (x + \frac {2 a^{2} c d - \frac {a \left (a d + b c\right )^{2}}{b}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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