Optimal. Leaf size=41 \[ \frac {\log \left ((a+b x)^2+c\right )}{2 b^2}-\frac {a \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b^2 \sqrt {c}} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {371, 635, 203, 260} \[ \frac {\log \left ((a+b x)^2+c\right )}{2 b^2}-\frac {a \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b^2 \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 371
Rule 635
Rubi steps
\begin {align*} \int \frac {x}{c+(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-a+x}{c+x^2} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{c+x^2} \, dx,x,a+b x\right )}{b^2}-\frac {a \operatorname {Subst}\left (\int \frac {1}{c+x^2} \, dx,x,a+b x\right )}{b^2}\\ &=-\frac {a \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b^2 \sqrt {c}}+\frac {\log \left (c+(a+b x)^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.93 \[ \frac {\log \left ((a+b x)^2+c\right )-\frac {2 a \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c}}}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 136, normalized size = 3.32 \[ \left [-\frac {a \sqrt {-c} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 2 \, {\left (b x + a\right )} \sqrt {-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) - c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{2} c}, -\frac {2 \, a \sqrt {c} \arctan \left (\frac {b x + a}{\sqrt {c}}\right ) - c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{2} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 43, normalized size = 1.05 \[ -\frac {a \arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{b^{2} \sqrt {c}} + \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 54, normalized size = 1.32 \[ -\frac {a \arctan \left (\frac {2 b^{2} x +2 b a}{2 b \sqrt {c}}\right )}{b^{2} \sqrt {c}}+\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+c \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 50, normalized size = 1.22 \[ -\frac {a \arctan \left (\frac {b^{2} x + a b}{b \sqrt {c}}\right )}{b^{2} \sqrt {c}} + \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.08, size = 46, normalized size = 1.12 \[ \frac {\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+c\right )}{2\,b^2}-\frac {a\,\mathrm {atan}\left (\frac {a}{\sqrt {c}}+\frac {b\,x}{\sqrt {c}}\right )}{b^2\,\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.25, size = 124, normalized size = 3.02 \[ \left (- \frac {a \sqrt {- c}}{2 b^{2} c} + \frac {1}{2 b^{2}}\right ) \log {\left (x + \frac {a^{2} - 2 b^{2} c \left (- \frac {a \sqrt {- c}}{2 b^{2} c} + \frac {1}{2 b^{2}}\right ) + c}{a b} \right )} + \left (\frac {a \sqrt {- c}}{2 b^{2} c} + \frac {1}{2 b^{2}}\right ) \log {\left (x + \frac {a^{2} - 2 b^{2} c \left (\frac {a \sqrt {- c}}{2 b^{2} c} + \frac {1}{2 b^{2}}\right ) + c}{a b} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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