Optimal. Leaf size=42 \[ \frac {A \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {B \log \left (a+c x^2\right )}{2 c} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {635, 205, 260} \[ \frac {A \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {B \log \left (a+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rubi steps
\begin {align*} \int \frac {A+B x}{a+c x^2} \, dx &=A \int \frac {1}{a+c x^2} \, dx+B \int \frac {x}{a+c x^2} \, dx\\ &=\frac {A \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {B \log \left (a+c x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.00 \[ \frac {A \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {B \log \left (a+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 98, normalized size = 2.33 \[ \left [\frac {B a \log \left (c x^{2} + a\right ) - \sqrt {-a c} A \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{2 \, a c}, \frac {B a \log \left (c x^{2} + a\right ) + 2 \, \sqrt {a c} A \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{2 \, a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 31, normalized size = 0.74 \[ \frac {A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} + \frac {B \log \left (c x^{2} + a\right )}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 32, normalized size = 0.76 \[ \frac {A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}+\frac {B \ln \left (c \,x^{2}+a \right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 31, normalized size = 0.74 \[ \frac {A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} + \frac {B \log \left (c x^{2} + a\right )}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 32, normalized size = 0.76 \[ \frac {B\,\ln \left (c\,x^2+a\right )}{2\,c}+\frac {A\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.26, size = 124, normalized size = 2.95 \[ \left (- \frac {A \sqrt {- a c^{3}}}{2 a c^{2}} + \frac {B}{2 c}\right ) \log {\left (x + \frac {- B a + 2 a c \left (- \frac {A \sqrt {- a c^{3}}}{2 a c^{2}} + \frac {B}{2 c}\right )}{A c} \right )} + \left (\frac {A \sqrt {- a c^{3}}}{2 a c^{2}} + \frac {B}{2 c}\right ) \log {\left (x + \frac {- B a + 2 a c \left (\frac {A \sqrt {- a c^{3}}}{2 a c^{2}} + \frac {B}{2 c}\right )}{A c} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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