Optimal. Leaf size=55 \[ \frac {(A c-a C) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {B \log \left (a+c x^2\right )}{2 c}+\frac {C x}{c} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1810, 635, 205, 260} \[ \frac {(A c-a C) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {B \log \left (a+c x^2\right )}{2 c}+\frac {C x}{c} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 1810
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{a+c x^2} \, dx &=\int \left (\frac {C}{c}+\frac {A c-a C+B c x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {C x}{c}+\frac {\int \frac {A c-a C+B c x}{a+c x^2} \, dx}{c}\\ &=\frac {C x}{c}+B \int \frac {x}{a+c x^2} \, dx+\frac {(A c-a C) \int \frac {1}{a+c x^2} \, dx}{c}\\ &=\frac {C x}{c}+\frac {(A c-a C) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {B \log \left (a+c x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 1.02 \[ -\frac {(a C-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {B \log \left (a+c x^2\right )}{2 c}+\frac {C x}{c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 125, normalized size = 2.27 \[ \left [\frac {2 \, C a c x + B a c \log \left (c x^{2} + a\right ) + {\left (C a - A c\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{2 \, a c^{2}}, \frac {2 \, C a c x + B a c \log \left (c x^{2} + a\right ) - 2 \, {\left (C a - A c\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{2 \, a c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 48, normalized size = 0.87 \[ \frac {C x}{c} + \frac {B \log \left (c x^{2} + a\right )}{2 \, c} - \frac {{\left (C a - A c\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 59, normalized size = 1.07 \[ \frac {A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}-\frac {C a \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {B \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {C x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 48, normalized size = 0.87 \[ \frac {C x}{c} + \frac {B \log \left (c x^{2} + a\right )}{2 \, c} - \frac {{\left (C a - A c\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.73, size = 56, normalized size = 1.02 \[ \frac {B\,\ln \left (c\,x^2+a\right )}{2\,c}+\frac {C\,x}{c}+\frac {A\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {c}}-\frac {C\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.49, size = 156, normalized size = 2.84 \[ \frac {C x}{c} + \left (\frac {B}{2 c} - \frac {\sqrt {- a c^{3}} \left (- A c + C a\right )}{2 a c^{3}}\right ) \log {\left (x + \frac {B a - 2 a c \left (\frac {B}{2 c} - \frac {\sqrt {- a c^{3}} \left (- A c + C a\right )}{2 a c^{3}}\right )}{- A c + C a} \right )} + \left (\frac {B}{2 c} + \frac {\sqrt {- a c^{3}} \left (- A c + C a\right )}{2 a c^{3}}\right ) \log {\left (x + \frac {B a - 2 a c \left (\frac {B}{2 c} + \frac {\sqrt {- a c^{3}} \left (- A c + C a\right )}{2 a c^{3}}\right )}{- A c + C a} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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