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 = 0.0525633, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 1810
Rule 635
Rule 205
Rule 260
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*}
Mathematica [A] time = 0.037828, 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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Maple [A] time = 0.046, size = 59, normalized size = 1.1 \begin{align*}{\frac{Cx}{c}}+{\frac{B\ln \left ( c{x}^{2}+a \right ) }{2\,c}}+{A\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{aC}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73884, size = 296, normalized size = 5.38 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.674107, size = 156, normalized size = 2.84 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16636, size = 65, normalized size = 1.18 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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