Optimal. Leaf size=129 \[ -\frac{\left (-4 a B c-4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2}}+\frac{\sqrt{a+b x+c x^2} (4 A c+b B+2 B c x)}{4 c}-\sqrt{a} A \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right ) \]
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Rubi [A] time = 0.103875, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {814, 843, 621, 206, 724} \[ -\frac{\left (-4 a B c-4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2}}+\frac{\sqrt{a+b x+c x^2} (4 A c+b B+2 B c x)}{4 c}-\sqrt{a} A \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 814
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a+b x+c x^2}}{x} \, dx &=\frac{(b B+4 A c+2 B c x) \sqrt{a+b x+c x^2}}{4 c}-\frac{\int \frac{-4 a A c+\frac{1}{2} \left (b^2 B-4 A b c-4 a B c\right ) x}{x \sqrt{a+b x+c x^2}} \, dx}{4 c}\\ &=\frac{(b B+4 A c+2 B c x) \sqrt{a+b x+c x^2}}{4 c}+(a A) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx-\frac{\left (b^2 B-4 A b c-4 a B c\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 c}\\ &=\frac{(b B+4 A c+2 B c x) \sqrt{a+b x+c x^2}}{4 c}-(2 a A) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )-\frac{\left (b^2 B-4 A b c-4 a B c\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{4 c}\\ &=\frac{(b B+4 A c+2 B c x) \sqrt{a+b x+c x^2}}{4 c}-\sqrt{a} A \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )-\frac{\left (b^2 B-4 A b c-4 a B c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.175193, size = 127, normalized size = 0.98 \[ \frac{\left (4 a B c+4 A b c+b^2 (-B)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{8 c^{3/2}}+\frac{\sqrt{a+x (b+c x)} (4 A c+b B+2 B c x)}{4 c}-\sqrt{a} A \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 184, normalized size = 1.4 \begin{align*}{\frac{Bx}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{bB}{4\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{aB}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{b}^{2}B}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+A\sqrt{c{x}^{2}+bx+a}+{\frac{Ab}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-A\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) \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 = 7.39224, size = 1575, normalized size = 12.21 \begin{align*} \left [\frac{8 \, A \sqrt{a} c^{2} \log \left (-\frac{8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{2}}\right ) -{\left (B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \,{\left (2 \, B c^{2} x + B b c + 4 \, A c^{2}\right )} \sqrt{c x^{2} + b x + a}}{16 \, c^{2}}, \frac{4 \, A \sqrt{a} c^{2} \log \left (-\frac{8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{2}}\right ) +{\left (B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \,{\left (2 \, B c^{2} x + B b c + 4 \, A c^{2}\right )} \sqrt{c x^{2} + b x + a}}{8 \, c^{2}}, \frac{16 \, A \sqrt{-a} c^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{2} + a b x + a^{2}\right )}}\right ) -{\left (B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \,{\left (2 \, B c^{2} x + B b c + 4 \, A c^{2}\right )} \sqrt{c x^{2} + b x + a}}{16 \, c^{2}}, \frac{8 \, A \sqrt{-a} c^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{2} + a b x + a^{2}\right )}}\right ) +{\left (B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \,{\left (2 \, B c^{2} x + B b c + 4 \, A c^{2}\right )} \sqrt{c x^{2} + b x + a}}{8 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \sqrt{a + b x + c x^{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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