Optimal. Leaf size=75 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2} d^2}-\frac{\sqrt{a+b x+c x^2}}{2 c d^2 (b+2 c x)} \]
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Rubi [A] time = 0.0287825, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {684, 621, 206} \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2} d^2}-\frac{\sqrt{a+b x+c x^2}}{2 c d^2 (b+2 c x)} \]
Antiderivative was successfully verified.
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Rule 684
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^2} \, dx &=-\frac{\sqrt{a+b x+c x^2}}{2 c d^2 (b+2 c x)}+\frac{\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{4 c d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{2 c d^2 (b+2 c x)}+\frac{\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 )}{2 c d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{2 c d^2 (b+2 c x)}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2} d^2}\\ \end{align*}
Mathematica [A] time = 0.299273, size = 114, normalized size = 1.52 \[ \frac{\sqrt{a+x (b+c x)} \left (\frac{\sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )}{\sqrt{4 a-\frac{b^2}{c}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}-\frac{2 \sqrt{c}}{b+2 c x}\right )}{4 c^{3/2} d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.195, size = 291, normalized size = 3.9 \begin{align*} -{\frac{1}{c{d}^{2} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}}+{\frac{x}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}+{\frac{b}{2\,c{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}+{\frac{a}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ( \left ( x+{\frac{b}{2\,c}} \right ) \sqrt{c}+\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{b}^{2}}{4\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ( \left ( x+{\frac{b}{2\,c}} \right ) \sqrt{c}+\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){c}^{-{\frac{3}{2}}}} \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 = 2.71201, size = 450, normalized size = 6. \begin{align*} \left [\frac{{\left (2 \, c x + b\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 \, \sqrt{c x^{2} + b x + a} c}{8 \,{\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )}}, -\frac{{\left (2 \, c x + b\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 \, \sqrt{c x^{2} + b x + a} c}{4 \,{\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )}}\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*} \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19959, size = 261, normalized size = 3.48 \begin{align*} -\frac{1}{4} \, d^{2}{\left (\frac{{\left (\frac{c \arctan \left (\frac{\sqrt{-\frac{b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \sqrt{-\frac{b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}\right )} \mathrm{sgn}\left (\frac{1}{2 \, c d x + b d}\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (d\right )}{c^{2} d^{4}{\left | c \right |}} - \frac{{\left (c \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) + \sqrt{-c} \sqrt{c}\right )} \mathrm{sgn}\left (\frac{1}{2 \, c d x + b d}\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (d\right )}{\sqrt{-c} c^{2} d^{4}{\left | c \right |}}\right )}{\left | c \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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