Optimal. Leaf size=75 \[ \frac{d^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}}+d^2 (b+2 c x) \sqrt{a+b x+c x^2} \]
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Rubi [A] time = 0.0297058, 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 = {692, 621, 206} \[ \frac{d^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}}+d^2 (b+2 c x) \sqrt{a+b x+c x^2} \]
Antiderivative was successfully verified.
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Rule 692
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx &=d^2 (b+2 c x) \sqrt{a+b x+c x^2}+\frac{1}{2} \left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx\\ &=d^2 (b+2 c x) \sqrt{a+b x+c x^2}+\left (\left (b^2-4 a c\right ) d^2\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 )\\ &=d^2 (b+2 c x) \sqrt{a+b x+c x^2}+\frac{\left (b^2-4 a c\right ) d^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0551479, size = 71, normalized size = 0.95 \[ d^2 \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{2 \sqrt{c}}+(b+2 c x) \sqrt{a+x (b+c x)}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 108, normalized size = 1.4 \begin{align*} 2\,{d}^{2}cx\sqrt{c{x}^{2}+bx+a}+{d}^{2}b\sqrt{c{x}^{2}+bx+a}+{\frac{{b}^{2}{d}^{2}}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-2\,{d}^{2}\sqrt{c}a\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \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 = 3.02853, size = 462, normalized size = 6.16 \begin{align*} \left [-\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{c} d^{2} \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 \, c^{2} d^{2} x + b c d^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \, c}, -\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{-c} d^{2} \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 \, c^{2} d^{2} x + b c d^{2}\right )} \sqrt{c x^{2} + b x + a}}{2 \, c}\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*} d^{2} \left (\int \frac{b^{2}}{\sqrt{a + b x + c x^{2}}}\, dx + \int \frac{4 c^{2} x^{2}}{\sqrt{a + b x + c x^{2}}}\, dx + \int \frac{4 b c x}{\sqrt{a + b x + c x^{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15866, size = 105, normalized size = 1.4 \begin{align*}{\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt{c x^{2} + b x + a} - \frac{{\left (b^{2} d^{2} - 4 \, a c d^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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