Optimal. Leaf size=123 \[ -\frac{d^2 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32 c^{3/2}}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16 c}+\frac{d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{8 c} \]
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Rubi [A] time = 0.0559653, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {685, 692, 621, 206} \[ -\frac{d^2 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32 c^{3/2}}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16 c}+\frac{d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{8 c} \]
Antiderivative was successfully verified.
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Rule 685
Rule 692
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (b d+2 c d x)^2 \sqrt{a+b x+c x^2} \, dx &=\frac{d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{8 c}-\frac{\left (b^2-4 a c\right ) \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx}{16 c}\\ &=-\frac{\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt{a+b x+c x^2}}{16 c}+\frac{d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{8 c}-\frac{\left (\left (b^2-4 a c\right )^2 d^2\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{32 c}\\ &=-\frac{\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt{a+b x+c x^2}}{16 c}+\frac{d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{8 c}-\frac{\left (\left (b^2-4 a c\right )^2 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 )}{16 c}\\ &=-\frac{\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt{a+b x+c x^2}}{16 c}+\frac{d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{8 c}-\frac{\left (b^2-4 a c\right )^2 d^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.371687, size = 140, normalized size = 1.14 \[ \frac{d^2 \sqrt{a+x (b+c x)} \left (2 (b+2 c x) \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )-\frac{c^{3/2} \sqrt{4 a-\frac{b^2}{c}} (a+x (b+c x)) \sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )}{\left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/2}}\right )}{32 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 230, normalized size = 1.9 \begin{align*}{d}^{2}cx \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}+{\frac{{d}^{2}b}{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}{d}^{2}x}{8}\sqrt{c{x}^{2}+bx+a}}+{\frac{{d}^{2}{b}^{3}}{16\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{2}{d}^{2}a}{4}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{d}^{2}{b}^{4}}{32}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{c{d}^{2}ax}{2}\sqrt{c{x}^{2}+bx+a}}-{\frac{a{d}^{2}b}{4}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{2}{d}^{2}}{2}\sqrt{c}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\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 = 2.29423, size = 702, normalized size = 5.71 \begin{align*} \left [\frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (16 \, c^{4} d^{2} x^{3} + 24 \, b c^{3} d^{2} x^{2} + 2 \,{\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{2} x +{\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a}}{64 \, c^{2}}, \frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (16 \, c^{4} d^{2} x^{3} + 24 \, b c^{3} d^{2} x^{2} + 2 \,{\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{2} x +{\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a}}{32 \, 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*} d^{2} \left (\int b^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 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 = 1.18944, size = 209, normalized size = 1.7 \begin{align*} \frac{1}{16} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \, c^{2} d^{2} x + 3 \, b c d^{2}\right )} x + \frac{5 \, b^{2} c^{3} d^{2} + 4 \, a c^{4} d^{2}}{c^{3}}\right )} x + \frac{b^{3} c^{2} d^{2} + 4 \, a b c^{3} d^{2}}{c^{3}}\right )} + \frac{{\left (b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{32 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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