Optimal. Leaf size=122 \[ \frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^2}-\frac{e \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (-b e+8 c d+6 c e x)}{12 c} \]
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Rubi [A] time = 0.0904475, antiderivative size = 122, 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 = {779, 612, 621, 206} \[ \frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^2}-\frac{e \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (-b e+8 c d+6 c e x)}{12 c} \]
Antiderivative was successfully verified.
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Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (b+2 c x) (d+e x) \sqrt{a+b x+c x^2} \, dx &=\frac{(8 c d-b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac{\left (\left (b^2-4 a c\right ) e\right ) \int \sqrt{a+b x+c x^2} \, dx}{8 c}\\ &=\frac{\left (b^2-4 a c\right ) e (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^2}+\frac{(8 c d-b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac{\left (\left (b^2-4 a c\right )^2 e\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{64 c^2}\\ &=\frac{\left (b^2-4 a c\right ) e (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^2}+\frac{(8 c d-b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac{\left (\left (b^2-4 a c\right )^2 e\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 )}{32 c^2}\\ &=\frac{\left (b^2-4 a c\right ) e (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^2}+\frac{(8 c d-b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac{\left (b^2-4 a c\right )^2 e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.150707, size = 134, normalized size = 1.1 \[ \frac{\sqrt{a+x (b+c x)} \left (4 a c (-5 b e+16 c d+6 c e x)-2 b^2 c e x+3 b^3 e+8 b c^2 x (8 d+5 e x)+16 c^3 x^2 (4 d+3 e x)\right )}{96 c^2}-\frac{e \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{64 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 235, normalized size = 1.9 \begin{align*}{\frac{ex}{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{be}{12\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}ex}{16\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{3}e}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{2}ea}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{e{b}^{4}}{64}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{aex}{4}\sqrt{c{x}^{2}+bx+a}}-{\frac{abe}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{e{a}^{2}}{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{2\,d}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\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 = 1.31944, size = 799, normalized size = 6.55 \begin{align*} \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{c} e \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 (48 \, c^{4} e x^{3} + 64 \, a c^{3} d + 8 \,{\left (8 \, c^{4} d + 5 \, b c^{3} e\right )} x^{2} +{\left (3 \, b^{3} c - 20 \, a b c^{2}\right )} e + 2 \,{\left (32 \, b c^{3} d -{\left (b^{2} c^{2} - 12 \, a c^{3}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{384 \, c^{3}}, \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-c} e \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 (48 \, c^{4} e x^{3} + 64 \, a c^{3} d + 8 \,{\left (8 \, c^{4} d + 5 \, b c^{3} e\right )} x^{2} +{\left (3 \, b^{3} c - 20 \, a b c^{2}\right )} e + 2 \,{\left (32 \, b c^{3} d -{\left (b^{2} c^{2} - 12 \, a c^{3}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{192 \, c^{3}}\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 \left (b + 2 c x\right ) \left (d + e x\right ) \sqrt{a + b x + c x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1841, size = 230, normalized size = 1.89 \begin{align*} \frac{1}{96} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, c x e + \frac{8 \, c^{4} d + 5 \, b c^{3} e}{c^{3}}\right )} x + \frac{32 \, b c^{3} d - b^{2} c^{2} e + 12 \, a c^{3} e}{c^{3}}\right )} x + \frac{64 \, a c^{3} d + 3 \, b^{3} c e - 20 \, a b c^{2} e}{c^{3}}\right )} + \frac{{\left (b^{4} e - 8 \, a b^{2} c e + 16 \, a^{2} c^{2} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{64 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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