Optimal. Leaf size=122 \[ -\frac{e \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{5/2}}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^2}+\frac{\left (a+b x+c x^2\right )^{3/2} (-b e+8 c d+6 c e x)}{12 c} \]
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Rubi [A] time = 0.217473, 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 \[ -\frac{e \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{5/2}}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^2}+\frac{\left (a+b x+c x^2\right )^{3/2} (-b e+8 c d+6 c e x)}{12 c} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x)*(d + e*x)*Sqrt[a + b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 20.4766, size = 114, normalized size = 0.93 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (b e - 8 c d - 6 c e x\right )}{12 c} + \frac{e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{32 c^{2}} - \frac{e \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{64 c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)*(e*x+d)*(c*x**2+b*x+a)**(1/2),x)
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Mathematica [A] time = 0.269724, size = 132, normalized size = 1.08 \[ \frac{\sqrt{a+x (b+c x)} \left (4 a c (-5 b e+16 c d+6 c e x)+3 b^3 e-2 b^2 c e x+8 b c^2 x (8 d+5 e x)+16 c^3 x^2 (4 d+3 e x)\right )}{96 c^2}-\frac{e \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{64 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x)*(d + e*x)*Sqrt[a + b*x + c*x^2],x]
[Out]
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Maple [B] time = 0.01, size = 235, normalized size = 1.9 \[ -{\frac{be}{12\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{2\,d}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}ex}{16\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{3}e}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{ae{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{{b}^{4}e}{64}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{ex}{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{aex}{4}\sqrt{c{x}^{2}+bx+a}}-{\frac{bea}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{e{a}^{2}}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(e*x+d)*(c*x^2+b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.313735, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) + 4 \,{\left (48 \, c^{3} e x^{3} + 64 \, a c^{2} d + 8 \,{\left (8 \, c^{3} d + 5 \, b c^{2} e\right )} x^{2} +{\left (3 \, b^{3} - 20 \, a b c\right )} e + 2 \,{\left (32 \, b c^{2} d -{\left (b^{2} c - 12 \, a c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c}}{384 \, c^{\frac{5}{2}}}, -\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) - 2 \,{\left (48 \, c^{3} e x^{3} + 64 \, a c^{2} d + 8 \,{\left (8 \, c^{3} d + 5 \, b c^{2} e\right )} x^{2} +{\left (3 \, b^{3} - 20 \, a b c\right )} e + 2 \,{\left (32 \, b c^{2} d -{\left (b^{2} c - 12 \, a c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c}}{192 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)*(e*x + d),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (b + 2 c x\right ) \left (d + e x\right ) \sqrt{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(e*x+d)*(c*x**2+b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.282369, size = 230, normalized size = 1.89 \[ \frac{1}{96} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, c x e + \frac{8 \, c^{4} d + 5 \, b c^{3} e}{c^{3}}\right )} x + \frac{32 \, b c^{3} d - b^{2} c^{2} e + 12 \, a c^{3} e}{c^{3}}\right )} x + \frac{64 \, a c^{3} d + 3 \, b^{3} c e - 20 \, a b c^{2} e}{c^{3}}\right )} + \frac{{\left (b^{4} e - 8 \, a b^{2} c e + 16 \, a^{2} c^{2} e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{64 \, c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)*(e*x + d),x, algorithm="giac")
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