Optimal. Leaf size=115 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 c} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.125053, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*Sqrt[a + b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.8335, size = 104, normalized size = 0.9 \[ \frac{e \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{8 c^{2}} + \frac{\left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**2+b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.211466, size = 112, normalized size = 0.97 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 c (2 a e+c x (3 d+2 e x))-3 b^2 e+2 b c (3 d+e x)\right )+3 \left (b^2-4 a c\right ) (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{48 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*Sqrt[a + b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.009, size = 229, normalized size = 2. \[{\frac{dx}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{bd}{4\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{ad}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{b}^{2}d}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{e}{3\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{bex}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{b}^{2}e}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{bea}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{e{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^2+b*x+a)^(1/2),x)
[Out]
_______________________________________________________________________________________
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)*(e*x + d),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.239606, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, c^{2} e x^{2} + 6 \, b c d -{\left (3 \, b^{2} - 8 \, a c\right )} e + 2 \,{\left (6 \, c^{2} d + b c e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{96 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (8 \, c^{2} e x^{2} + 6 \, b c d -{\left (3 \, b^{2} - 8 \, a c\right )} e + 2 \,{\left (6 \, c^{2} d + b c e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 3 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{48 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(e*x + d),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right ) \sqrt{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(c*x**2+b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.221383, size = 174, normalized size = 1.51 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \, x e + \frac{6 \, c^{2} d + b c e}{c^{2}}\right )} x + \frac{6 \, b c d - 3 \, b^{2} e + 8 \, a c e}{c^{2}}\right )} + \frac{{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(e*x + d),x, algorithm="giac")
[Out]