Optimal. Leaf size=144 \[ -\frac{\left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-8 A b c+5 b^2 B\right )}{64 c^3}-\frac{\left (a+b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.171423, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{\left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-8 A b c+5 b^2 B\right )}{64 c^3}-\frac{\left (a+b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]
Antiderivative was successfully verified.
[In] Int[x*(A + B*x)*Sqrt[a + b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.7786, size = 146, normalized size = 1.01 \[ \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (4 A c - \frac{5 B b}{2} + 3 B c x\right )}{12 c^{2}} + \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} \left (- 8 A b c - 4 B a c + 5 B b^{2}\right )}{64 c^{3}} - \frac{\left (- 4 a c + b^{2}\right ) \left (- 8 A b c - 4 B a c + 5 B b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x+A)*(c*x**2+b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.219877, size = 154, normalized size = 1.07 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 b c (2 c x (2 A+B x)-13 a B)+8 c^2 \left (8 a A+3 a B x+8 A c x^2+6 B c x^3\right )-2 b^2 c (12 A+5 B x)+15 b^3 B\right )-3 \left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{384 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x*(A + B*x)*Sqrt[a + b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.011, size = 352, normalized size = 2.4 \[{\frac{A}{3\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{Abx}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{b}^{2}A}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{abA}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{A{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{Bx}{4\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Bb}{24\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}Bx}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,B{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,a{b}^{2}B}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{5\,{b}^{4}B}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{aBx}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{abB}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x+A)*(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)*(B*x + A)*x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.318547, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (48 \, B c^{3} x^{3} + 15 \, B b^{3} + 64 \, A a c^{2} + 8 \,{\left (B b c^{2} + 8 \, A c^{3}\right )} x^{2} - 4 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} c - 2 \,{\left (5 \, B b^{2} c - 4 \,{\left (3 \, B a + 2 \, A b\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (5 \, B b^{4} + 16 \,{\left (B a^{2} + 2 \, A a b\right )} c^{2} - 8 \,{\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{768 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (48 \, B c^{3} x^{3} + 15 \, B b^{3} + 64 \, A a c^{2} + 8 \,{\left (B b c^{2} + 8 \, A c^{3}\right )} x^{2} - 4 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} c - 2 \,{\left (5 \, B b^{2} c - 4 \,{\left (3 \, B a + 2 \, A b\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 3 \,{\left (5 \, B b^{4} + 16 \,{\left (B a^{2} + 2 \, A a b\right )} c^{2} - 8 \,{\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{384 \, \sqrt{-c} c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)*x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \left (A + B x\right ) \sqrt{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x+A)*(c*x**2+b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.281998, size = 240, normalized size = 1.67 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, B x + \frac{B b c^{2} + 8 \, A c^{3}}{c^{3}}\right )} x - \frac{5 \, B b^{2} c - 12 \, B a c^{2} - 8 \, A b c^{2}}{c^{3}}\right )} x + \frac{15 \, B b^{3} - 52 \, B a b c - 24 \, A b^{2} c + 64 \, A a c^{2}}{c^{3}}\right )} + \frac{{\left (5 \, B b^{4} - 24 \, B a b^{2} c - 8 \, A b^{3} c + 16 \, B a^{2} c^{2} + 32 \, A a b c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)*x,x, algorithm="giac")
[Out]