Optimal. Leaf size=79 \[ -\frac{\sqrt{1-d^2 x^2} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac{b \sin ^{-1}(d x)}{2 d^3}-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.252987, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{\sqrt{1-d^2 x^2} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac{b \sin ^{-1}(d x)}{2 d^3}-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2} \]
Antiderivative was successfully verified.
[In] Int[(x*(a + b*x + c*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 26.4531, size = 68, normalized size = 0.86 \[ \frac{b \operatorname{asin}{\left (d x \right )}}{2 d^{3}} - \frac{c x^{2} \sqrt{- d^{2} x^{2} + 1}}{3 d^{2}} - \frac{\sqrt{- d^{2} x^{2} + 1} \left (6 a d^{2} + 3 b d^{2} x + 4 c\right )}{6 d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(c*x**2+b*x+a)/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0918098, size = 57, normalized size = 0.72 \[ \frac{3 b d \sin ^{-1}(d x)-\sqrt{1-d^2 x^2} \left (3 d^2 (2 a+b x)+2 c \left (d^2 x^2+2\right )\right )}{6 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(a + b*x + c*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0., size = 139, normalized size = 1.8 \[ -{\frac{{\it csgn} \left ( d \right ) }{6\,{d}^{4}}\sqrt{-dx+1}\sqrt{dx+1} \left ( 2\,{\it csgn} \left ( d \right ){x}^{2}c{d}^{2}\sqrt{-{d}^{2}{x}^{2}+1}+3\,{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}xb{d}^{2}+6\,{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}a{d}^{2}+4\,{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}c-3\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) bd \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c*x^2+b*x+a)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.5115, size = 134, normalized size = 1.7 \[ -\frac{\sqrt{-d^{2} x^{2} + 1} c x^{2}}{3 \, d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1} b x}{2 \, d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1} a}{d^{2}} + \frac{b \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1} c}{3 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*x/(sqrt(d*x + 1)*sqrt(-d*x + 1)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.233505, size = 306, normalized size = 3.87 \[ -\frac{2 \, c d^{5} x^{6} + 3 \, b d^{5} x^{5} - 15 \, b d^{3} x^{3} - 12 \, a d^{3} x^{2} + 6 \,{\left (a d^{5} - c d^{3}\right )} x^{4} + 12 \, b d x + 3 \,{\left (2 \, c d^{3} x^{4} + 3 \, b d^{3} x^{3} + 4 \, a d^{3} x^{2} - 4 \, b d x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 6 \,{\left (3 \, b d^{2} x^{2} -{\left (b d^{2} x^{2} - 4 \, b\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 4 \, b\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{6 \,{\left (3 \, d^{5} x^{2} - 4 \, d^{3} -{\left (d^{5} x^{2} - 4 \, d^{3}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*x/(sqrt(d*x + 1)*sqrt(-d*x + 1)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 108.768, size = 313, normalized size = 3.96 \[ - \frac{i a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{a{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{i b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} + \frac{b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} - \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c*x**2+b*x+a)/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.215714, size = 123, normalized size = 1.56 \[ \frac{6 \, b d^{10} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right ) -{\left (6 \, a d^{11} - 3 \, b d^{10} + 6 \, c d^{9} +{\left (2 \,{\left (d x + 1\right )} c d^{9} + 3 \, b d^{10} - 4 \, c d^{9}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}}{3840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*x/(sqrt(d*x + 1)*sqrt(-d*x + 1)),x, algorithm="giac")
[Out]