Optimal. Leaf size=80 \[ a \sqrt{d x-c} \sqrt{c+d x}-a c \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )+\frac{b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.286411, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ a \sqrt{d x-c} \sqrt{c+d x}-a c \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )+\frac{b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.3838, size = 65, normalized size = 0.81 \[ - a c \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )} + a \sqrt{- c + d x} \sqrt{c + d x} + \frac{b \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(d*x-c)**(1/2)*(d*x+c)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.142888, size = 75, normalized size = 0.94 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2-b c^2+b d^2 x^2\right )}{3 d^2}+a c \tan ^{-1}\left (\frac{c}{\sqrt{d x-c} \sqrt{c+d x}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.02, size = 174, normalized size = 2.2 \[{\frac{1}{3\,{d}^{2}}\sqrt{dx-c}\sqrt{dx+c} \left ({x}^{2}b{d}^{2}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+3\,a{c}^{2}\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){d}^{2}+3\,a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{2}\sqrt{-{c}^{2}}-b{c}^{2}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2)/x,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((b*x^2 + a)*sqrt(d*x + c)*sqrt(d*x - c)/x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.250421, size = 383, normalized size = 4.79 \[ -\frac{4 \, b d^{6} x^{6} - b c^{6} + 3 \, a c^{4} d^{2} - 3 \,{\left (3 \, b c^{2} d^{4} - 4 \, a d^{6}\right )} x^{4} + 3 \,{\left (2 \, b c^{4} d^{2} - 5 \, a c^{2} d^{4}\right )} x^{2} -{\left (4 \, b d^{5} x^{5} -{\left (7 \, b c^{2} d^{3} - 12 \, a d^{5}\right )} x^{3} + 3 \,{\left (b c^{4} d - 3 \, a c^{2} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} + 6 \,{\left (4 \, a c d^{5} x^{3} - 3 \, a c^{3} d^{3} x -{\left (4 \, a c d^{4} x^{2} - a c^{3} d^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{3 \,{\left (4 \, d^{5} x^{3} - 3 \, c^{2} d^{3} x -{\left (4 \, d^{4} x^{2} - c^{2} d^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*sqrt(d*x + c)*sqrt(d*x - c)/x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right ) \sqrt{- c + d x} \sqrt{c + d x}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)*(d*x-c)**(1/2)*(d*x+c)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.224712, size = 109, normalized size = 1.36 \[ 2 \, a c \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right ) + \frac{1}{1920} \,{\left (3 \, a d^{6} +{\left ({\left (d x + c\right )} b d^{4} - 2 \, b c d^{4}\right )}{\left (d x + c\right )}\right )} \sqrt{d x + c} \sqrt{d x - c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*sqrt(d*x + c)*sqrt(d*x - c)/x,x, algorithm="giac")
[Out]