Optimal. Leaf size=76 \[ \frac{\left (a d^2+2 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{2 c^3}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{2 c^2 x^2} \]
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Rubi [A] time = 0.260603, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{\left (a d^2+2 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{2 c^3}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{2 c^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/(x^3*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 13.3616, size = 63, normalized size = 0.83 \[ \frac{a \sqrt{- c + d x} \sqrt{c + d x}}{2 c^{2} x^{2}} + \frac{\left (a d^{2} + 2 b c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )}}{2 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)/x**3/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [C] time = 0.139132, size = 103, normalized size = 1.36 \[ \frac{a c \sqrt{d x-c} \sqrt{c+d x}-i x^2 \left (a d^2+2 b c^2\right ) \log \left (\frac{4 c^2 \left (\sqrt{d x-c} \sqrt{c+d x}-i c\right )}{x \left (a d^2+2 b c^2\right )}\right )}{2 c^3 x^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)/(x^3*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
[Out]
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Maple [B] time = 0.029, size = 158, normalized size = 2.1 \[ -{\frac{1}{2\,{c}^{2}{x}^{2}}\sqrt{dx-c}\sqrt{dx+c} \left ( \ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{2}a{d}^{2}+2\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{2}b{c}^{2}-a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}\sqrt{-{c}^{2}} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/x^3/(d*x-c)^(1/2)/(d*x+c)^(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((b*x^2 + a)/(sqrt(d*x + c)*sqrt(d*x - c)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.245142, size = 279, normalized size = 3.67 \[ -\frac{2 \, a c d^{3} x^{3} - 2 \, a c^{3} d x -{\left (2 \, a c d^{2} x^{2} - a c^{3}\right )} \sqrt{d x + c} \sqrt{d x - c} + 2 \,{\left (2 \,{\left (2 \, b c^{2} d + a d^{3}\right )} \sqrt{d x + c} \sqrt{d x - c} x^{3} - 2 \,{\left (2 \, b c^{2} d^{2} + a d^{4}\right )} x^{4} +{\left (2 \, b c^{4} + a c^{2} d^{2}\right )} x^{2}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{2 \,{\left (2 \, c^{3} d^{2} x^{4} - 2 \, \sqrt{d x + c} \sqrt{d x - c} c^{3} d x^{3} - c^{5} x^{2}\right )}} \]
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^3),x, algorithm="fricas")
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Sympy [A] time = 68.5575, size = 162, normalized size = 2.13 \[ - \frac{a d^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} + \frac{i a d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} - \frac{b{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)/x**3/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.229743, size = 190, normalized size = 2.5 \[ -\frac{\frac{{\left (2 \, b c^{2} d + a d^{3}\right )} \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c^{3}} + \frac{2 \,{\left (a d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 4 \, a c^{2} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{2} c^{2}}}{d} \]
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^3),x, algorithm="giac")
[Out]