Optimal. Leaf size=57 \[ \frac{a \sqrt{d x-c} \sqrt{c+d x}}{c^2 x}+\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d} \]
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Rubi [A] time = 0.227326, antiderivative size = 57, 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 \[ \frac{a \sqrt{d x-c} \sqrt{c+d x}}{c^2 x}+\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/(x^2*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
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Rubi in Sympy [A] time = 11.5265, size = 46, normalized size = 0.81 \[ \frac{a \sqrt{- c + d x} \sqrt{c + d x}}{c^{2} x} + \frac{2 b \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c + d x}} \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)/x**2/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0726016, size = 60, normalized size = 1.05 \[ \frac{a \sqrt{d x-c} \sqrt{c+d x}}{c^2 x}+\frac{b \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )}{d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)/(x^2*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
[Out]
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Maple [C] time = 0.029, size = 97, normalized size = 1.7 \[{\frac{{\it csgn} \left ( d \right ) }{{c}^{2}xd}\sqrt{dx-c}\sqrt{dx+c} \left ( \ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) xb{c}^{2}+a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/x^2/(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^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237346, size = 117, normalized size = 2.05 \[ \frac{a d -{\left (b d x^{2} - \sqrt{d x + c} \sqrt{d x - c} b x\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{d^{2} x^{2} - \sqrt{d x + c} \sqrt{d x - c} d x} \]
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^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 51.5059, size = 165, normalized size = 2.89 \[ - \frac{a d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} - \frac{i a d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)/x**2/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.224909, size = 89, normalized size = 1.56 \[ \frac{\frac{16 \, a d^{2}}{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}} - b{\rm ln}\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4}\right )}{2 \, d} \]
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^2),x, algorithm="giac")
[Out]