Optimal. Leaf size=68 \[ \frac{\left (2 a d^2+b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^3}+\frac{b x \sqrt{d x-c} \sqrt{c+d x}}{2 d^2} \]
[Out]
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Rubi [A] time = 0.115793, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{\left (2 a d^2+b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^3}+\frac{b x \sqrt{d x-c} \sqrt{c+d x}}{2 d^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/(Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 13.259, size = 75, normalized size = 1.1 \[ \frac{2 a \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )}}{d} + \frac{b c^{2} \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )}}{d^{3}} + \frac{b x \sqrt{- c + d x} \sqrt{c + d x}}{2 d^{2}} \]
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)
[Out]
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Mathematica [A] time = 0.0665225, size = 71, normalized size = 1.04 \[ \frac{\left (2 a d^2+b c^2\right ) \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )+b d x \sqrt{d x-c} \sqrt{c+d x}}{2 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)/(Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
[Out]
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Maple [C] time = 0.025, size = 124, normalized size = 1.8 \[{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{3}}\sqrt{dx-c}\sqrt{dx+c} \left ( bx\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d+b{c}^{2}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) +2\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) a{d}^{2} \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/(d*x-c)^(1/2)/(d*x+c)^(1/2),x)
[Out]
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Maxima [A] time = 1.39622, size = 140, normalized size = 2.06 \[ \frac{a \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{\sqrt{d^{2}}} + \frac{b c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{2}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} b x}{2 \, d^{2}} \]
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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237688, size = 261, normalized size = 3.84 \[ -\frac{2 \, b d^{4} x^{4} - 2 \, b c^{2} d^{2} x^{2} -{\left (2 \, b d^{3} x^{3} - b c^{2} d x\right )} \sqrt{d x + c} \sqrt{d x - c} -{\left (b c^{4} + 2 \, a c^{2} d^{2} + 2 \,{\left (b c^{2} d + 2 \, a d^{3}\right )} \sqrt{d x + c} \sqrt{d x - c} x - 2 \,{\left (b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{2}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{2 \,{\left (2 \, d^{5} x^{2} - 2 \, \sqrt{d x + c} \sqrt{d x - c} d^{4} x - c^{2} d^{3}\right )}} \]
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, algorithm="fricas")
[Out]
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Sympy [A] time = 43.6661, size = 199, normalized size = 2.93 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{b c^{2}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i b c^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} \]
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)
[Out]
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GIAC/XCAS [A] time = 0.255644, size = 107, normalized size = 1.57 \[ \frac{{\left ({\left (d x + c\right )} b d^{4} - b c d^{4}\right )} \sqrt{d x + c} \sqrt{d x - c} - 2 \,{\left (b c^{2} d^{4} + 2 \, a d^{6}\right )}{\rm ln}\left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{384 \, d} \]
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, algorithm="giac")
[Out]