Optimal. Leaf size=75 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (2 a d^2+3 b c^2\right )}{3 c^4 x}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{3 c^2 x^3} \]
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Rubi [A] time = 0.254355, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (2 a d^2+3 b c^2\right )}{3 c^4 x}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{3 c^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/(x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 11.9889, size = 63, normalized size = 0.84 \[ \frac{a \sqrt{- c + d x} \sqrt{c + d x}}{3 c^{2} x^{3}} + \frac{\sqrt{- c + d x} \sqrt{c + d x} \left (2 a d^{2} + 3 b c^{2}\right )}{3 c^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)/x**4/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0741589, size = 56, normalized size = 0.75 \[ \sqrt{d x-c} \sqrt{c+d x} \left (\frac{2 a d^2+3 b c^2}{3 c^4 x}+\frac{a}{3 c^2 x^3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)/(x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
[Out]
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Maple [A] time = 0.008, size = 49, normalized size = 0.7 \[{\frac{2\,a{d}^{2}{x}^{2}+3\,b{c}^{2}{x}^{2}+a{c}^{2}}{3\,{x}^{3}{c}^{4}}\sqrt{dx+c}\sqrt{dx-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/x^4/(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^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237519, size = 162, normalized size = 2.16 \[ \frac{6 \, b d^{2} x^{4} - a c^{2} - 3 \,{\left (b c^{2} - a d^{2}\right )} x^{2} - 3 \,{\left (2 \, b d x^{3} + a d x\right )} \sqrt{d x + c} \sqrt{d x - c}}{3 \,{\left (4 \, d^{3} x^{6} - 3 \, c^{2} d x^{4} -{\left (4 \, d^{2} x^{5} - c^{2} x^{3}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]
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^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 108.996, size = 170, normalized size = 2.27 \[ - \frac{a d^{3}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{9}{4}, \frac{11}{4}, 1 & \frac{5}{2}, \frac{5}{2}, 3 \\2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} - \frac{i a d^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, 1 & \\\frac{7}{4}, \frac{9}{4} & \frac{3}{2}, 2, 2, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} - \frac{b d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} - \frac{i b d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)/x**4/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.234151, size = 185, normalized size = 2.47 \[ \frac{8 \,{\left (3 \, b d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{8} + 24 \, b c^{2} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 24 \, a d^{4}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 48 \, b c^{4} d^{2} + 32 \, a c^{2} d^{4}\right )}}{3 \,{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{3} d} \]
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^4),x, algorithm="giac")
[Out]