Optimal. Leaf size=87 \[ \frac{\left (4 a c^2+3 b\right ) \cosh ^{-1}(c x)}{8 c^5}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (4 a c^2+3 b\right )}{8 c^4}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{4 c^2} \]
[Out]
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Rubi [A] time = 0.261869, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{\left (4 a c^2+3 b\right ) \cosh ^{-1}(c x)}{8 c^5}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (4 a c^2+3 b\right )}{8 c^4}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{4 c^2} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]
[Out]
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Rubi in Sympy [A] time = 12.7528, size = 80, normalized size = 0.92 \[ \frac{b x^{3} \sqrt{c x - 1} \sqrt{c x + 1}}{4 c^{2}} + \frac{x \left (4 a c^{2} + 3 b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{8 c^{4}} + \frac{\left (4 a c^{2} + 3 b\right ) \operatorname{acosh}{\left (c x \right )}}{8 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0936331, size = 83, normalized size = 0.95 \[ \frac{c x \sqrt{c x-1} \sqrt{c x+1} \left (4 a c^2+b \left (2 c^2 x^2+3\right )\right )+\left (4 a c^2+3 b\right ) \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )}{8 c^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]
[Out]
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Maple [C] time = 0.028, size = 147, normalized size = 1.7 \[{\frac{{\it csgn} \left ( c \right ) }{8\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1} \left ( 2\,b{x}^{3}\sqrt{{c}^{2}{x}^{2}-1}{c}^{3}{\it csgn} \left ( c \right ) +4\,ax\sqrt{{c}^{2}{x}^{2}-1}{c}^{3}{\it csgn} \left ( c \right ) +3\,bx\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) c+4\,a\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ){c}^{2}+3\,b\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x^2+a)/(c*x-1)^(1/2)/(c*x+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.38053, size = 177, normalized size = 2.03 \[ \frac{\sqrt{c^{2} x^{2} - 1} b x^{3}}{4 \, c^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} a x}{2 \, c^{2}} + \frac{a \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{2 \, \sqrt{c^{2}} c^{2}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} b x}{8 \, c^{4}} + \frac{3 \, b \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{8 \, \sqrt{c^{2}} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*x^2/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.243741, size = 406, normalized size = 4.67 \[ -\frac{16 \, b c^{8} x^{8} + 32 \, a c^{8} x^{6} - 4 \,{\left (12 \, a c^{6} + 7 \, b c^{4}\right )} x^{4} + 4 \,{\left (4 \, a c^{4} + 3 \, b c^{2}\right )} x^{2} -{\left (16 \, b c^{7} x^{7} + 8 \,{\left (4 \, a c^{7} + b c^{5}\right )} x^{5} - 2 \,{\left (16 \, a c^{5} + 11 \, b c^{3}\right )} x^{3} +{\left (4 \, a c^{3} + 3 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (8 \,{\left (4 \, a c^{6} + 3 \, b c^{4}\right )} x^{4} + 4 \, a c^{2} - 8 \,{\left (4 \, a c^{4} + 3 \, b c^{2}\right )} x^{2} - 4 \,{\left (2 \,{\left (4 \, a c^{5} + 3 \, b c^{3}\right )} x^{3} -{\left (4 \, a c^{3} + 3 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 3 \, b\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{8 \,{\left (8 \, c^{9} x^{4} - 8 \, c^{7} x^{2} + c^{5} - 4 \,{\left (2 \, c^{8} x^{3} - c^{6} x\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*x^2/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 86.6118, size = 212, normalized size = 2.44 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{7}{4}, - \frac{5}{4} & - \frac{3}{2}, - \frac{3}{2}, -1, 1 \\-2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 1 & \\- \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, -2, -2, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.249062, size = 151, normalized size = 1.74 \[ -\frac{{\left (4 \, a c^{18} + 5 \, b c^{16} -{\left (4 \, a c^{18} + 9 \, b c^{16} + 2 \,{\left ({\left (c x + 1\right )} b c^{16} - 3 \, b c^{16}\right )}{\left (c x + 1\right )}\right )}{\left (c x + 1\right )}\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 2 \,{\left (4 \, a c^{18} + 3 \, b c^{16}\right )}{\rm ln}\left ({\left | -\sqrt{c x + 1} + \sqrt{c x - 1} \right |}\right )}{114688 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*x^2/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="giac")
[Out]