Optimal. Leaf size=103 \[ \frac{2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^6}+\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^4}+\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{5 c^2} \]
[Out]
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Rubi [A] time = 0.26647, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^6}+\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^4}+\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{5 c^2} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]
[Out]
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Rubi in Sympy [A] time = 13.3902, size = 95, normalized size = 0.92 \[ \frac{b x^{4} \sqrt{c x - 1} \sqrt{c x + 1}}{5 c^{2}} + \frac{x^{2} \left (5 a c^{2} + 4 b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{15 c^{4}} + \frac{2 \left (5 a c^{2} + 4 b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{15 c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0631832, size = 61, normalized size = 0.59 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2 \left (c^2 x^2+2\right )+b \left (3 c^4 x^4+4 c^2 x^2+8\right )\right )}{15 c^6} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]
[Out]
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Maple [A] time = 0.009, size = 57, normalized size = 0.6 \[{\frac{3\,b{x}^{4}{c}^{4}+5\,a{c}^{4}{x}^{2}+4\,b{c}^{2}{x}^{2}+10\,a{c}^{2}+8\,b}{15\,{c}^{6}}\sqrt{cx-1}\sqrt{cx+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x^2+a)/(c*x-1)^(1/2)/(c*x+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.39394, size = 128, normalized size = 1.24 \[ \frac{\sqrt{c^{2} x^{2} - 1} b x^{4}}{5 \, c^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} a x^{2}}{3 \, c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} b x^{2}}{15 \, c^{4}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1} a}{3 \, c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} b}{15 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*x^3/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253568, size = 333, normalized size = 3.23 \[ -\frac{48 \, b c^{10} x^{10} + 20 \,{\left (4 \, a c^{10} - b c^{8}\right )} x^{8} + 5 \,{\left (4 \, a c^{8} + 11 \, b c^{6}\right )} x^{6} - 5 \,{\left (43 \, a c^{6} + 35 \, b c^{4}\right )} x^{4} - 10 \, a c^{2} + 25 \,{\left (5 \, a c^{4} + 4 \, b c^{2}\right )} x^{2} -{\left (48 \, b c^{9} x^{9} + 4 \,{\left (20 \, a c^{9} + b c^{7}\right )} x^{7} + 3 \,{\left (20 \, a c^{7} + 21 \, b c^{5}\right )} x^{5} - 35 \,{\left (5 \, a c^{5} + 4 \, b c^{3}\right )} x^{3} + 10 \,{\left (5 \, a c^{3} + 4 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 8 \, b}{15 \,{\left (16 \, c^{11} x^{5} - 20 \, c^{9} x^{3} + 5 \, c^{7} x -{\left (16 \, c^{10} x^{4} - 12 \, c^{8} x^{2} + c^{6}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*x^3/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 89.8548, size = 216, normalized size = 2.1 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{9}{4}, - \frac{7}{4} & -2, -2, - \frac{3}{2}, 1 \\- \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{6}} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} -3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, -2, 1 & \\- \frac{11}{4}, - \frac{9}{4} & -3, - \frac{5}{2}, - \frac{5}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.21793, size = 130, normalized size = 1.26 \[ \frac{{\left (15 \, a c^{27} + 15 \, b c^{25} -{\left (10 \, a c^{27} + 20 \, b c^{25} -{\left (5 \, a c^{27} + 22 \, b c^{25} + 3 \,{\left ({\left (c x + 1\right )} b c^{25} - 4 \, b c^{25}\right )}{\left (c x + 1\right )}\right )}{\left (c x + 1\right )}\right )}{\left (c x + 1\right )}\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{276480 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*x^3/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="giac")
[Out]