Optimal. Leaf size=103 \[ \frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^4}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^6}+\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{5 c^2} \]
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Rubi [A] time = 0.0749051, 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, Rules used = {460, 100, 12, 74} \[ \frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^4}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^6}+\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{5 c^2} \]
Antiderivative was successfully verified.
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Rule 460
Rule 100
Rule 12
Rule 74
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx &=\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x}}{5 c^2}-\frac{1}{5} \left (-5 a-\frac{4 b}{c^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{\left (4 b+5 a c^2\right ) x^2 \sqrt{-1+c x} \sqrt{1+c x}}{15 c^4}+\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x}}{5 c^2}+\frac{\left (4 b+5 a c^2\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{15 c^4}\\ &=\frac{\left (4 b+5 a c^2\right ) x^2 \sqrt{-1+c x} \sqrt{1+c x}}{15 c^4}+\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x}}{5 c^2}+\frac{\left (2 \left (4 b+5 a c^2\right )\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{15 c^4}\\ &=\frac{2 \left (4 b+5 a c^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}{15 c^6}+\frac{\left (4 b+5 a c^2\right ) x^2 \sqrt{-1+c x} \sqrt{1+c x}}{15 c^4}+\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x}}{5 c^2}\\ \end{align*}
Mathematica [A] time = 0.0415032, size = 70, normalized size = 0.68 \[ \frac{\left (c^2 x^2-1\right ) \left (5 a c^2 \left (c^2 x^2+2\right )+b \left (3 c^4 x^4+4 c^2 x^2+8\right )\right )}{15 c^6 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 57, normalized size = 0.6 \begin{align*}{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946007, size = 128, normalized size = 1.24 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54227, size = 128, normalized size = 1.24 \begin{align*} \frac{{\left (3 \, b c^{4} x^{4} + 10 \, a c^{2} +{\left (5 \, a c^{4} + 4 \, b c^{2}\right )} x^{2} + 8 \, b\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{15 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 45.7106, size = 216, normalized size = 2.1 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18833, size = 130, normalized size = 1.26 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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