Optimal. Leaf size=125 \[ \frac{x^3 \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{24 c^4}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{16 c^6}+\frac{\left (6 a c^2+5 b\right ) \cosh ^{-1}(c x)}{16 c^7}+\frac{b x^5 \sqrt{c x-1} \sqrt{c x+1}}{6 c^2} \]
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Rubi [A] time = 0.0848769, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {460, 100, 12, 90, 52} \[ \frac{x^3 \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{24 c^4}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{16 c^6}+\frac{\left (6 a c^2+5 b\right ) \cosh ^{-1}(c x)}{16 c^7}+\frac{b x^5 \sqrt{c x-1} \sqrt{c x+1}}{6 c^2} \]
Antiderivative was successfully verified.
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Rule 460
Rule 100
Rule 12
Rule 90
Rule 52
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx &=\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}-\frac{1}{6} \left (-6 a-\frac{5 b}{c^2}\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{\left (5 b+6 a c^2\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{24 c^4}+\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}+\frac{\left (5 b+6 a c^2\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{24 c^4}\\ &=\frac{\left (5 b+6 a c^2\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{24 c^4}+\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}+\frac{\left (5 b+6 a c^2\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c^4}\\ &=\frac{\left (5 b+6 a c^2\right ) x \sqrt{-1+c x} \sqrt{1+c x}}{16 c^6}+\frac{\left (5 b+6 a c^2\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{24 c^4}+\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}+\frac{\left (5 b+6 a c^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c^6}\\ &=\frac{\left (5 b+6 a c^2\right ) x \sqrt{-1+c x} \sqrt{1+c x}}{16 c^6}+\frac{\left (5 b+6 a c^2\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{24 c^4}+\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}+\frac{\left (5 b+6 a c^2\right ) \cosh ^{-1}(c x)}{16 c^7}\\ \end{align*}
Mathematica [A] time = 0.103118, size = 117, normalized size = 0.94 \[ \frac{c x \left (c^2 x^2-1\right ) \left (6 a c^2 \left (2 c^2 x^2+3\right )+b \left (8 c^4 x^4+10 c^2 x^2+15\right )\right )+3 \sqrt{c^2 x^2-1} \left (6 a c^2+5 b\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{48 c^7 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.077, size = 191, normalized size = 1.5 \begin{align*}{\frac{{\it csgn} \left ( c \right ) }{48\,{c}^{7}}\sqrt{cx-1}\sqrt{cx+1} \left ( 8\,{\it csgn} \left ( c \right ){x}^{5}b{c}^{5}\sqrt{{c}^{2}{x}^{2}-1}+12\,{\it csgn} \left ( c \right ){x}^{3}a{c}^{5}\sqrt{{c}^{2}{x}^{2}-1}+10\,\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ){c}^{3}{x}^{3}b+18\,\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ){c}^{3}xa+15\,\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) cxb+18\,\ln \left ( \left ( \sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) +cx \right ){\it csgn} \left ( c \right ) \right ) a{c}^{2}+15\,\ln \left ( \left ( \sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) +cx \right ){\it csgn} \left ( c \right ) \right ) b \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961023, size = 231, normalized size = 1.85 \begin{align*} \frac{\sqrt{c^{2} x^{2} - 1} b x^{5}}{6 \, c^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} a x^{3}}{4 \, c^{2}} + \frac{5 \, \sqrt{c^{2} x^{2} - 1} b x^{3}}{24 \, c^{4}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} a x}{8 \, c^{4}} + \frac{3 \, a \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{8 \, \sqrt{c^{2}} c^{4}} + \frac{5 \, \sqrt{c^{2} x^{2} - 1} b x}{16 \, c^{6}} + \frac{5 \, b \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{16 \, \sqrt{c^{2}} c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54684, size = 224, normalized size = 1.79 \begin{align*} \frac{{\left (8 \, b c^{5} x^{5} + 2 \,{\left (6 \, a c^{5} + 5 \, b c^{3}\right )} x^{3} + 3 \,{\left (6 \, a c^{3} + 5 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 3 \,{\left (6 \, a c^{2} + 5 \, b\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{48 \, c^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 59.9497, size = 216, normalized size = 1.73 \begin{align*} \frac{a{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 a{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}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{11}{4}, - \frac{9}{4} & - \frac{5}{2}, - \frac{5}{2}, -2, 1 \\-3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, -2, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{7}} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{7}{2}, - \frac{13}{4}, -3, - \frac{11}{4}, - \frac{5}{2}, 1 & \\- \frac{13}{4}, - \frac{11}{4} & - \frac{7}{2}, -3, -3, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23898, size = 205, normalized size = 1.64 \begin{align*} -\frac{{\left (30 \, a c^{38} + 33 \, b c^{36} -{\left (54 \, a c^{38} + 85 \, b c^{36} - 2 \,{\left (18 \, a c^{38} + 55 \, b c^{36} -{\left (6 \, a c^{38} + 45 \, b c^{36} + 4 \,{\left ({\left (c x + 1\right )} b c^{36} - 5 \, b c^{36}\right )}{\left (c x + 1\right )}\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} + 6 \,{\left (6 \, a c^{38} + 5 \, b c^{36}\right )} \log \left ({\left | -\sqrt{c x + 1} + \sqrt{c x - 1} \right |}\right )}{34603008 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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