Optimal. Leaf size=125 \[ \frac {\left (5 b+6 a c^2\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{16 c^6}+\frac {\left (5 b+6 a c^2\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{24 c^4}+\frac {b x^5 \sqrt {-1+c x} \sqrt {1+c x}}{6 c^2}+\frac {\left (5 b+6 a c^2\right ) \cosh ^{-1}(c x)}{16 c^7} \]
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Rubi [A]
time = 0.06, antiderivative size = 125, normalized size of antiderivative = 1.00, 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 = {471, 102, 12,
92, 54} \begin {gather*} \frac {\left (6 a c^2+5 b\right ) \cosh ^{-1}(c x)}{16 c^7}+\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (6 a c^2+5 b\right )}{16 c^6}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} \left (6 a c^2+5 b\right )}{24 c^4}+\frac {b x^5 \sqrt {c x-1} \sqrt {c x+1}}{6 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 54
Rule 92
Rule 102
Rule 471
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*}
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Mathematica [A]
time = 0.21, size = 96, normalized size = 0.77 \begin {gather*} \frac {c x \sqrt {-1+c x} \sqrt {1+c x} \left (6 a c^2 \left (3+2 c^2 x^2\right )+b \left (15+10 c^2 x^2+8 c^4 x^4\right )\right )+6 \left (5 b+6 a c^2\right ) \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{48 c^7} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.29, size = 191, normalized size = 1.53
method | result | size |
risch | \(\frac {x \left (8 b \,x^{4} c^{4}+12 a \,c^{4} x^{2}+10 b \,c^{2} x^{2}+18 c^{2} a +15 b \right ) \sqrt {c x +1}\, \sqrt {c x -1}}{48 c^{6}}+\frac {\left (\frac {3 \ln \left (\frac {c^{2} x}{\sqrt {c^{2}}}+\sqrt {c^{2} x^{2}-1}\right ) a}{8 c^{4} \sqrt {c^{2}}}+\frac {5 \ln \left (\frac {c^{2} x}{\sqrt {c^{2}}}+\sqrt {c^{2} x^{2}-1}\right ) b}{16 c^{6} \sqrt {c^{2}}}\right ) \sqrt {\left (c x +1\right ) \left (c x -1\right )}}{\sqrt {c x -1}\, \sqrt {c x +1}}\) | \(156\) |
default | \(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (8 \,\mathrm {csgn}\left (c \right ) b \,c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+12 \,\mathrm {csgn}\left (c \right ) a \,c^{5} x^{3} \sqrt {c^{2} x^{2}-1}+10 \sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\left (c \right ) c^{3} b \,x^{3}+18 \sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\left (c \right ) c^{3} a x +15 \sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\left (c \right ) c b x +18 \ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\left (c \right )+c x \right ) \mathrm {csgn}\left (c \right )\right ) a \,c^{2}+15 \ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\left (c \right )+c x \right ) \mathrm {csgn}\left (c \right )\right ) b \right ) \mathrm {csgn}\left (c \right )}{48 c^{7} \sqrt {c^{2} x^{2}-1}}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 153, normalized size = 1.22 \begin {gather*} \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} c\right )}{8 \, c^{5}} + \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} c\right )}{16 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.28, size = 96, normalized size = 0.77 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 172, normalized size = 1.38 \begin {gather*} \frac {{\left ({\left (2 \, {\left ({\left (c x + 1\right )} {\left (4 \, {\left (c x + 1\right )} {\left (\frac {{\left (c x + 1\right )} b}{c^{6}} - \frac {5 \, b}{c^{6}}\right )} + \frac {3 \, {\left (2 \, a c^{38} + 15 \, b c^{36}\right )}}{c^{42}}\right )} - \frac {18 \, a c^{38} + 55 \, b c^{36}}{c^{42}}\right )} {\left (c x + 1\right )} + \frac {54 \, a c^{38} + 85 \, b c^{36}}{c^{42}}\right )} {\left (c x + 1\right )} - \frac {3 \, {\left (10 \, a c^{38} + 11 \, b c^{36}\right )}}{c^{42}}\right )} \sqrt {c x + 1} \sqrt {c x - 1} - \frac {6 \, {\left (6 \, a c^{2} + 5 \, b\right )} \log \left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}{c^{6}}}{48 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 32.63, size = 1154, normalized size = 9.23 \begin {gather*} -\frac {-\frac {175\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^3}{12\,{\left (\sqrt {c\,x+1}-1\right )}^3}+\frac {311\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^5}{4\,{\left (\sqrt {c\,x+1}-1\right )}^5}+\frac {8361\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^7}{4\,{\left (\sqrt {c\,x+1}-1\right )}^7}+\frac {42259\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^9}{6\,{\left (\sqrt {c\,x+1}-1\right )}^9}+\frac {25295\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{11}}{2\,{\left (\sqrt {c\,x+1}-1\right )}^{11}}+\frac {25295\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{13}}{2\,{\left (\sqrt {c\,x+1}-1\right )}^{13}}+\frac {42259\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{15}}{6\,{\left (\sqrt {c\,x+1}-1\right )}^{15}}+\frac {8361\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{17}}{4\,{\left (\sqrt {c\,x+1}-1\right )}^{17}}+\frac {311\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{19}}{4\,{\left (\sqrt {c\,x+1}-1\right )}^{19}}-\frac {175\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{21}}{12\,{\left (\sqrt {c\,x+1}-1\right )}^{21}}+\frac {5\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{23}}{4\,{\left (\sqrt {c\,x+1}-1\right )}^{23}}+\frac {5\,b\,\left (\sqrt {c\,x-1}-\mathrm {i}\right )}{4\,\left (\sqrt {c\,x+1}-1\right )}}{c^7-\frac {12\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {c\,x+1}-1\right )}^2}+\frac {66\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {c\,x+1}-1\right )}^4}-\frac {220\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {c\,x+1}-1\right )}^6}+\frac {495\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {c\,x+1}-1\right )}^8}-\frac {792\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {c\,x+1}-1\right )}^{10}}+\frac {924\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {c\,x+1}-1\right )}^{12}}-\frac {792\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {c\,x+1}-1\right )}^{14}}+\frac {495\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {c\,x+1}-1\right )}^{16}}-\frac {220\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{18}}{{\left (\sqrt {c\,x+1}-1\right )}^{18}}+\frac {66\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{20}}{{\left (\sqrt {c\,x+1}-1\right )}^{20}}-\frac {12\,c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{22}}{{\left (\sqrt {c\,x+1}-1\right )}^{22}}+\frac {c^7\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{24}}{{\left (\sqrt {c\,x+1}-1\right )}^{24}}}+\frac {\frac {23\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^3}{2\,{\left (\sqrt {c\,x+1}-1\right )}^3}+\frac {333\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {c\,x+1}-1\right )}^5}+\frac {671\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {c\,x+1}-1\right )}^7}+\frac {671\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^9}{2\,{\left (\sqrt {c\,x+1}-1\right )}^9}+\frac {333\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{11}}{2\,{\left (\sqrt {c\,x+1}-1\right )}^{11}}+\frac {23\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{13}}{2\,{\left (\sqrt {c\,x+1}-1\right )}^{13}}-\frac {3\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{15}}{2\,{\left (\sqrt {c\,x+1}-1\right )}^{15}}-\frac {3\,a\,\left (\sqrt {c\,x-1}-\mathrm {i}\right )}{2\,\left (\sqrt {c\,x+1}-1\right )}}{c^5-\frac {8\,c^5\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {c\,x+1}-1\right )}^2}+\frac {28\,c^5\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {c\,x+1}-1\right )}^4}-\frac {56\,c^5\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {c\,x+1}-1\right )}^6}+\frac {70\,c^5\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {c\,x+1}-1\right )}^8}-\frac {56\,c^5\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {c\,x+1}-1\right )}^{10}}+\frac {28\,c^5\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {c\,x+1}-1\right )}^{12}}-\frac {8\,c^5\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {c\,x+1}-1\right )}^{14}}+\frac {c^5\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {c\,x+1}-1\right )}^{16}}}+\frac {3\,a\,\mathrm {atanh}\left (\frac {\sqrt {c\,x-1}-\mathrm {i}}{\sqrt {c\,x+1}-1}\right )}{2\,c^5}+\frac {5\,b\,\mathrm {atanh}\left (\frac {\sqrt {c\,x-1}-\mathrm {i}}{\sqrt {c\,x+1}-1}\right )}{4\,c^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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