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]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {460, 90, 52} \[ \frac {x \sqrt {c x-1} \sqrt {c x+1} \left (4 a c^2+3 b\right )}{8 c^4}+\frac {\left (4 a c^2+3 b\right ) \cosh ^{-1}(c x)}{8 c^5}+\frac {b x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 90
Rule 460
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx &=\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x}}{4 c^2}-\frac {1}{4} \left (-4 a-\frac {3 b}{c^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {\left (3 b+4 a c^2\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{8 c^4}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x}}{4 c^2}+\frac {\left (3 b+4 a c^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^4}\\ &=\frac {\left (3 b+4 a c^2\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{8 c^4}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x}}{4 c^2}+\frac {\left (3 b+4 a c^2\right ) \cosh ^{-1}(c x)}{8 c^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 98, normalized size = 1.13 \[ \frac {c x \left (c^2 x^2-1\right ) \left (4 a c^2+b \left (2 c^2 x^2+3\right )\right )+\sqrt {c^2 x^2-1} \left (4 a c^2+3 b\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{8 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.01, size = 77, normalized size = 0.89 \[ \frac {{\left (2 \, b c^{3} x^{3} + {\left (4 \, a c^{3} + 3 \, b c\right )} x\right )} \sqrt {c x + 1} \sqrt {c x - 1} - {\left (4 \, a c^{2} + 3 \, b\right )} \log \left (-c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{8 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 121, normalized size = 1.39 \[ \frac {{\left ({\left (c x + 1\right )} {\left (2 \, {\left (c x + 1\right )} {\left (\frac {{\left (c x + 1\right )} b}{c^{4}} - \frac {3 \, b}{c^{4}}\right )} + \frac {4 \, a c^{18} + 9 \, b c^{16}}{c^{20}}\right )} - \frac {4 \, a c^{18} + 5 \, b c^{16}}{c^{20}}\right )} \sqrt {c x + 1} \sqrt {c x - 1} - \frac {2 \, {\left (4 \, a c^{2} + 3 \, b\right )} \log \left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}{c^{4}}}{8 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.08, size = 147, normalized size = 1.69 \[ \frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} x^{3} \mathrm {csgn}\relax (c )+4 \sqrt {c^{2} x^{2}-1}\, a \,c^{3} x \,\mathrm {csgn}\relax (c )+4 a \,c^{2} \ln \left (\left (c x +\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\relax (c )\right ) \mathrm {csgn}\relax (c )\right )+3 \sqrt {c^{2} x^{2}-1}\, b c x \,\mathrm {csgn}\relax (c )+3 b \ln \left (\left (c x +\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\relax (c )\right ) \mathrm {csgn}\relax (c )\right )\right ) \mathrm {csgn}\relax (c )}{8 \sqrt {c^{2} x^{2}-1}\, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 113, normalized size = 1.30 \[ \frac {\sqrt {c^{2} x^{2} - 1} b x^{3}}{4 \, c^{2}} + \frac {\sqrt {c^{2} x^{2} - 1} a x}{2 \, c^{2}} + \frac {a \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{2 \, c^{3}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} b x}{8 \, c^{4}} + \frac {3 \, b \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{8 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 22.50, size = 720, normalized size = 8.28 \[ \frac {\frac {23\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^3}{2\,{\left (\sqrt {c\,x+1}-1\right )}^3}+\frac {333\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {c\,x+1}-1\right )}^5}+\frac {671\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {c\,x+1}-1\right )}^7}+\frac {671\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^9}{2\,{\left (\sqrt {c\,x+1}-1\right )}^9}+\frac {333\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{11}}{2\,{\left (\sqrt {c\,x+1}-1\right )}^{11}}+\frac {23\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{13}}{2\,{\left (\sqrt {c\,x+1}-1\right )}^{13}}-\frac {3\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^{15}}{2\,{\left (\sqrt {c\,x+1}-1\right )}^{15}}-\frac {3\,b\,\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 {\frac {14\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {c\,x+1}-1\right )}^3}+\frac {14\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {c\,x+1}-1\right )}^5}+\frac {2\,a\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {c\,x+1}-1\right )}^7}+\frac {2\,a\,\left (\sqrt {c\,x-1}-\mathrm {i}\right )}{\sqrt {c\,x+1}-1}}{c^3-\frac {4\,c^3\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {c\,x+1}-1\right )}^2}+\frac {6\,c^3\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {c\,x+1}-1\right )}^4}-\frac {4\,c^3\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {c\,x+1}-1\right )}^6}+\frac {c^3\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {c\,x+1}-1\right )}^8}}+\frac {2\,a\,\mathrm {atanh}\left (\frac {\sqrt {c\,x-1}-\mathrm {i}}{\sqrt {c\,x+1}-1}\right )}{c^3}+\frac {3\,b\,\mathrm {atanh}\left (\frac {\sqrt {c\,x-1}-\mathrm {i}}{\sqrt {c\,x+1}-1}\right )}{2\,c^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________