Optimal. Leaf size=86 \[ -\frac {3 \left (15-2 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac {45}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {3}{2} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1266, 827, 858,
221, 272, 65, 213} \begin {gather*} -\frac {3}{2} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )+\frac {45}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {\left (2-3 x^2\right ) \left (x^4+5\right )^{3/2}}{4 x^4}-\frac {3 \left (15-2 x^2\right ) \sqrt {x^4+5}}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 213
Rule 221
Rule 272
Rule 827
Rule 858
Rule 1266
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \left (5+x^4\right )^{3/2}}{x^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(2+3 x) \left (5+x^2\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}-\frac {3}{16} \text {Subst}\left (\int \frac {(-60-8 x) \sqrt {5+x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3 \left (15-2 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac {3}{32} \text {Subst}\left (\int \frac {80+120 x}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {3 \left (15-2 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac {15}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {5+x^2}} \, dx,x,x^2\right )+\frac {45}{4} \text {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {3 \left (15-2 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac {45}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {15}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {5+x}} \, dx,x,x^4\right )\\ &=-\frac {3 \left (15-2 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac {45}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {15}{2} \text {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {5+x^4}\right )\\ &=-\frac {3 \left (15-2 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac {45}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {3}{2} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.19, size = 81, normalized size = 0.94 \begin {gather*} \frac {\sqrt {5+x^4} \left (-10-30 x^2+4 x^4+3 x^6\right )}{4 x^4}+\frac {45}{4} \tanh ^{-1}\left (\frac {x^2}{\sqrt {5+x^4}}\right )+3 \sqrt {5} \tanh ^{-1}\left (\frac {x^2-\sqrt {5+x^4}}{\sqrt {5}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.27, size = 73, normalized size = 0.85
method | result | size |
default | \(\sqrt {x^{4}+5}-\frac {5 \sqrt {x^{4}+5}}{2 x^{4}}-\frac {3 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{2}+\frac {3 x^{2} \sqrt {x^{4}+5}}{4}+\frac {45 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{4}-\frac {15 \sqrt {x^{4}+5}}{2 x^{2}}\) | \(73\) |
elliptic | \(\sqrt {x^{4}+5}-\frac {5 \sqrt {x^{4}+5}}{2 x^{4}}-\frac {3 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{2}+\frac {3 x^{2} \sqrt {x^{4}+5}}{4}+\frac {45 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{4}-\frac {15 \sqrt {x^{4}+5}}{2 x^{2}}\) | \(73\) |
risch | \(-\frac {5 \left (3 x^{6}+x^{4}+15 x^{2}+5\right )}{2 x^{4} \sqrt {x^{4}+5}}+\frac {3 x^{2} \sqrt {x^{4}+5}}{4}+\frac {45 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{4}+\sqrt {x^{4}+5}-\frac {3 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{2}\) | \(76\) |
trager | \(\frac {\left (3 x^{6}+4 x^{4}-30 x^{2}-10\right ) \sqrt {x^{4}+5}}{4 x^{4}}+\frac {3 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-5\right )-\sqrt {x^{4}+5}}{x^{2}}\right )}{2}+\frac {45 \ln \left (-x^{2}-\sqrt {x^{4}+5}\right )}{4}\) | \(79\) |
meijerg | \(\frac {3 \sqrt {5}\, \left (\frac {5 \sqrt {\pi }\, \left (-\frac {12 x^{4}}{5}+8\right )}{6 x^{4}}-\frac {5 \sqrt {\pi }\, \left (8-\frac {16 x^{4}}{5}\right ) \sqrt {1+\frac {x^{4}}{5}}}{6 x^{4}}-4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{4}}{5}}}{2}\right )+2 \left (1-2 \ln \left (2\right )+4 \ln \left (x \right )-\ln \left (5\right )\right ) \sqrt {\pi }-\frac {20 \sqrt {\pi }}{3 x^{4}}\right )}{8 \sqrt {\pi }}+\frac {-\frac {15 \sqrt {\pi }\, \sqrt {5}\, \left (-\frac {x^{4}}{10}+1\right ) \sqrt {1+\frac {x^{4}}{5}}}{2 x^{2}}+\frac {45 \sqrt {\pi }\, \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{4}}{\sqrt {\pi }}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 123, normalized size = 1.43 \begin {gather*} \frac {3}{4} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{\sqrt {5} + \sqrt {x^{4} + 5}}\right ) + \sqrt {x^{4} + 5} - \frac {15 \, \sqrt {x^{4} + 5}}{2 \, x^{2}} + \frac {15 \, \sqrt {x^{4} + 5}}{4 \, x^{2} {\left (\frac {x^{4} + 5}{x^{4}} - 1\right )}} - \frac {5 \, \sqrt {x^{4} + 5}}{2 \, x^{4}} + \frac {45}{8} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) - \frac {45}{8} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 82, normalized size = 0.95 \begin {gather*} \frac {6 \, \sqrt {5} x^{4} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{x^{2}}\right ) - 45 \, x^{4} \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) - 30 \, x^{4} + {\left (3 \, x^{6} + 4 \, x^{4} - 30 \, x^{2} - 10\right )} \sqrt {x^{4} + 5}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 6.47, size = 133, normalized size = 1.55 \begin {gather*} \frac {3 x^{6}}{4 \sqrt {x^{4} + 5}} - \frac {15 x^{2}}{4 \sqrt {x^{4} + 5}} + \sqrt {x^{4} + 5} + \frac {\sqrt {5} \log {\left (x^{4} \right )}}{2} - \sqrt {5} \log {\left (\sqrt {\frac {x^{4}}{5} + 1} + 1 \right )} - \frac {\sqrt {5} \operatorname {asinh}{\left (\frac {\sqrt {5}}{x^{2}} \right )}}{2} + \frac {45 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{4} - \frac {5 \sqrt {1 + \frac {5}{x^{4}}}}{2 x^{2}} - \frac {75}{2 x^{2} \sqrt {x^{4} + 5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 146 vs.
\(2 (68) = 136\).
time = 2.73, size = 146, normalized size = 1.70 \begin {gather*} \frac {1}{4} \, \sqrt {x^{4} + 5} {\left (3 \, x^{2} + 4\right )} + \frac {3}{2} \, \sqrt {5} \log \left (-\frac {x^{2} + \sqrt {5} - \sqrt {x^{4} + 5}}{x^{2} - \sqrt {5} - \sqrt {x^{4} + 5}}\right ) + \frac {5 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{3} + 15 \, {\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{2} + 5 \, x^{2} - 5 \, \sqrt {x^{4} + 5} - 75\right )}}{{\left ({\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{2} - 5\right )}^{2}} - \frac {45}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.55, size = 71, normalized size = 0.83 \begin {gather*} \frac {45\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{4}+\sqrt {x^4+5}\,\left (\frac {3\,x^2}{4}+1\right )-\frac {15\,\sqrt {x^4+5}}{2\,x^2}-\frac {5\,\sqrt {x^4+5}}{2\,x^4}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {5}\,\sqrt {x^4+5}\,1{}\mathrm {i}}{5}\right )\,3{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________