Optimal. Leaf size=267 \[ -\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}+\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{3/8} b^{5/8}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} (-a)^{3/8} b^{5/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {301, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ -\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}+\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{3/8} b^{5/8}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} (-a)^{3/8} b^{5/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 205
Rule 208
Rule 211
Rule 212
Rule 301
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^4}{a+b x^8} \, dx &=-\frac {\int \frac {1}{\sqrt {-a}-\sqrt {b} x^4} \, dx}{2 \sqrt {b}}+\frac {\int \frac {1}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{2 \sqrt {b}}\\ &=-\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 \sqrt [4]{-a} \sqrt {b}}-\frac {\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 \sqrt [4]{-a} \sqrt {b}}+\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 \sqrt [4]{-a} \sqrt {b}}+\frac {\int \frac {\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 \sqrt [4]{-a} \sqrt {b}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}+\frac {\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 \sqrt [4]{-a} b^{3/4}}+\frac {\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 \sqrt [4]{-a} b^{3/4}}-\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{b}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}-\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{b}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}-\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}+\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{3/8} b^{5/8}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{3/8} b^{5/8}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{3/8} b^{5/8}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{3/8} b^{5/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}-\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}+\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{3/8} b^{5/8}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 324, normalized size = 1.21 \[ -\frac {\cos \left (\frac {\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\cos \left (\frac {\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sin \left (\frac {\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sin \left (\frac {\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+2 \sin \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right )+2 \sin \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right )+2 \cos \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \cos \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac {\pi }{8}\right )\right )}{8 a^{3/8} b^{5/8}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.70, size = 438, normalized size = 1.64 \[ \frac {1}{4} \, \sqrt {2} \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{8}} \arctan \left (-\sqrt {2} a b^{2} x \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {3}{8}} + \sqrt {2} \sqrt {\sqrt {2} a^{2} b^{3} x \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {5}{8}} - a b \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{4}} + x^{2}} a b^{2} \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {3}{8}} + 1\right ) + \frac {1}{4} \, \sqrt {2} \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{8}} \arctan \left (-\sqrt {2} a b^{2} x \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {3}{8}} + \sqrt {2} \sqrt {-\sqrt {2} a^{2} b^{3} x \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {5}{8}} - a b \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{4}} + x^{2}} a b^{2} \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {3}{8}} - 1\right ) + \frac {1}{16} \, \sqrt {2} \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a^{2} b^{3} x \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {5}{8}} - a b \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{4}} + x^{2}\right ) - \frac {1}{16} \, \sqrt {2} \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a^{2} b^{3} x \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {5}{8}} - a b \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{4}} + x^{2}\right ) - \frac {1}{2} \, \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{8}} \arctan \left (-a b^{2} x \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {3}{8}} + \sqrt {-a b \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{4}} + x^{2}} a b^{2} \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {3}{8}}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{8}} \log \left (a^{2} b^{3} \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {5}{8}} + x\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {1}{8}} \log \left (-a^{2} b^{3} \left (-\frac {1}{a^{3} b^{5}}\right )^{\frac {5}{8}} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.37, size = 437, normalized size = 1.64 \[ -\frac {\left (\frac {a}{b}\right )^{\frac {5}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {5}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {5}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {5}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {5}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {5}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {5}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {5}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.01, size = 27, normalized size = 0.10 \[ \frac {\ln \left (-\RootOf \left (b \,\textit {\_Z}^{8}+a \right )+x \right )}{8 b \RootOf \left (b \,\textit {\_Z}^{8}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{b x^{8} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.17, size = 110, normalized size = 0.41 \[ -\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x}{{\left (-a\right )}^{1/8}}\right )}{4\,{\left (-a\right )}^{3/8}\,b^{5/8}}+\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{3/8}\,b^{5/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{3/8}\,b^{5/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{3/8}\,b^{5/8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.34, size = 29, normalized size = 0.11 \[ \operatorname {RootSum} {\left (16777216 t^{8} a^{3} b^{5} + 1, \left (t \mapsto t \log {\left (- 32768 t^{5} a^{2} b^{3} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________