Integrand size = 15, antiderivative size = 75 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^5} \, dx=-\frac {\sqrt [4]{a+b x^4}}{4 x^4}-\frac {b \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 43, 65, 218, 212, 209} \[ \int \frac {\sqrt [4]{a+b x^4}}{x^5} \, dx=-\frac {b \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{4 x^4} \]
[In]
[Out]
Rule 43
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [4]{a+b x}}{x^2} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [4]{a+b x^4}}{4 x^4}+\frac {1}{16} b \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/4}} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [4]{a+b x^4}}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right ) \\ & = -\frac {\sqrt [4]{a+b x^4}}{4 x^4}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 \sqrt {a}}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 \sqrt {a}} \\ & = -\frac {\sqrt [4]{a+b x^4}}{4 x^4}-\frac {b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^5} \, dx=-\frac {\sqrt [4]{a+b x^4}}{4 x^4}-\frac {b \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}} \]
[In]
[Out]
Time = 4.46 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.13
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {-\left (b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right ) b \,x^{4}+2 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) b \,x^{4}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{\frac {3}{4}}}{16 x^{4} a^{\frac {3}{4}}}\) | \(85\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.28 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^5} \, dx=-\frac {\left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} b + a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) + i \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} b + i \, a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) - i \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} b - i \, a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) - \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} b - a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) + 4 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{16 \, x^{4}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^5} \, dx=- \frac {\sqrt [4]{b} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac {7}{4}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^5} \, dx=-\frac {b \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{8 \, a^{\frac {3}{4}}} + \frac {b \log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{16 \, a^{\frac {3}{4}}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{4 \, x^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (55) = 110\).
Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.76 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^5} \, dx=\frac {\frac {2 \, \sqrt {2} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} b^{2} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{a} - \frac {8 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b}{x^{4}}}{32 \, b} \]
[In]
[Out]
Time = 5.88 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^5} \, dx=-\frac {{\left (b\,x^4+a\right )}^{1/4}}{4\,x^4}-\frac {b\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{3/4}}-\frac {b\,\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{3/4}} \]
[In]
[Out]