Integrand size = 24, antiderivative size = 86 \[ \int \frac {-b+a x^8}{x^6 \left (b+a x^4\right )^{3/4}} \, dx=\frac {\left (b-4 a x^4\right ) \sqrt [4]{b+a x^4}}{5 b x^5}-\frac {1}{2} \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \]
[Out]
Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1503, 277, 270, 338, 304, 209, 212} \[ \int \frac {-b+a x^8}{x^6 \left (b+a x^4\right )^{3/4}} \, dx=-\frac {1}{2} \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {1}{2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\frac {4 a \sqrt [4]{a x^4+b}}{5 b x}+\frac {\sqrt [4]{a x^4+b}}{5 x^5} \]
[In]
[Out]
Rule 209
Rule 212
Rule 270
Rule 277
Rule 304
Rule 338
Rule 1503
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b}{x^6 \left (b+a x^4\right )^{3/4}}+\frac {a x^2}{\left (b+a x^4\right )^{3/4}}\right ) \, dx \\ & = a \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx-b \int \frac {1}{x^6 \left (b+a x^4\right )^{3/4}} \, dx \\ & = \frac {\sqrt [4]{b+a x^4}}{5 x^5}+\frac {1}{5} (4 a) \int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx+a \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {\sqrt [4]{b+a x^4}}{5 x^5}-\frac {4 a \sqrt [4]{b+a x^4}}{5 b x}+\frac {1}{2} \sqrt {a} \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{2} \sqrt {a} \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {\sqrt [4]{b+a x^4}}{5 x^5}-\frac {4 a \sqrt [4]{b+a x^4}}{5 b x}-\frac {1}{2} \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {-b+a x^8}{x^6 \left (b+a x^4\right )^{3/4}} \, dx=\frac {\left (b-4 a x^4\right ) \sqrt [4]{b+a x^4}}{5 b x^5}-\frac {1}{2} \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \]
[In]
[Out]
Time = 1.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.29
method | result | size |
pseudoelliptic | \(\frac {5 b \,a^{\frac {1}{4}} \ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) x^{5}+10 b \,a^{\frac {1}{4}} \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) x^{5}-16 a \left (a \,x^{4}+b \right )^{\frac {1}{4}} x^{4}+4 b \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{20 b \,x^{5}}\) | \(111\) |
[In]
[Out]
Timed out. \[ \int \frac {-b+a x^8}{x^6 \left (b+a x^4\right )^{3/4}} \, dx=\text {Timed out} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.22 \[ \int \frac {-b+a x^8}{x^6 \left (b+a x^4\right )^{3/4}} \, dx=- \frac {a^{\frac {5}{4}} \sqrt [4]{1 + \frac {b}{a x^{4}}} \Gamma \left (- \frac {5}{4}\right )}{4 b \Gamma \left (\frac {3}{4}\right )} + \frac {\sqrt [4]{a} \sqrt [4]{1 + \frac {b}{a x^{4}}} \Gamma \left (- \frac {5}{4}\right )}{16 x^{4} \Gamma \left (\frac {3}{4}\right )} + \frac {a x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 b^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.24 \[ \int \frac {-b+a x^8}{x^6 \left (b+a x^4\right )^{3/4}} \, dx=\frac {1}{4} \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {3}{4}}}\right )} - \frac {\frac {5 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} a}{x} - \frac {{\left (a x^{4} + b\right )}^{\frac {5}{4}}}{x^{5}}}{5 \, b} \]
[In]
[Out]
\[ \int \frac {-b+a x^8}{x^6 \left (b+a x^4\right )^{3/4}} \, dx=\int { \frac {a x^{8} - b}{{\left (a x^{4} + b\right )}^{\frac {3}{4}} x^{6}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {-b+a x^8}{x^6 \left (b+a x^4\right )^{3/4}} \, dx=-\int \frac {b-a\,x^8}{x^6\,{\left (a\,x^4+b\right )}^{3/4}} \,d x \]
[In]
[Out]