Integrand size = 17, antiderivative size = 123 \[ \int \frac {1}{x \left (-b+a x^2\right )^{3/4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{-\sqrt {b}+\sqrt {-b+a x^2}}\right )}{\sqrt {2} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} b^{3/4}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.59, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {272, 65, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{x \left (-b+a x^2\right )^{3/4}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} b^{3/4}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {a x^2-b}+\sqrt {b}\right )}{2 \sqrt {2} b^{3/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {a x^2-b}+\sqrt {b}\right )}{2 \sqrt {2} b^{3/4}} \]
[In]
[Out]
Rule 65
Rule 210
Rule 217
Rule 272
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x (-b+a x)^{3/4}} \, dx,x,x^2\right ) \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{a} \\ & = \frac {\text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{a \sqrt {b}}+\frac {\text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{a \sqrt {b}} \\ & = -\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \sqrt {2} b^{3/4}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \sqrt {2} b^{3/4}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \sqrt {b}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \sqrt {b}} \\ & = -\frac {\log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2} b^{3/4}}+\frac {\log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2} b^{3/4}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{3/4}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{3/4}} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{3/4}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{3/4}}-\frac {\log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2} b^{3/4}}+\frac {\log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2} b^{3/4}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x \left (-b+a x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {b}+\sqrt {-b+a x^2}}\right )}{\sqrt {2} b^{3/4}} \]
[In]
[Out]
Time = 0.92 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {-b^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{2}-b}-\sqrt {b}}{b^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{2}-b}-\sqrt {b}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )\right )}{4 b^{\frac {3}{4}}}\) | \(142\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \left (-b+a x^2\right )^{3/4}} \, dx=\frac {1}{2} \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (b \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} i \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (i \, b \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} i \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (-i \, b \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (-b \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right ) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.34 \[ \int \frac {1}{x \left (-b+a x^2\right )^{3/4}} \, dx=- \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{2}}} \right )}}{2 a^{\frac {3}{4}} x^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x \left (-b+a x^2\right )^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{2 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{2 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{4 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{4 \, b^{\frac {3}{4}}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x \left (-b+a x^2\right )^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{2 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{2 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{4 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{4 \, b^{\frac {3}{4}}} \]
[In]
[Out]
Time = 5.99 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.36 \[ \int \frac {1}{x \left (-b+a x^2\right )^{3/4}} \, dx=-\frac {\mathrm {atan}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )+\mathrm {atanh}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{{\left (-b\right )}^{3/4}} \]
[In]
[Out]