Integrand size = 17, antiderivative size = 145 \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{x} \, dx=\frac {1}{3} \left (-b+a x^4\right )^{3/4}+\frac {b^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}}{-\sqrt {b}+\sqrt {-b+a x^4}}\right )}{2 \sqrt {2}}+\frac {b^{3/4} \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^4}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2}} \]
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Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.50, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {272, 52, 65, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{x} \, dx=\frac {b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}\right )}{2 \sqrt {2}}-\frac {b^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}+1\right )}{2 \sqrt {2}}-\frac {b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}+\sqrt {a x^4-b}+\sqrt {b}\right )}{4 \sqrt {2}}+\frac {b^{3/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}+\sqrt {a x^4-b}+\sqrt {b}\right )}{4 \sqrt {2}}+\frac {1}{3} \left (a x^4-b\right )^{3/4} \]
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Rule 52
Rule 65
Rule 210
Rule 272
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {(-b+a x)^{3/4}}{x} \, dx,x,x^4\right ) \\ & = \frac {1}{3} \left (-b+a x^4\right )^{3/4}-\frac {1}{4} b \text {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^4\right ) \\ & = \frac {1}{3} \left (-b+a x^4\right )^{3/4}-\frac {b \text {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^4}\right )}{a} \\ & = \frac {1}{3} \left (-b+a x^4\right )^{3/4}+\frac {b \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^4}\right )}{2 a}-\frac {b \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^4}\right )}{2 a} \\ & = \frac {1}{3} \left (-b+a x^4\right )^{3/4}-\frac {b^{3/4} \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^4}\right )}{4 \sqrt {2}}-\frac {b^{3/4} \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^4}\right )}{4 \sqrt {2}}-\frac {1}{4} b \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^4}\right )-\frac {1}{4} b \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^4}\right ) \\ & = \frac {1}{3} \left (-b+a x^4\right )^{3/4}-\frac {b^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}+\sqrt {-b+a x^4}\right )}{4 \sqrt {2}}+\frac {b^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}+\sqrt {-b+a x^4}\right )}{4 \sqrt {2}}-\frac {b^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt {2}}+\frac {b^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt {2}} \\ & = \frac {1}{3} \left (-b+a x^4\right )^{3/4}+\frac {b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt {2}}-\frac {b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt {2}}-\frac {b^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}+\sqrt {-b+a x^4}\right )}{4 \sqrt {2}}+\frac {b^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}+\sqrt {-b+a x^4}\right )}{4 \sqrt {2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{x} \, dx=\frac {1}{12} \left (4 \left (-b+a x^4\right )^{3/4}-3 \sqrt {2} b^{3/4} \arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^4}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}}\right )+3 \sqrt {2} b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}}{\sqrt {b}+\sqrt {-b+a x^4}}\right )\right ) \]
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Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.10
method | result | size |
pseudoelliptic | \(\frac {\left (a \,x^{4}-b \right )^{\frac {3}{4}}}{3}-\frac {\ln \left (\frac {\sqrt {a \,x^{4}-b}-b^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}{\sqrt {a \,x^{4}-b}+b^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}\right ) b^{\frac {3}{4}} \sqrt {2}}{8}-\frac {\arctan \left (\frac {\sqrt {2}\, \left (a \,x^{4}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) b^{\frac {3}{4}} \sqrt {2}}{4}+\frac {\arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{4}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) b^{\frac {3}{4}} \sqrt {2}}{4}\) | \(159\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.05 \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{x} \, dx=-\frac {1}{4} \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{2} + \left (-b^{3}\right )^{\frac {3}{4}}\right ) + \frac {1}{4} i \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{2} + i \, \left (-b^{3}\right )^{\frac {3}{4}}\right ) - \frac {1}{4} i \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{2} - i \, \left (-b^{3}\right )^{\frac {3}{4}}\right ) + \frac {1}{4} \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{2} - \left (-b^{3}\right )^{\frac {3}{4}}\right ) + \frac {1}{3} \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} \]
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Result contains complex when optimal does not.
Time = 0.85 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.32 \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{x} \, dx=- \frac {a^{\frac {3}{4}} x^{3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac {1}{4}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{x} \, dx=-\frac {1}{8} \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} b + \frac {1}{3} \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} \]
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Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{x} \, dx=-\frac {1}{4} \, \sqrt {2} b^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} b^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) + \frac {1}{8} \, \sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right ) - \frac {1}{8} \, \sqrt {2} b^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right ) + \frac {1}{3} \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} \]
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Time = 6.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.44 \[ \int \frac {\left (-b+a x^4\right )^{3/4}}{x} \, dx=\frac {{\left (a\,x^4-b\right )}^{3/4}}{3}+\frac {{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (a\,x^4-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{2}-\frac {{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (a\,x^4-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{2} \]
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