Integrand size = 17, antiderivative size = 150 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=-\frac {\sqrt [4]{-b+a x^3}}{3 x^3}-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{-\sqrt {b}+\sqrt {-b+a x^3}}\right )}{6 \sqrt {2} b^{3/4}}+\frac {a \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^3}}\right )}{6 \sqrt {2} b^{3/4}} \]
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Time = 0.12 (sec) , antiderivative size = 225, 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, 43, 65, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=-\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}}+\frac {a \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{6 \sqrt {2} b^{3/4}}-\frac {a \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {a x^3-b}+\sqrt {b}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {a x^3-b}+\sqrt {b}\right )}{12 \sqrt {2} b^{3/4}}-\frac {\sqrt [4]{a x^3-b}}{3 x^3} \]
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Rule 43
Rule 65
Rule 210
Rule 217
Rule 272
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [4]{-b+a x}}{x^2} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [4]{-b+a x^3}}{3 x^3}+\frac {1}{12} a \text {Subst}\left (\int \frac {1}{x (-b+a x)^{3/4}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [4]{-b+a x^3}}{3 x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right ) \\ & = -\frac {\sqrt [4]{-b+a x^3}}{3 x^3}+\frac {\text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{6 \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^3}\right )}{6 \sqrt {b}} \\ & = -\frac {\sqrt [4]{-b+a x^3}}{3 x^3}-\frac {a \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^3}\right )}{12 \sqrt {2} b^{3/4}}-\frac {a \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^3}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 \sqrt {b}}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 \sqrt {b}} \\ & = -\frac {\sqrt [4]{-b+a x^3}}{3 x^3}-\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}}-\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}} \\ & = -\frac {\sqrt [4]{-b+a x^3}}{3 x^3}-\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}}+\frac {a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}}-\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=\frac {-4 b^{3/4} \sqrt [4]{-b+a x^3}+\sqrt {2} a x^3 \arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^3}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}\right )+\sqrt {2} a x^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{\sqrt {b}+\sqrt {-b+a x^3}}\right )}{12 b^{3/4} x^3} \]
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Time = 0.46 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.20
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\frac {-b^{\frac {1}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{3}-b}-\sqrt {b}}{b^{\frac {1}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{3}-b}-\sqrt {b}}\right ) \sqrt {2}\, a \,x^{3}+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{3}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{3}-2 \arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{3}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{3}-8 \left (a \,x^{3}-b \right )^{\frac {1}{4}} b^{\frac {3}{4}}}{24 x^{3} b^{\frac {3}{4}}}\) | \(180\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=\frac {\left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a + \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) + i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a + i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) - i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a - i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) - \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a - \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) - 4 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}}{12 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=- \frac {\sqrt [4]{a} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 x^{\frac {9}{4}} \Gamma \left (\frac {7}{4}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=\frac {\sqrt {2} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{12 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{12 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} a \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{24 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} a \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{24 \, b^{\frac {3}{4}}} - \frac {{\left (a x^{3} - b\right )}^{\frac {1}{4}}}{3 \, x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=\frac {\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {\sqrt {2} a^{2} \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {\sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {8 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}} a}{x^{3}}}{24 \, a} \]
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Time = 6.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=-\frac {{\left (a\,x^3-b\right )}^{1/4}}{3\,x^3}-\frac {a\,\mathrm {atan}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{3/4}}-\frac {a\,\mathrm {atanh}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{3/4}} \]
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