Integrand size = 24, antiderivative size = 97 \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=-\frac {4 \left (b x+a x^4\right )^{3/4}}{9 x^3}+\frac {2}{3} a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )+\frac {2}{3} a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.47, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2077, 2036, 335, 281, 246, 218, 212, 209, 2039} \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3+b} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a x^4+b x}}+\frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3+b} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a x^4+b x}}-\frac {4 \left (a x^4+b x\right )^{3/4}}{9 x^3} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 281
Rule 335
Rule 2036
Rule 2039
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{\sqrt [4]{b x+a x^4}}+\frac {b}{x^3 \sqrt [4]{b x+a x^4}}\right ) \, dx \\ & = a \int \frac {1}{\sqrt [4]{b x+a x^4}} \, dx+b \int \frac {1}{x^3 \sqrt [4]{b x+a x^4}} \, dx \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}} \\ & = -\frac {4 \left (b x+a x^4\right )^{3/4}}{9 x^3}+\frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{b+a x^3} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{b+a x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}} \\ \end{align*}
Time = 4.44 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.19 \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\frac {2 \left (x \left (b+a x^3\right )\right )^{3/4} \left (-2 \left (b+a x^3\right )^{3/4}+3 a^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )+3 a^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )\right )}{9 x^3 \left (b+a x^3\right )^{3/4}} \]
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Time = 0.98 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(\frac {3 \ln \left (\frac {-a^{\frac {1}{4}} x -{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x -{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}\right ) a^{\frac {3}{4}} x^{3}-6 \arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{x \,a^{\frac {1}{4}}}\right ) a^{\frac {3}{4}} x^{3}-4 {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {3}{4}}}{9 x^{3}}\) | \(98\) |
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Timed out. \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\text {Timed out} \]
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\[ \int \frac {b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\int \frac {a x^{3} + b}{x^{3} \sqrt [4]{x \left (a x^{3} + b\right )}}\, dx \]
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\[ \int \frac {b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\int { \frac {a x^{3} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (77) = 154\).
Time = 0.29 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.91 \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{6} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right ) + \frac {1}{6} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right ) - \frac {4}{9} \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}} \]
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Time = 6.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.60 \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\frac {4\,a\,x\,{\left (\frac {a\,x^3}{b}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -\frac {a\,x^3}{b}\right )}{3\,{\left (a\,x^4+b\,x\right )}^{1/4}}-\frac {4\,{\left (a\,x^4+b\,x\right )}^{3/4}}{9\,x^3} \]
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