Integrand size = 19, antiderivative size = 81 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2} \, dx=-\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.67, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2045, 2057, 65, 338, 304, 209, 212} \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2} \, dx=-\frac {2 \sqrt [4]{a} x^{9/4} (a x+b)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{\left (a x^4+b x^3\right )^{3/4}}+\frac {2 \sqrt [4]{a} x^{9/4} (a x+b)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{\left (a x^4+b x^3\right )^{3/4}}-\frac {4 \sqrt [4]{a x^4+b x^3}}{x} \]
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Rule 65
Rule 209
Rule 212
Rule 304
Rule 338
Rule 2045
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \sqrt [4]{b x^3+a x^4}}{x}+a \int \frac {x^2}{\left (b x^3+a x^4\right )^{3/4}} \, dx \\ & = -\frac {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (a x^{9/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{\left (b x^3+a x^4\right )^{3/4}} \\ & = -\frac {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (4 a x^{9/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\left (b x^3+a x^4\right )^{3/4}} \\ & = -\frac {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (4 a x^{9/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\left (b x^3+a x^4\right )^{3/4}} \\ & = -\frac {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (2 \sqrt {a} x^{9/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\left (b x^3+a x^4\right )^{3/4}}-\frac {\left (2 \sqrt {a} x^{9/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\left (b x^3+a x^4\right )^{3/4}} \\ & = -\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} x^{9/4} (b+a x)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\left (b x^3+a x^4\right )^{3/4}}+\frac {2 \sqrt [4]{a} x^{9/4} (b+a x)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\left (b x^3+a x^4\right )^{3/4}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2} \, dx=-\frac {2 x^2 (b+a x)^{3/4} \left (2 \sqrt [4]{b+a x}+\sqrt [4]{a} \sqrt [4]{x} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-\sqrt [4]{a} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )}{\left (x^3 (b+a x)\right )^{3/4}} \]
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Time = 2.66 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05
method | result | size |
pseudoelliptic | \(\frac {x \left (\ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )\right ) a^{\frac {1}{4}}-4 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\) | \(85\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2} \, dx=\frac {a^{\frac {1}{4}} x \log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - a^{\frac {1}{4}} x \log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, a^{\frac {1}{4}} x \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, a^{\frac {1}{4}} x \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x} \]
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\[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (67) = 134\).
Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.26 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2} \, dx=\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) - \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) - 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} \]
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Time = 5.99 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{x^2} \, dx=-\frac {4\,{\left (a\,x^4+b\,x^3\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ -\frac {a\,x}{b}\right )}{x\,{\left (\frac {a\,x}{b}+1\right )}^{1/4}} \]
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