Integrand size = 19, antiderivative size = 85 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}} \]
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Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {65, 338, 304, 211, 214} \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}} \]
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Rule 65
Rule 211
Rule 214
Rule 304
Rule 338
Rubi steps \begin{align*} \text {integral}& = \frac {4 \text {Subst}\left (\int \frac {x^2}{\left (c-\frac {a d}{b}+\frac {d x^4}{b}\right )^{3/4}} \, dx,x,\sqrt [4]{a+b x}\right )}{b} \\ & = \frac {4 \text {Subst}\left (\int \frac {x^2}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{b} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{\sqrt {d}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{\sqrt {d}} \\ & = -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )\right )}{\sqrt [4]{b} d^{3/4}} \]
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\[\int \frac {1}{\left (b x +a \right )^{\frac {1}{4}} \left (d x +c \right )^{\frac {3}{4}}}d x\]
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.61 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=\left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) - \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) + i \, \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, b d x + i \, a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) - i \, \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, b d x - i \, a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) \]
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\[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=\int \frac {1}{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{1/4}\,{\left (c+d\,x\right )}^{3/4}} \,d x \]
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