Integrand size = 20, antiderivative size = 279 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {1+b x}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {1+b x}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}} \]
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Time = 0.22 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{b x+1}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{b x+1}}+1\right )}{\sqrt [4]{a} b^{3/4}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{b x+1}}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {b x+1}}+\sqrt {a}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{b x+1}}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {b x+1}}+\sqrt {a}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}} \]
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Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \text {Subst}\left (\int \frac {x^2}{\left (1+\frac {b}{a}-\frac {b x^4}{a}\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{a} \\ & = -\frac {4 \text {Subst}\left (\int \frac {x^2}{1+\frac {b x^4}{a}} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{a} \\ & = \frac {2 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{1+\frac {b x^4}{a}} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{a \sqrt {b}}-\frac {2 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{1+\frac {b x^4}{a}} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{a \sqrt {b}} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{b}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{b}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}} \\ & = -\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {1+b x}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {1+b x}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}}-\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}+\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}} \\ & = \frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {1+b x}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {1+b x}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x} \sqrt [4]{1+b x}}{\sqrt {b} \sqrt {1-a x}-\sqrt {a} \sqrt {1+b x}}\right )+\text {arctanh}\left (\frac {\sqrt {b} \sqrt {1-a x}+\sqrt {a} \sqrt {1+b x}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x} \sqrt [4]{1+b x}}\right )\right )}{\sqrt [4]{a} b^{3/4}} \]
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\[\int \frac {1}{\left (-a x +1\right )^{\frac {1}{4}} \left (b x +1\right )^{\frac {3}{4}}}d x\]
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=-\left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a b x - b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) + \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a b x - b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} - {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) - i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, a b x - i \, b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) + i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, a b x + i \, b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) \]
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\[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\int \frac {1}{\sqrt [4]{- a x + 1} \left (b x + 1\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\int { \frac {1}{{\left (-a x + 1\right )}^{\frac {1}{4}} {\left (b x + 1\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\int { \frac {1}{{\left (-a x + 1\right )}^{\frac {1}{4}} {\left (b x + 1\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\int \frac {1}{{\left (1-a\,x\right )}^{1/4}\,{\left (b\,x+1\right )}^{3/4}} \,d x \]
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