Integrand size = 19, antiderivative size = 332 \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}+\frac {2 \sqrt [6]{d} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt [6]{d} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}}+\frac {\sqrt [6]{d} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}} \]
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Time = 0.37 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {49, 65, 338, 302, 648, 632, 210, 642, 214} \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\frac {\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{b^{7/6}}+\frac {2 \sqrt [6]{d} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt [6]{d} \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}+\frac {\sqrt [6]{d} \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}} \]
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Rule 49
Rule 65
Rule 210
Rule 214
Rule 302
Rule 338
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {d \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx}{b} \\ & = -\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {(6 d) \text {Subst}\left (\int \frac {x^4}{\left (c-\frac {a d}{b}+\frac {d x^6}{b}\right )^{5/6}} \, dx,x,\sqrt [6]{a+b x}\right )}{b^2} \\ & = -\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {(6 d) \text {Subst}\left (\int \frac {x^4}{1-\frac {d x^6}{b}} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{b^2} \\ & = -\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {\left (2 \sqrt [3]{d}\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt [6]{b}}{2}-\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{b^{7/6}}+\frac {\left (2 \sqrt [3]{d}\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt [6]{b}}{2}+\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{b^{7/6}}+\frac {\left (2 \sqrt [3]{d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{b} \\ & = -\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt [6]{d} \text {Subst}\left (\int \frac {-\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}}+\frac {\sqrt [6]{d} \text {Subst}\left (\int \frac {\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}}-\frac {\left (3 \sqrt [3]{d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b}-\frac {\left (3 \sqrt [3]{d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b} \\ & = -\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt [6]{d} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}}+\frac {\sqrt [6]{d} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}}-\frac {\left (3 \sqrt [6]{d}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}+\frac {\left (3 \sqrt [6]{d}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}} \\ & = -\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {\sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{b^{7/6}}-\frac {\sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{b^{7/6}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt [6]{d} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}}+\frac {\sqrt [6]{d} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\frac {-\frac {6 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{a+b x}}+\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}-\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+2 \sqrt [6]{d} \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+\sqrt [6]{d} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{b^{7/6}} \]
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\[\int \frac {\left (d x +c \right )^{\frac {1}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (241) = 482\).
Time = 0.24 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\frac {{\left (b^{2} x + a b + \sqrt {-3} {\left (b^{2} x + a b\right )}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b^{2} x + a b + \sqrt {-3} {\left (b^{2} x + a b\right )}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} + 2 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) - {\left (b^{2} x + a b + \sqrt {-3} {\left (b^{2} x + a b\right )}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b^{2} x + a b + \sqrt {-3} {\left (b^{2} x + a b\right )}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} - 2 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) + {\left (b^{2} x + a b - \sqrt {-3} {\left (b^{2} x + a b\right )}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b^{2} x + a b - \sqrt {-3} {\left (b^{2} x + a b\right )}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} + 2 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) - {\left (b^{2} x + a b - \sqrt {-3} {\left (b^{2} x + a b\right )}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b^{2} x + a b - \sqrt {-3} {\left (b^{2} x + a b\right )}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} - 2 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) + 2 \, {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) - 2 \, {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) - 12 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{2 \, {\left (b^{2} x + a b\right )}} \]
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\[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\int \frac {\sqrt [6]{c + d x}}{\left (a + b x\right )^{\frac {7}{6}}}\, dx \]
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\[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \]
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\[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\int \frac {{\left (c+d\,x\right )}^{1/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \]
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