Integrand size = 19, antiderivative size = 66 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx=\frac {6 (a+b x)^{7/6}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {36 b (a+b x)^{7/6}}{91 (b c-a d)^2 (c+d x)^{7/6}} \]
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Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx=\frac {36 b (a+b x)^{7/6}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac {6 (a+b x)^{7/6}}{13 (c+d x)^{13/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{7/6}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {(6 b) \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx}{13 (b c-a d)} \\ & = \frac {6 (a+b x)^{7/6}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {36 b (a+b x)^{7/6}}{91 (b c-a d)^2 (c+d x)^{7/6}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx=\frac {6 (a+b x)^{7/6} (13 b c-7 a d+6 b d x)}{91 (b c-a d)^2 (c+d x)^{13/6}} \]
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Time = 0.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {7}{6}} \left (-6 b d x +7 a d -13 b c \right )}{91 \left (d x +c \right )^{\frac {13}{6}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(54\) |
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (54) = 108\).
Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.65 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx=\frac {6 \, {\left (6 \, b^{2} d x^{2} + 13 \, a b c - 7 \, a^{2} d + {\left (13 \, b^{2} c - a b d\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{91 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{3} + 3 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2} + 3 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x\right )}} \]
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Timed out. \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \]
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\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \]
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Time = 0.69 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.08 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx=\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {36\,b^2\,x^2\,{\left (a+b\,x\right )}^{1/6}}{91\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {\left (42\,a^2\,d-78\,a\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{91\,d^3\,{\left (a\,d-b\,c\right )}^2}+\frac {x\,\left (78\,b^2\,c-6\,a\,b\,d\right )\,{\left (a+b\,x\right )}^{1/6}}{91\,d^3\,{\left (a\,d-b\,c\right )}^2}\right )}{x^3+\frac {c^3}{d^3}+\frac {3\,c\,x^2}{d}+\frac {3\,c^2\,x}{d^2}} \]
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