Integrand size = 19, antiderivative size = 66 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx=\frac {6 \sqrt [6]{a+b x}}{7 (b c-a d) (c+d x)^{7/6}}+\frac {36 b \sqrt [6]{a+b x}}{7 (b c-a d)^2 \sqrt [6]{c+d x}} \]
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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} \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx=\frac {36 b \sqrt [6]{a+b x}}{7 \sqrt [6]{c+d x} (b c-a d)^2}+\frac {6 \sqrt [6]{a+b x}}{7 (c+d x)^{7/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 \sqrt [6]{a+b x}}{7 (b c-a d) (c+d x)^{7/6}}+\frac {(6 b) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx}{7 (b c-a d)} \\ & = \frac {6 \sqrt [6]{a+b x}}{7 (b c-a d) (c+d x)^{7/6}}+\frac {36 b \sqrt [6]{a+b x}}{7 (b c-a d)^2 \sqrt [6]{c+d x}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx=\frac {6 \sqrt [6]{a+b x} (7 b c-a d+6 b d x)}{7 (b c-a d)^2 (c+d x)^{7/6}} \]
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Time = 0.80 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {1}{6}} \left (-6 b d x +a d -7 b c \right )}{7 \left (d x +c \right )^{\frac {7}{6}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(53\) |
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (54) = 108\).
Time = 0.22 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx=\frac {6 \, {\left (6 \, b d x + 7 \, b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{7 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{6}} \left (c + d x\right )^{\frac {13}{6}}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {13}{6}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {13}{6}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{13/6}} \,d x \]
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