Integrand size = 19, antiderivative size = 66 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx=\frac {6 (a+b x)^{13/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {36 b (a+b x)^{13/6}}{247 (b c-a d)^2 (c+d x)^{13/6}} \]
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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} \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx=\frac {36 b (a+b x)^{13/6}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac {6 (a+b x)^{13/6}}{19 (c+d x)^{19/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{13/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {(6 b) \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx}{19 (b c-a d)} \\ & = \frac {6 (a+b x)^{13/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {36 b (a+b x)^{13/6}}{247 (b c-a d)^2 (c+d x)^{13/6}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx=\frac {6 (a+b x)^{13/6} (19 b c-13 a d+6 b d x)}{247 (b c-a d)^2 (c+d x)^{19/6}} \]
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Time = 0.77 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {13}{6}} \left (-6 b d x +13 a d -19 b c \right )}{247 \left (d x +c \right )^{\frac {19}{6}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(54\) |
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Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (54) = 108\).
Time = 0.23 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.56 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx=\frac {6 \, {\left (6 \, b^{3} d x^{3} + 19 \, a^{2} b c - 13 \, a^{3} d + {\left (19 \, b^{3} c - a b^{2} d\right )} x^{2} + 2 \, {\left (19 \, a b^{2} c - 10 \, a^{2} b d\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{247 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} x^{4} + 4 \, {\left (b^{2} c^{3} d^{3} - 2 \, a b c^{2} d^{4} + a^{2} c d^{5}\right )} x^{3} + 6 \, {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 4 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {25}{6}}} \,d x } \]
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\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {25}{6}}} \,d x } \]
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Time = 0.90 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.86 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx=-\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {\left (78\,a^3\,d-114\,a^2\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{247\,d^4\,{\left (a\,d-b\,c\right )}^2}-\frac {36\,b^3\,x^3\,{\left (a+b\,x\right )}^{1/6}}{247\,d^3\,{\left (a\,d-b\,c\right )}^2}-\frac {x^2\,\left (114\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (a+b\,x\right )}^{1/6}}{247\,d^4\,{\left (a\,d-b\,c\right )}^2}+\frac {12\,a\,b\,x\,\left (10\,a\,d-19\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{247\,d^4\,{\left (a\,d-b\,c\right )}^2}\right )}{x^4+\frac {c^4}{d^4}+\frac {4\,c\,x^3}{d}+\frac {4\,c^3\,x}{d^3}+\frac {6\,c^2\,x^2}{d^2}} \]
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