Integrand size = 19, antiderivative size = 101 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx=\frac {6 (a+b x)^{11/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {72 b (a+b x)^{11/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac {432 b^2 (a+b x)^{11/6}}{4301 (b c-a d)^3 (c+d x)^{11/6}} \]
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Time = 0.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx=\frac {432 b^2 (a+b x)^{11/6}}{4301 (c+d x)^{11/6} (b c-a d)^3}+\frac {72 b (a+b x)^{11/6}}{391 (c+d x)^{17/6} (b c-a d)^2}+\frac {6 (a+b x)^{11/6}}{23 (c+d x)^{23/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{11/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {(12 b) \int \frac {(a+b x)^{5/6}}{(c+d x)^{23/6}} \, dx}{23 (b c-a d)} \\ & = \frac {6 (a+b x)^{11/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {72 b (a+b x)^{11/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac {\left (72 b^2\right ) \int \frac {(a+b x)^{5/6}}{(c+d x)^{17/6}} \, dx}{391 (b c-a d)^2} \\ & = \frac {6 (a+b x)^{11/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {72 b (a+b x)^{11/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac {432 b^2 (a+b x)^{11/6}}{4301 (b c-a d)^3 (c+d x)^{11/6}} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx=\frac {6 (a+b x)^{11/6} \left (187 a^2 d^2-22 a b d (23 c+6 d x)+b^2 \left (391 c^2+276 c d x+72 d^2 x^2\right )\right )}{4301 (b c-a d)^3 (c+d x)^{23/6}} \]
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Time = 0.75 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {11}{6}} \left (72 d^{2} x^{2} b^{2}-132 x a b \,d^{2}+276 x \,b^{2} c d +187 a^{2} d^{2}-506 a b c d +391 b^{2} c^{2}\right )}{4301 \left (d x +c \right )^{\frac {23}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
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Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (83) = 166\).
Time = 0.24 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.35 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx=\frac {6 \, {\left (72 \, b^{3} d^{2} x^{3} + 391 \, a b^{2} c^{2} - 506 \, a^{2} b c d + 187 \, a^{3} d^{2} + 12 \, {\left (23 \, b^{3} c d - 5 \, a b^{2} d^{2}\right )} x^{2} + {\left (391 \, b^{3} c^{2} - 230 \, a b^{2} c d + 55 \, a^{2} b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{4301 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3} + {\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} x^{4} + 4 \, {\left (b^{3} c^{4} d^{3} - 3 \, a b^{2} c^{3} d^{4} + 3 \, a^{2} b c^{2} d^{5} - a^{3} c d^{6}\right )} x^{3} + 6 \, {\left (b^{3} c^{5} d^{2} - 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5}\right )} x^{2} + 4 \, {\left (b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}\right )} x\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {29}{6}}} \,d x } \]
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\[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {29}{6}}} \,d x } \]
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Time = 0.90 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx=-\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {{\left (a+b\,x\right )}^{5/6}\,\left (1122\,a^3\,d^2-3036\,a^2\,b\,c\,d+2346\,a\,b^2\,c^2\right )}{4301\,d^4\,{\left (a\,d-b\,c\right )}^3}+\frac {432\,b^3\,x^3\,{\left (a+b\,x\right )}^{5/6}}{4301\,d^2\,{\left (a\,d-b\,c\right )}^3}+\frac {x\,{\left (a+b\,x\right )}^{5/6}\,\left (330\,a^2\,b\,d^2-1380\,a\,b^2\,c\,d+2346\,b^3\,c^2\right )}{4301\,d^4\,{\left (a\,d-b\,c\right )}^3}-\frac {72\,b^2\,x^2\,\left (5\,a\,d-23\,b\,c\right )\,{\left (a+b\,x\right )}^{5/6}}{4301\,d^3\,{\left (a\,d-b\,c\right )}^3}\right )}{x^4+\frac {c^4}{d^4}+\frac {4\,c\,x^3}{d}+\frac {4\,c^3\,x}{d^3}+\frac {6\,c^2\,x^2}{d^2}} \]
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