Integrand size = 19, antiderivative size = 98 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{17/6}} \, dx=-\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{11/6}}-\frac {72 d (a+b x)^{5/6}}{11 (b c-a d)^2 (c+d x)^{11/6}}-\frac {432 b d (a+b x)^{5/6}}{55 (b c-a d)^3 (c+d x)^{5/6}} \]
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Time = 0.02 (sec) , antiderivative size = 98, 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} \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{17/6}} \, dx=-\frac {432 b d (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^3}-\frac {72 d (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)^2}-\frac {6}{\sqrt [6]{a+b x} (c+d x)^{11/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{11/6}}-\frac {(12 d) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx}{b c-a d} \\ & = -\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{11/6}}-\frac {72 d (a+b x)^{5/6}}{11 (b c-a d)^2 (c+d x)^{11/6}}-\frac {(72 b d) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{11 (b c-a d)^2} \\ & = -\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{11/6}}-\frac {72 d (a+b x)^{5/6}}{11 (b c-a d)^2 (c+d x)^{11/6}}-\frac {432 b d (a+b x)^{5/6}}{55 (b c-a d)^3 (c+d x)^{5/6}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{17/6}} \, dx=-\frac {6 \left (-5 a^2 d^2+2 a b d (11 c+6 d x)+b^2 \left (55 c^2+132 c d x+72 d^2 x^2\right )\right )}{55 (b c-a d)^3 \sqrt [6]{a+b x} (c+d x)^{11/6}} \]
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Time = 0.96 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.07
method | result | size |
gosper | \(-\frac {6 \left (-72 d^{2} x^{2} b^{2}-12 x a b \,d^{2}-132 x \,b^{2} c d +5 a^{2} d^{2}-22 a b c d -55 b^{2} c^{2}\right )}{55 \left (b x +a \right )^{\frac {1}{6}} \left (d x +c \right )^{\frac {11}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
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Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (82) = 164\).
Time = 0.24 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.79 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{17/6}} \, dx=-\frac {6 \, {\left (72 \, b^{2} d^{2} x^{2} + 55 \, b^{2} c^{2} + 22 \, a b c d - 5 \, a^{2} d^{2} + 12 \, {\left (11 \, b^{2} c d + a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{55 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]
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Timed out. \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{17/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{17/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {17}{6}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{17/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {17}{6}}} \,d x } \]
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Time = 1.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{17/6}} \, dx=\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {432\,b^2\,x^2}{55\,{\left (a\,d-b\,c\right )}^3}+\frac {-30\,a^2\,d^2+132\,a\,b\,c\,d+330\,b^2\,c^2}{55\,d^2\,{\left (a\,d-b\,c\right )}^3}+\frac {72\,b\,x\,\left (a\,d+11\,b\,c\right )}{55\,d\,{\left (a\,d-b\,c\right )}^3}\right )}{x^2\,{\left (a+b\,x\right )}^{1/6}+\frac {c^2\,{\left (a+b\,x\right )}^{1/6}}{d^2}+\frac {2\,c\,x\,{\left (a+b\,x\right )}^{1/6}}{d}} \]
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