Integrand size = 19, antiderivative size = 136 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\frac {6 (a+b x)^{5/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {108 b (a+b x)^{5/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac {1296 b^2 (a+b x)^{5/6}}{4301 (b c-a d)^3 (c+d x)^{11/6}}+\frac {7776 b^3 (a+b x)^{5/6}}{21505 (b c-a d)^4 (c+d x)^{5/6}} \]
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Time = 0.02 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\frac {7776 b^3 (a+b x)^{5/6}}{21505 (c+d x)^{5/6} (b c-a d)^4}+\frac {1296 b^2 (a+b x)^{5/6}}{4301 (c+d x)^{11/6} (b c-a d)^3}+\frac {108 b (a+b x)^{5/6}}{391 (c+d x)^{17/6} (b c-a d)^2}+\frac {6 (a+b x)^{5/6}}{23 (c+d x)^{23/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{5/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {(18 b) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx}{23 (b c-a d)} \\ & = \frac {6 (a+b x)^{5/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {108 b (a+b x)^{5/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac {\left (216 b^2\right ) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx}{391 (b c-a d)^2} \\ & = \frac {6 (a+b x)^{5/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {108 b (a+b x)^{5/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac {1296 b^2 (a+b x)^{5/6}}{4301 (b c-a d)^3 (c+d x)^{11/6}}+\frac {\left (1296 b^3\right ) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{4301 (b c-a d)^3} \\ & = \frac {6 (a+b x)^{5/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac {108 b (a+b x)^{5/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac {1296 b^2 (a+b x)^{5/6}}{4301 (b c-a d)^3 (c+d x)^{11/6}}+\frac {7776 b^3 (a+b x)^{5/6}}{21505 (b c-a d)^4 (c+d x)^{5/6}} \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\frac {6 (a+b x)^{5/6} \left (-935 a^3 d^3+165 a^2 b d^2 (23 c+6 d x)-15 a b^2 d \left (391 c^2+276 c d x+72 d^2 x^2\right )+b^3 \left (4301 c^3+7038 c^2 d x+4968 c d^2 x^2+1296 d^3 x^3\right )\right )}{21505 (b c-a d)^4 (c+d x)^{23/6}} \]
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Time = 0.92 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {5}{6}} \left (-1296 d^{3} x^{3} b^{3}+1080 x^{2} a \,b^{2} d^{3}-4968 x^{2} b^{3} c \,d^{2}-990 x \,a^{2} b \,d^{3}+4140 x a \,b^{2} c \,d^{2}-7038 x \,b^{3} c^{2} d +935 a^{3} d^{3}-3795 a^{2} b c \,d^{2}+5865 a \,b^{2} c^{2} d -4301 b^{3} c^{3}\right )}{21505 \left (d x +c \right )^{\frac {23}{6}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (112) = 224\).
Time = 0.26 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.09 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\frac {6 \, {\left (1296 \, b^{3} d^{3} x^{3} + 4301 \, b^{3} c^{3} - 5865 \, a b^{2} c^{2} d + 3795 \, a^{2} b c d^{2} - 935 \, a^{3} d^{3} + 216 \, {\left (23 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (391 \, b^{3} c^{2} d - 230 \, a b^{2} c d^{2} + 55 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{21505 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4} + {\left (b^{4} c^{4} d^{4} - 4 \, a b^{3} c^{3} d^{5} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}\right )} x^{4} + 4 \, {\left (b^{4} c^{5} d^{3} - 4 \, a b^{3} c^{4} d^{4} + 6 \, a^{2} b^{2} c^{3} d^{5} - 4 \, a^{3} b c^{2} d^{6} + a^{4} c d^{7}\right )} x^{3} + 6 \, {\left (b^{4} c^{6} d^{2} - 4 \, a b^{3} c^{5} d^{3} + 6 \, a^{2} b^{2} c^{4} d^{4} - 4 \, a^{3} b c^{3} d^{5} + a^{4} c^{2} d^{6}\right )} x^{2} + 4 \, {\left (b^{4} c^{7} d - 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{5} d^{3} - 4 \, a^{3} b c^{4} d^{4} + a^{4} c^{3} d^{5}\right )} x\right )}} \]
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Timed out. \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {29}{6}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {29}{6}}} \,d x } \]
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Time = 1.38 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.15 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx=\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {7776\,b^4\,x^4}{21505\,d\,{\left (a\,d-b\,c\right )}^4}-\frac {5610\,a^4\,d^3-22770\,a^3\,b\,c\,d^2+35190\,a^2\,b^2\,c^2\,d-25806\,a\,b^3\,c^3}{21505\,d^4\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (330\,a^3\,b\,d^3-2070\,a^2\,b^2\,c\,d^2+7038\,a\,b^3\,c^2\,d+25806\,b^4\,c^3\right )}{21505\,d^4\,{\left (a\,d-b\,c\right )}^4}+\frac {1296\,b^3\,x^3\,\left (a\,d+23\,b\,c\right )}{21505\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b^2\,x^2\,\left (-5\,a^2\,d^2+46\,a\,b\,c\,d+391\,b^2\,c^2\right )}{21505\,d^3\,{\left (a\,d-b\,c\right )}^4}\right )}{x^4\,{\left (a+b\,x\right )}^{1/6}+\frac {c^4\,{\left (a+b\,x\right )}^{1/6}}{d^4}+\frac {6\,c^2\,x^2\,{\left (a+b\,x\right )}^{1/6}}{d^2}+\frac {4\,c\,x^3\,{\left (a+b\,x\right )}^{1/6}}{d}+\frac {4\,c^3\,x\,{\left (a+b\,x\right )}^{1/6}}{d^3}} \]
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