Integrand size = 19, antiderivative size = 136 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\frac {6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {108 b (a+b x)^{7/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {1296 b^2 (a+b x)^{7/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}}+\frac {7776 b^3 (a+b x)^{7/6}}{43225 (b c-a d)^4 (c+d x)^{7/6}} \]
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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} \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\frac {7776 b^3 (a+b x)^{7/6}}{43225 (c+d x)^{7/6} (b c-a d)^4}+\frac {1296 b^2 (a+b x)^{7/6}}{6175 (c+d x)^{13/6} (b c-a d)^3}+\frac {108 b (a+b x)^{7/6}}{475 (c+d x)^{19/6} (b c-a d)^2}+\frac {6 (a+b x)^{7/6}}{25 (c+d x)^{25/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {(18 b) \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx}{25 (b c-a d)} \\ & = \frac {6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {108 b (a+b x)^{7/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {\left (216 b^2\right ) \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx}{475 (b c-a d)^2} \\ & = \frac {6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {108 b (a+b x)^{7/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {1296 b^2 (a+b x)^{7/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}}+\frac {\left (1296 b^3\right ) \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx}{6175 (b c-a d)^3} \\ & = \frac {6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {108 b (a+b x)^{7/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {1296 b^2 (a+b x)^{7/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}}+\frac {7776 b^3 (a+b x)^{7/6}}{43225 (b c-a d)^4 (c+d x)^{7/6}} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\frac {6 (a+b x)^{7/6} \left (-1729 a^3 d^3+273 a^2 b d^2 (25 c+6 d x)-21 a b^2 d \left (475 c^2+300 c d x+72 d^2 x^2\right )+b^3 \left (6175 c^3+8550 c^2 d x+5400 c d^2 x^2+1296 d^3 x^3\right )\right )}{43225 (b c-a d)^4 (c+d x)^{25/6}} \]
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Time = 0.89 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {7}{6}} \left (-1296 d^{3} x^{3} b^{3}+1512 x^{2} a \,b^{2} d^{3}-5400 x^{2} b^{3} c \,d^{2}-1638 x \,a^{2} b \,d^{3}+6300 x a \,b^{2} c \,d^{2}-8550 x \,b^{3} c^{2} d +1729 a^{3} d^{3}-6825 a^{2} b c \,d^{2}+9975 a \,b^{2} c^{2} d -6175 b^{3} c^{3}\right )}{43225 \left (d x +c \right )^{\frac {25}{6}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (112) = 224\).
Time = 0.26 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.92 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\frac {6 \, {\left (1296 \, b^{4} d^{3} x^{4} + 6175 \, a b^{3} c^{3} - 9975 \, a^{2} b^{2} c^{2} d + 6825 \, a^{3} b c d^{2} - 1729 \, a^{4} d^{3} + 216 \, {\left (25 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 18 \, {\left (475 \, b^{4} c^{2} d - 50 \, a b^{3} c d^{2} + 7 \, a^{2} b^{2} d^{3}\right )} x^{2} + {\left (6175 \, b^{4} c^{3} - 1425 \, a b^{3} c^{2} d + 525 \, a^{2} b^{2} c d^{2} - 91 \, a^{3} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{43225 \, {\left (b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4} + {\left (b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}\right )} x^{5} + 5 \, {\left (b^{4} c^{5} d^{4} - 4 \, a b^{3} c^{4} d^{5} + 6 \, a^{2} b^{2} c^{3} d^{6} - 4 \, a^{3} b c^{2} d^{7} + a^{4} c d^{8}\right )} x^{4} + 10 \, {\left (b^{4} c^{6} d^{3} - 4 \, a b^{3} c^{5} d^{4} + 6 \, a^{2} b^{2} c^{4} d^{5} - 4 \, a^{3} b c^{3} d^{6} + a^{4} c^{2} d^{7}\right )} x^{3} + 10 \, {\left (b^{4} c^{7} d^{2} - 4 \, a b^{3} c^{6} d^{3} + 6 \, a^{2} b^{2} c^{5} d^{4} - 4 \, a^{3} b c^{4} d^{5} + a^{4} c^{3} d^{6}\right )} x^{2} + 5 \, {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )} x\right )}} \]
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Timed out. \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {31}{6}}} \,d x } \]
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\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {31}{6}}} \,d x } \]
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Time = 1.29 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.22 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx=\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {7776\,b^4\,x^4\,{\left (a+b\,x\right )}^{1/6}}{43225\,d^2\,{\left (a\,d-b\,c\right )}^4}-\frac {{\left (a+b\,x\right )}^{1/6}\,\left (10374\,a^4\,d^3-40950\,a^3\,b\,c\,d^2+59850\,a^2\,b^2\,c^2\,d-37050\,a\,b^3\,c^3\right )}{43225\,d^5\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,{\left (a+b\,x\right )}^{1/6}\,\left (-546\,a^3\,b\,d^3+3150\,a^2\,b^2\,c\,d^2-8550\,a\,b^3\,c^2\,d+37050\,b^4\,c^3\right )}{43225\,d^5\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b^2\,x^2\,{\left (a+b\,x\right )}^{1/6}\,\left (7\,a^2\,d^2-50\,a\,b\,c\,d+475\,b^2\,c^2\right )}{43225\,d^4\,{\left (a\,d-b\,c\right )}^4}-\frac {1296\,b^3\,x^3\,\left (a\,d-25\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{43225\,d^3\,{\left (a\,d-b\,c\right )}^4}\right )}{x^5+\frac {c^5}{d^5}+\frac {5\,c\,x^4}{d}+\frac {5\,c^4\,x}{d^4}+\frac {10\,c^2\,x^3}{d^2}+\frac {10\,c^3\,x^2}{d^3}} \]
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