Integrand size = 19, antiderivative size = 134 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx=-\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac {108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac {1296 b d (a+b x)^{5/6}}{187 (b c-a d)^3 (c+d x)^{11/6}}-\frac {7776 b^2 d (a+b x)^{5/6}}{935 (b c-a d)^4 (c+d x)^{5/6}} \]
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Time = 0.03 (sec) , antiderivative size = 134, 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} \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx=-\frac {7776 b^2 d (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^4}-\frac {1296 b d (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^3}-\frac {108 d (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)^2}-\frac {6}{\sqrt [6]{a+b x} (c+d x)^{17/6} (b c-a d)} \]
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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)^{17/6}}-\frac {(18 d) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx}{b c-a d} \\ & = -\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac {108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac {(216 b d) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx}{17 (b c-a d)^2} \\ & = -\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac {108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac {1296 b d (a+b x)^{5/6}}{187 (b c-a d)^3 (c+d x)^{11/6}}-\frac {\left (1296 b^2 d\right ) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{187 (b c-a d)^3} \\ & = -\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac {108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac {1296 b d (a+b x)^{5/6}}{187 (b c-a d)^3 (c+d x)^{11/6}}-\frac {7776 b^2 d (a+b x)^{5/6}}{935 (b c-a d)^4 (c+d x)^{5/6}} \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx=-\frac {6 \left (55 a^3 d^3-15 a^2 b d^2 (17 c+6 d x)+3 a b^2 d \left (187 c^2+204 c d x+72 d^2 x^2\right )+b^3 \left (935 c^3+3366 c^2 d x+3672 c d^2 x^2+1296 d^3 x^3\right )\right )}{935 (b c-a d)^4 \sqrt [6]{a+b x} (c+d x)^{17/6}} \]
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Time = 0.95 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.28
method | result | size |
gosper | \(-\frac {6 \left (1296 d^{3} x^{3} b^{3}+216 x^{2} a \,b^{2} d^{3}+3672 x^{2} b^{3} c \,d^{2}-90 x \,a^{2} b \,d^{3}+612 x a \,b^{2} c \,d^{2}+3366 x \,b^{3} c^{2} d +55 a^{3} d^{3}-255 a^{2} b c \,d^{2}+561 a \,b^{2} c^{2} d +935 b^{3} c^{3}\right )}{935 \left (b x +a \right )^{\frac {1}{6}} \left (d x +c \right )^{\frac {17}{6}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (112) = 224\).
Time = 0.26 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.41 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx=-\frac {6 \, {\left (1296 \, b^{3} d^{3} x^{3} + 935 \, b^{3} c^{3} + 561 \, a b^{2} c^{2} d - 255 \, a^{2} b c d^{2} + 55 \, a^{3} d^{3} + 216 \, {\left (17 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (187 \, b^{3} c^{2} d + 34 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{935 \, {\left (a b^{4} c^{7} - 4 \, a^{2} b^{3} c^{6} d + 6 \, a^{3} b^{2} c^{5} d^{2} - 4 \, a^{4} b c^{4} d^{3} + a^{5} c^{3} d^{4} + {\left (b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{4} + {\left (3 \, b^{5} c^{5} d^{2} - 11 \, a b^{4} c^{4} d^{3} + 14 \, a^{2} b^{3} c^{3} d^{4} - 6 \, a^{3} b^{2} c^{2} d^{5} - a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{3} + 3 \, {\left (b^{5} c^{6} d - 3 \, a b^{4} c^{5} d^{2} + 2 \, a^{2} b^{3} c^{4} d^{3} + 2 \, a^{3} b^{2} c^{3} d^{4} - 3 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{2} + {\left (b^{5} c^{7} - a b^{4} c^{6} d - 6 \, a^{2} b^{3} c^{5} d^{2} + 14 \, a^{3} b^{2} c^{4} d^{3} - 11 \, a^{4} b c^{3} d^{4} + 3 \, a^{5} c^{2} d^{5}\right )} x\right )}} \]
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Timed out. \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {23}{6}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {23}{6}}} \,d x } \]
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Time = 1.45 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.56 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx=-\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {7776\,b^3\,x^3}{935\,{\left (a\,d-b\,c\right )}^4}+\frac {330\,a^3\,d^3-1530\,a^2\,b\,c\,d^2+3366\,a\,b^2\,c^2\,d+5610\,b^3\,c^3}{935\,d^3\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b\,x\,\left (-5\,a^2\,d^2+34\,a\,b\,c\,d+187\,b^2\,c^2\right )}{935\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {1296\,b^2\,x^2\,\left (a\,d+17\,b\,c\right )}{935\,d\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,{\left (a+b\,x\right )}^{1/6}+\frac {c^3\,{\left (a+b\,x\right )}^{1/6}}{d^3}+\frac {3\,c\,x^2\,{\left (a+b\,x\right )}^{1/6}}{d}+\frac {3\,c^2\,x\,{\left (a+b\,x\right )}^{1/6}}{d^2}} \]
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