Integrand size = 19, antiderivative size = 101 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx=\frac {6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {72 b (a+b x)^{13/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {432 b^2 (a+b x)^{13/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}} \]
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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} \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx=\frac {432 b^2 (a+b x)^{13/6}}{6175 (c+d x)^{13/6} (b c-a d)^3}+\frac {72 b (a+b x)^{13/6}}{475 (c+d x)^{19/6} (b c-a d)^2}+\frac {6 (a+b x)^{13/6}}{25 (c+d x)^{25/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {(12 b) \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx}{25 (b c-a d)} \\ & = \frac {6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {72 b (a+b x)^{13/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {\left (72 b^2\right ) \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx}{475 (b c-a d)^2} \\ & = \frac {6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {72 b (a+b x)^{13/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {432 b^2 (a+b x)^{13/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx=\frac {6 (a+b x)^{13/6} \left (247 a^2 d^2-26 a b d (25 c+6 d x)+b^2 \left (475 c^2+300 c d x+72 d^2 x^2\right )\right )}{6175 (b c-a d)^3 (c+d x)^{25/6}} \]
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Time = 0.92 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {13}{6}} \left (72 d^{2} x^{2} b^{2}-156 x a b \,d^{2}+300 x \,b^{2} c d +247 a^{2} d^{2}-650 a b c d +475 b^{2} c^{2}\right )}{6175 \left (d x +c \right )^{\frac {25}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (83) = 166\).
Time = 0.24 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.23 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx=\frac {6 \, {\left (72 \, b^{4} d^{2} x^{4} + 475 \, a^{2} b^{2} c^{2} - 650 \, a^{3} b c d + 247 \, a^{4} d^{2} + 12 \, {\left (25 \, b^{4} c d - a b^{3} d^{2}\right )} x^{3} + {\left (475 \, b^{4} c^{2} - 50 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (475 \, a b^{3} c^{2} - 500 \, a^{2} b^{2} c d + 169 \, a^{3} b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{6175 \, {\left (b^{3} c^{8} - 3 \, a b^{2} c^{7} d + 3 \, a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3} + {\left (b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}\right )} x^{5} + 5 \, {\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} x^{4} + 10 \, {\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} x^{3} + 10 \, {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{2} + 5 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\right )} x\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {31}{6}}} \,d x } \]
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\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {31}{6}}} \,d x } \]
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Time = 1.34 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.75 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx=-\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {{\left (a+b\,x\right )}^{1/6}\,\left (1482\,a^4\,d^2-3900\,a^3\,b\,c\,d+2850\,a^2\,b^2\,c^2\right )}{6175\,d^5\,{\left (a\,d-b\,c\right )}^3}+\frac {432\,b^4\,x^4\,{\left (a+b\,x\right )}^{1/6}}{6175\,d^3\,{\left (a\,d-b\,c\right )}^3}+\frac {x^2\,{\left (a+b\,x\right )}^{1/6}\,\left (42\,a^2\,b^2\,d^2-300\,a\,b^3\,c\,d+2850\,b^4\,c^2\right )}{6175\,d^5\,{\left (a\,d-b\,c\right )}^3}-\frac {72\,b^3\,x^3\,\left (a\,d-25\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{6175\,d^4\,{\left (a\,d-b\,c\right )}^3}+\frac {12\,a\,b\,x\,{\left (a+b\,x\right )}^{1/6}\,\left (169\,a^2\,d^2-500\,a\,b\,c\,d+475\,b^2\,c^2\right )}{6175\,d^5\,{\left (a\,d-b\,c\right )}^3}\right )}{x^5+\frac {c^5}{d^5}+\frac {5\,c\,x^4}{d}+\frac {5\,c^4\,x}{d^4}+\frac {10\,c^2\,x^3}{d^2}+\frac {10\,c^3\,x^2}{d^3}} \]
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