Integrand size = 19, antiderivative size = 136 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\frac {6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac {108 b (a+b x)^{13/6}}{775 (b c-a d)^2 (c+d x)^{25/6}}+\frac {1296 b^2 (a+b x)^{13/6}}{14725 (b c-a d)^3 (c+d x)^{19/6}}+\frac {7776 b^3 (a+b x)^{13/6}}{191425 (b c-a d)^4 (c+d x)^{13/6}} \]
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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} \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\frac {7776 b^3 (a+b x)^{13/6}}{191425 (c+d x)^{13/6} (b c-a d)^4}+\frac {1296 b^2 (a+b x)^{13/6}}{14725 (c+d x)^{19/6} (b c-a d)^3}+\frac {108 b (a+b x)^{13/6}}{775 (c+d x)^{25/6} (b c-a d)^2}+\frac {6 (a+b x)^{13/6}}{31 (c+d x)^{31/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac {(18 b) \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx}{31 (b c-a d)} \\ & = \frac {6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac {108 b (a+b x)^{13/6}}{775 (b c-a d)^2 (c+d x)^{25/6}}+\frac {\left (216 b^2\right ) \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx}{775 (b c-a d)^2} \\ & = \frac {6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac {108 b (a+b x)^{13/6}}{775 (b c-a d)^2 (c+d x)^{25/6}}+\frac {1296 b^2 (a+b x)^{13/6}}{14725 (b c-a d)^3 (c+d x)^{19/6}}+\frac {\left (1296 b^3\right ) \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx}{14725 (b c-a d)^3} \\ & = \frac {6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac {108 b (a+b x)^{13/6}}{775 (b c-a d)^2 (c+d x)^{25/6}}+\frac {1296 b^2 (a+b x)^{13/6}}{14725 (b c-a d)^3 (c+d x)^{19/6}}+\frac {7776 b^3 (a+b x)^{13/6}}{191425 (b c-a d)^4 (c+d x)^{13/6}} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\frac {6 (a+b x)^{13/6} \left (-6175 a^3 d^3+741 a^2 b d^2 (31 c+6 d x)-39 a b^2 d \left (775 c^2+372 c d x+72 d^2 x^2\right )+b^3 \left (14725 c^3+13950 c^2 d x+6696 c d^2 x^2+1296 d^3 x^3\right )\right )}{191425 (b c-a d)^4 (c+d x)^{31/6}} \]
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Time = 1.42 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {13}{6}} \left (-1296 d^{3} x^{3} b^{3}+2808 x^{2} a \,b^{2} d^{3}-6696 x^{2} b^{3} c \,d^{2}-4446 x \,a^{2} b \,d^{3}+14508 x a \,b^{2} c \,d^{2}-13950 x \,b^{3} c^{2} d +6175 a^{3} d^{3}-22971 a^{2} b c \,d^{2}+30225 a \,b^{2} c^{2} d -14725 b^{3} c^{3}\right )}{191425 \left (d x +c \right )^{\frac {31}{6}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (112) = 224\).
Time = 0.26 (sec) , antiderivative size = 649, normalized size of antiderivative = 4.77 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\frac {6 \, {\left (1296 \, b^{5} d^{3} x^{5} + 14725 \, a^{2} b^{3} c^{3} - 30225 \, a^{3} b^{2} c^{2} d + 22971 \, a^{4} b c d^{2} - 6175 \, a^{5} d^{3} + 216 \, {\left (31 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4} + 18 \, {\left (775 \, b^{5} c^{2} d - 62 \, a b^{4} c d^{2} + 7 \, a^{2} b^{3} d^{3}\right )} x^{3} + {\left (14725 \, b^{5} c^{3} - 2325 \, a b^{4} c^{2} d + 651 \, a^{2} b^{3} c d^{2} - 91 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (14725 \, a b^{4} c^{3} - 23250 \, a^{2} b^{3} c^{2} d + 15717 \, a^{3} b^{2} c d^{2} - 3952 \, a^{4} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{191425 \, {\left (b^{4} c^{10} - 4 \, a b^{3} c^{9} d + 6 \, a^{2} b^{2} c^{8} d^{2} - 4 \, a^{3} b c^{7} d^{3} + a^{4} c^{6} d^{4} + {\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{6} + 6 \, {\left (b^{4} c^{5} d^{5} - 4 \, a b^{3} c^{4} d^{6} + 6 \, a^{2} b^{2} c^{3} d^{7} - 4 \, a^{3} b c^{2} d^{8} + a^{4} c d^{9}\right )} x^{5} + 15 \, {\left (b^{4} c^{6} d^{4} - 4 \, a b^{3} c^{5} d^{5} + 6 \, a^{2} b^{2} c^{4} d^{6} - 4 \, a^{3} b c^{3} d^{7} + a^{4} c^{2} d^{8}\right )} x^{4} + 20 \, {\left (b^{4} c^{7} d^{3} - 4 \, a b^{3} c^{6} d^{4} + 6 \, a^{2} b^{2} c^{5} d^{5} - 4 \, a^{3} b c^{4} d^{6} + a^{4} c^{3} d^{7}\right )} x^{3} + 15 \, {\left (b^{4} c^{8} d^{2} - 4 \, a b^{3} c^{7} d^{3} + 6 \, a^{2} b^{2} c^{6} d^{4} - 4 \, a^{3} b c^{5} d^{5} + a^{4} c^{4} d^{6}\right )} x^{2} + 6 \, {\left (b^{4} c^{9} d - 4 \, a b^{3} c^{8} d^{2} + 6 \, a^{2} b^{2} c^{7} d^{3} - 4 \, a^{3} b c^{6} d^{4} + a^{4} c^{5} d^{5}\right )} x\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {37}{6}}} \,d x } \]
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\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {37}{6}}} \,d x } \]
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Time = 1.36 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.83 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {7776\,b^5\,x^5\,{\left (a+b\,x\right )}^{1/6}}{191425\,d^3\,{\left (a\,d-b\,c\right )}^4}-\frac {{\left (a+b\,x\right )}^{1/6}\,\left (37050\,a^5\,d^3-137826\,a^4\,b\,c\,d^2+181350\,a^3\,b^2\,c^2\,d-88350\,a^2\,b^3\,c^3\right )}{191425\,d^6\,{\left (a\,d-b\,c\right )}^4}+\frac {x^2\,{\left (a+b\,x\right )}^{1/6}\,\left (-546\,a^3\,b^2\,d^3+3906\,a^2\,b^3\,c\,d^2-13950\,a\,b^4\,c^2\,d+88350\,b^5\,c^3\right )}{191425\,d^6\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,{\left (a+b\,x\right )}^{1/6}\,\left (-47424\,a^4\,b\,d^3+188604\,a^3\,b^2\,c\,d^2-279000\,a^2\,b^3\,c^2\,d+176700\,a\,b^4\,c^3\right )}{191425\,d^6\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b^3\,x^3\,{\left (a+b\,x\right )}^{1/6}\,\left (7\,a^2\,d^2-62\,a\,b\,c\,d+775\,b^2\,c^2\right )}{191425\,d^5\,{\left (a\,d-b\,c\right )}^4}-\frac {1296\,b^4\,x^4\,\left (a\,d-31\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{191425\,d^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^6+\frac {c^6}{d^6}+\frac {6\,c\,x^5}{d}+\frac {6\,c^5\,x}{d^5}+\frac {15\,c^2\,x^4}{d^2}+\frac {20\,c^3\,x^3}{d^3}+\frac {15\,c^4\,x^2}{d^4}} \]
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