Integrand size = 19, antiderivative size = 136 \[ \int \frac {1}{(a+b x)^{11/3} (c+d x)^{4/3}} \, dx=-\frac {3}{8 (b c-a d) (a+b x)^{8/3} \sqrt [3]{c+d x}}+\frac {27 d}{40 (b c-a d)^2 (a+b x)^{5/3} \sqrt [3]{c+d x}}-\frac {81 d^2}{40 (b c-a d)^3 (a+b x)^{2/3} \sqrt [3]{c+d x}}-\frac {243 d^3 \sqrt [3]{a+b x}}{40 (b c-a d)^4 \sqrt [3]{c+d x}} \]
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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} \[ \int \frac {1}{(a+b x)^{11/3} (c+d x)^{4/3}} \, dx=-\frac {243 d^3 \sqrt [3]{a+b x}}{40 \sqrt [3]{c+d x} (b c-a d)^4}-\frac {81 d^2}{40 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^3}+\frac {27 d}{40 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)^2}-\frac {3}{8 (a+b x)^{8/3} \sqrt [3]{c+d x} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{8 (b c-a d) (a+b x)^{8/3} \sqrt [3]{c+d x}}-\frac {(9 d) \int \frac {1}{(a+b x)^{8/3} (c+d x)^{4/3}} \, dx}{8 (b c-a d)} \\ & = -\frac {3}{8 (b c-a d) (a+b x)^{8/3} \sqrt [3]{c+d x}}+\frac {27 d}{40 (b c-a d)^2 (a+b x)^{5/3} \sqrt [3]{c+d x}}+\frac {\left (27 d^2\right ) \int \frac {1}{(a+b x)^{5/3} (c+d x)^{4/3}} \, dx}{20 (b c-a d)^2} \\ & = -\frac {3}{8 (b c-a d) (a+b x)^{8/3} \sqrt [3]{c+d x}}+\frac {27 d}{40 (b c-a d)^2 (a+b x)^{5/3} \sqrt [3]{c+d x}}-\frac {81 d^2}{40 (b c-a d)^3 (a+b x)^{2/3} \sqrt [3]{c+d x}}-\frac {\left (81 d^3\right ) \int \frac {1}{(a+b x)^{2/3} (c+d x)^{4/3}} \, dx}{40 (b c-a d)^3} \\ & = -\frac {3}{8 (b c-a d) (a+b x)^{8/3} \sqrt [3]{c+d x}}+\frac {27 d}{40 (b c-a d)^2 (a+b x)^{5/3} \sqrt [3]{c+d x}}-\frac {81 d^2}{40 (b c-a d)^3 (a+b x)^{2/3} \sqrt [3]{c+d x}}-\frac {243 d^3 \sqrt [3]{a+b x}}{40 (b c-a d)^4 \sqrt [3]{c+d x}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a+b x)^{11/3} (c+d x)^{4/3}} \, dx=-\frac {3 \left (40 a^3 d^3+60 a^2 b d^2 (c+3 d x)+24 a b^2 d \left (-c^2+3 c d x+9 d^2 x^2\right )+b^3 \left (5 c^3-9 c^2 d x+27 c d^2 x^2+81 d^3 x^3\right )\right )}{40 (b c-a d)^4 (a+b x)^{8/3} \sqrt [3]{c+d x}} \]
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Time = 0.72 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
method | result | size |
gosper | \(-\frac {3 \left (81 d^{3} x^{3} b^{3}+216 x^{2} a \,b^{2} d^{3}+27 x^{2} b^{3} c \,d^{2}+180 x \,a^{2} b \,d^{3}+72 x a \,b^{2} c \,d^{2}-9 x \,b^{3} c^{2} d +40 a^{3} d^{3}+60 a^{2} b c \,d^{2}-24 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right )}{40 \left (b x +a \right )^{\frac {8}{3}} \left (d x +c \right )^{\frac {1}{3}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (112) = 224\).
Time = 0.26 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.35 \[ \int \frac {1}{(a+b x)^{11/3} (c+d x)^{4/3}} \, dx=-\frac {3 \, {\left (81 \, b^{3} d^{3} x^{3} + 5 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 60 \, a^{2} b c d^{2} + 40 \, a^{3} d^{3} + 27 \, {\left (b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x^{2} - 9 \, {\left (b^{3} c^{2} d - 8 \, a b^{2} c d^{2} - 20 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{40 \, {\left (a^{3} b^{4} c^{5} - 4 \, a^{4} b^{3} c^{4} d + 6 \, a^{5} b^{2} c^{3} d^{2} - 4 \, a^{6} b c^{2} d^{3} + a^{7} c d^{4} + {\left (b^{7} c^{4} d - 4 \, a b^{6} c^{3} d^{2} + 6 \, a^{2} b^{5} c^{2} d^{3} - 4 \, a^{3} b^{4} c d^{4} + a^{4} b^{3} d^{5}\right )} x^{4} + {\left (b^{7} c^{5} - a b^{6} c^{4} d - 6 \, a^{2} b^{5} c^{3} d^{2} + 14 \, a^{3} b^{4} c^{2} d^{3} - 11 \, a^{4} b^{3} c d^{4} + 3 \, a^{5} b^{2} d^{5}\right )} x^{3} + 3 \, {\left (a b^{6} c^{5} - 3 \, a^{2} b^{5} c^{4} d + 2 \, a^{3} b^{4} c^{3} d^{2} + 2 \, a^{4} b^{3} c^{2} d^{3} - 3 \, a^{5} b^{2} c d^{4} + a^{6} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} c^{5} - 11 \, a^{3} b^{4} c^{4} d + 14 \, a^{4} b^{3} c^{3} d^{2} - 6 \, a^{5} b^{2} c^{2} d^{3} - a^{6} b c d^{4} + a^{7} d^{5}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{11/3} (c+d x)^{4/3}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {11}{3}} \left (c + d x\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{11/3} (c+d x)^{4/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {11}{3}} {\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{11/3} (c+d x)^{4/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {11}{3}} {\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{11/3} (c+d x)^{4/3}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{11/3}\,{\left (c+d\,x\right )}^{4/3}} \,d x \]
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