Integrand size = 19, antiderivative size = 136 \[ \int \frac {1}{(a+b x)^{14/3} \sqrt [3]{c+d x}} \, dx=-\frac {3 (c+d x)^{2/3}}{11 (b c-a d) (a+b x)^{11/3}}+\frac {27 d (c+d x)^{2/3}}{88 (b c-a d)^2 (a+b x)^{8/3}}-\frac {81 d^2 (c+d x)^{2/3}}{220 (b c-a d)^3 (a+b x)^{5/3}}+\frac {243 d^3 (c+d x)^{2/3}}{440 (b c-a d)^4 (a+b x)^{2/3}} \]
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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} \[ \int \frac {1}{(a+b x)^{14/3} \sqrt [3]{c+d x}} \, dx=\frac {243 d^3 (c+d x)^{2/3}}{440 (a+b x)^{2/3} (b c-a d)^4}-\frac {81 d^2 (c+d x)^{2/3}}{220 (a+b x)^{5/3} (b c-a d)^3}+\frac {27 d (c+d x)^{2/3}}{88 (a+b x)^{8/3} (b c-a d)^2}-\frac {3 (c+d x)^{2/3}}{11 (a+b x)^{11/3} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {3 (c+d x)^{2/3}}{11 (b c-a d) (a+b x)^{11/3}}-\frac {(9 d) \int \frac {1}{(a+b x)^{11/3} \sqrt [3]{c+d x}} \, dx}{11 (b c-a d)} \\ & = -\frac {3 (c+d x)^{2/3}}{11 (b c-a d) (a+b x)^{11/3}}+\frac {27 d (c+d x)^{2/3}}{88 (b c-a d)^2 (a+b x)^{8/3}}+\frac {\left (27 d^2\right ) \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx}{44 (b c-a d)^2} \\ & = -\frac {3 (c+d x)^{2/3}}{11 (b c-a d) (a+b x)^{11/3}}+\frac {27 d (c+d x)^{2/3}}{88 (b c-a d)^2 (a+b x)^{8/3}}-\frac {81 d^2 (c+d x)^{2/3}}{220 (b c-a d)^3 (a+b x)^{5/3}}-\frac {\left (81 d^3\right ) \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx}{220 (b c-a d)^3} \\ & = -\frac {3 (c+d x)^{2/3}}{11 (b c-a d) (a+b x)^{11/3}}+\frac {27 d (c+d x)^{2/3}}{88 (b c-a d)^2 (a+b x)^{8/3}}-\frac {81 d^2 (c+d x)^{2/3}}{220 (b c-a d)^3 (a+b x)^{5/3}}+\frac {243 d^3 (c+d x)^{2/3}}{440 (b c-a d)^4 (a+b x)^{2/3}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+b x)^{14/3} \sqrt [3]{c+d x}} \, dx=\frac {3 (c+d x)^{2/3} \left (220 a^3 d^3+132 a^2 b d^2 (-2 c+3 d x)+33 a b^2 d \left (5 c^2-6 c d x+9 d^2 x^2\right )+b^3 \left (-40 c^3+45 c^2 d x-54 c d^2 x^2+81 d^3 x^3\right )\right )}{440 (b c-a d)^4 (a+b x)^{11/3}} \]
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Time = 0.88 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
method | result | size |
gosper | \(\frac {3 \left (d x +c \right )^{\frac {2}{3}} \left (81 d^{3} x^{3} b^{3}+297 x^{2} a \,b^{2} d^{3}-54 x^{2} b^{3} c \,d^{2}+396 x \,a^{2} b \,d^{3}-198 x a \,b^{2} c \,d^{2}+45 x \,b^{3} c^{2} d +220 a^{3} d^{3}-264 a^{2} b c \,d^{2}+165 a \,b^{2} c^{2} d -40 b^{3} c^{3}\right )}{440 \left (b x +a \right )^{\frac {11}{3}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (112) = 224\).
Time = 0.30 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.09 \[ \int \frac {1}{(a+b x)^{14/3} \sqrt [3]{c+d x}} \, dx=\frac {3 \, {\left (81 \, b^{3} d^{3} x^{3} - 40 \, b^{3} c^{3} + 165 \, a b^{2} c^{2} d - 264 \, a^{2} b c d^{2} + 220 \, a^{3} d^{3} - 27 \, {\left (2 \, b^{3} c d^{2} - 11 \, a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (5 \, b^{3} c^{2} d - 22 \, a b^{2} c d^{2} + 44 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{440 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{14/3} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {14}{3}} \sqrt [3]{c + d x}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{14/3} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {14}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{14/3} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {14}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{14/3} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{14/3}\,{\left (c+d\,x\right )}^{1/3}} \,d x \]
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