Integrand size = 19, antiderivative size = 241 \[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=-\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}-\frac {14 b (b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}+\frac {7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}-\frac {14 \sqrt [3]{b} (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} d^{10/3}}-\frac {7 \sqrt [3]{b} (b c-a d)^2 \log (a+b x)}{9 d^{10/3}}-\frac {7 \sqrt [3]{b} (b c-a d)^2 \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 d^{10/3}} \]
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Time = 0.08 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {49, 52, 61} \[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=-\frac {14 \sqrt [3]{b} (b c-a d)^2 \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} d^{10/3}}-\frac {7 \sqrt [3]{b} (b c-a d)^2 \log (a+b x)}{9 d^{10/3}}-\frac {7 \sqrt [3]{b} (b c-a d)^2 \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 d^{10/3}}-\frac {14 b \sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{3 d^3}+\frac {7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}-\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}} \]
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Rule 49
Rule 52
Rule 61
Rubi steps \begin{align*} \text {integral}& = -\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}+\frac {(7 b) \int \frac {(a+b x)^{4/3}}{\sqrt [3]{c+d x}} \, dx}{d} \\ & = -\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}+\frac {7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}-\frac {(14 b (b c-a d)) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{3 d^2} \\ & = -\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}-\frac {14 b (b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}+\frac {7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}+\frac {\left (14 b (b c-a d)^2\right ) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{9 d^3} \\ & = -\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}-\frac {14 b (b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}+\frac {7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}-\frac {14 \sqrt [3]{b} (b c-a d)^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} d^{10/3}}-\frac {7 \sqrt [3]{b} (b c-a d)^2 \log (a+b x)}{9 d^{10/3}}-\frac {7 \sqrt [3]{b} (b c-a d)^2 \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 d^{10/3}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\frac {-\frac {3 \sqrt [3]{d} \sqrt [3]{a+b x} \left (18 a^2 d^2-a b d (49 c+13 d x)+b^2 \left (28 c^2+7 c d x-3 d^2 x^2\right )\right )}{\sqrt [3]{c+d x}}-28 \sqrt {3} \sqrt [3]{b} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )-28 \sqrt [3]{b} (b c-a d)^2 \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+14 \sqrt [3]{b} (b c-a d)^2 \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{18 d^{10/3}} \]
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\[\int \frac {\left (b x +a \right )^{\frac {7}{3}}}{\left (d x +c \right )^{\frac {4}{3}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (187) = 374\).
Time = 0.24 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=-\frac {28 \, \sqrt {3} {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} d \left (-\frac {b}{d}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b d x + b c\right )}}{3 \, {\left (b d x + b c\right )}}\right ) + 14 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{d x + c}\right ) - 28 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}\right ) - 3 \, {\left (3 \, b^{2} d^{2} x^{2} - 28 \, b^{2} c^{2} + 49 \, a b c d - 18 \, a^{2} d^{2} - {\left (7 \, b^{2} c d - 13 \, a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, {\left (d^{4} x + c d^{3}\right )}} \]
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\[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\int \frac {\left (a + b x\right )^{\frac {7}{3}}}{\left (c + d x\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^{7/3}}{{\left (c+d\,x\right )}^{4/3}} \,d x \]
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