Integrand size = 19, antiderivative size = 66 \[ \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx=-\frac {4}{3 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}-\frac {16 d \sqrt [4]{a+b x}}{3 (b c-a d)^2 \sqrt [4]{c+d x}} \]
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Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx=-\frac {16 d \sqrt [4]{a+b x}}{3 \sqrt [4]{c+d x} (b c-a d)^2}-\frac {4}{3 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {4}{3 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}-\frac {(4 d) \int \frac {1}{(a+b x)^{3/4} (c+d x)^{5/4}} \, dx}{3 (b c-a d)} \\ & = -\frac {4}{3 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}-\frac {16 d \sqrt [4]{a+b x}}{3 (b c-a d)^2 \sqrt [4]{c+d x}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx=-\frac {4 (b c+3 a d+4 b d x)}{3 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \]
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Time = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(-\frac {4 \left (4 b d x +3 a d +b c \right )}{3 \left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(53\) |
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (54) = 108\).
Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.91 \[ \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx=-\frac {4 \, {\left (4 \, b d x + b c + 3 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{3 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {7}{4}} \left (c + d x\right )^{\frac {5}{4}}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{7/4}\,{\left (c+d\,x\right )}^{5/4}} \,d x \]
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