Integrand size = 19, antiderivative size = 136 \[ \int \frac {1}{(a+b x)^{15/4} (c+d x)^{5/4}} \, dx=-\frac {4}{11 (b c-a d) (a+b x)^{11/4} \sqrt [4]{c+d x}}+\frac {48 d}{77 (b c-a d)^2 (a+b x)^{7/4} \sqrt [4]{c+d x}}-\frac {128 d^2}{77 (b c-a d)^3 (a+b x)^{3/4} \sqrt [4]{c+d x}}-\frac {512 d^3 \sqrt [4]{a+b x}}{77 (b c-a d)^4 \sqrt [4]{c+d x}} \]
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Time = 0.03 (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} \[ \int \frac {1}{(a+b x)^{15/4} (c+d x)^{5/4}} \, dx=-\frac {512 d^3 \sqrt [4]{a+b x}}{77 \sqrt [4]{c+d x} (b c-a d)^4}-\frac {128 d^2}{77 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^3}+\frac {48 d}{77 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)^2}-\frac {4}{11 (a+b x)^{11/4} \sqrt [4]{c+d x} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {4}{11 (b c-a d) (a+b x)^{11/4} \sqrt [4]{c+d x}}-\frac {(12 d) \int \frac {1}{(a+b x)^{11/4} (c+d x)^{5/4}} \, dx}{11 (b c-a d)} \\ & = -\frac {4}{11 (b c-a d) (a+b x)^{11/4} \sqrt [4]{c+d x}}+\frac {48 d}{77 (b c-a d)^2 (a+b x)^{7/4} \sqrt [4]{c+d x}}+\frac {\left (96 d^2\right ) \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx}{77 (b c-a d)^2} \\ & = -\frac {4}{11 (b c-a d) (a+b x)^{11/4} \sqrt [4]{c+d x}}+\frac {48 d}{77 (b c-a d)^2 (a+b x)^{7/4} \sqrt [4]{c+d x}}-\frac {128 d^2}{77 (b c-a d)^3 (a+b x)^{3/4} \sqrt [4]{c+d x}}-\frac {\left (128 d^3\right ) \int \frac {1}{(a+b x)^{3/4} (c+d x)^{5/4}} \, dx}{77 (b c-a d)^3} \\ & = -\frac {4}{11 (b c-a d) (a+b x)^{11/4} \sqrt [4]{c+d x}}+\frac {48 d}{77 (b c-a d)^2 (a+b x)^{7/4} \sqrt [4]{c+d x}}-\frac {128 d^2}{77 (b c-a d)^3 (a+b x)^{3/4} \sqrt [4]{c+d x}}-\frac {512 d^3 \sqrt [4]{a+b x}}{77 (b c-a d)^4 \sqrt [4]{c+d x}} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a+b x)^{15/4} (c+d x)^{5/4}} \, dx=-\frac {4 \left (77 a^3 d^3+77 a^2 b d^2 (c+4 d x)+11 a b^2 d \left (-3 c^2+8 c d x+32 d^2 x^2\right )+b^3 \left (7 c^3-12 c^2 d x+32 c d^2 x^2+128 d^3 x^3\right )\right )}{77 (b c-a d)^4 (a+b x)^{11/4} \sqrt [4]{c+d x}} \]
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Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
method | result | size |
gosper | \(-\frac {4 \left (128 d^{3} x^{3} b^{3}+352 x^{2} a \,b^{2} d^{3}+32 x^{2} b^{3} c \,d^{2}+308 x \,a^{2} b \,d^{3}+88 x a \,b^{2} c \,d^{2}-12 x \,b^{3} c^{2} d +77 a^{3} d^{3}+77 a^{2} b c \,d^{2}-33 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right )}{77 \left (b x +a \right )^{\frac {11}{4}} \left (d x +c \right )^{\frac {1}{4}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (112) = 224\).
Time = 0.53 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.36 \[ \int \frac {1}{(a+b x)^{15/4} (c+d x)^{5/4}} \, dx=-\frac {4 \, {\left (128 \, b^{3} d^{3} x^{3} + 7 \, b^{3} c^{3} - 33 \, a b^{2} c^{2} d + 77 \, a^{2} b c d^{2} + 77 \, a^{3} d^{3} + 32 \, {\left (b^{3} c d^{2} + 11 \, a b^{2} d^{3}\right )} x^{2} - 4 \, {\left (3 \, b^{3} c^{2} d - 22 \, a b^{2} c d^{2} - 77 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{77 \, {\left (a^{3} b^{4} c^{5} - 4 \, a^{4} b^{3} c^{4} d + 6 \, a^{5} b^{2} c^{3} d^{2} - 4 \, a^{6} b c^{2} d^{3} + a^{7} c d^{4} + {\left (b^{7} c^{4} d - 4 \, a b^{6} c^{3} d^{2} + 6 \, a^{2} b^{5} c^{2} d^{3} - 4 \, a^{3} b^{4} c d^{4} + a^{4} b^{3} d^{5}\right )} x^{4} + {\left (b^{7} c^{5} - a b^{6} c^{4} d - 6 \, a^{2} b^{5} c^{3} d^{2} + 14 \, a^{3} b^{4} c^{2} d^{3} - 11 \, a^{4} b^{3} c d^{4} + 3 \, a^{5} b^{2} d^{5}\right )} x^{3} + 3 \, {\left (a b^{6} c^{5} - 3 \, a^{2} b^{5} c^{4} d + 2 \, a^{3} b^{4} c^{3} d^{2} + 2 \, a^{4} b^{3} c^{2} d^{3} - 3 \, a^{5} b^{2} c d^{4} + a^{6} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} c^{5} - 11 \, a^{3} b^{4} c^{4} d + 14 \, a^{4} b^{3} c^{3} d^{2} - 6 \, a^{5} b^{2} c^{2} d^{3} - a^{6} b c d^{4} + a^{7} d^{5}\right )} x\right )}} \]
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Timed out. \[ \int \frac {1}{(a+b x)^{15/4} (c+d x)^{5/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+b x)^{15/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {15}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{15/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {15}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{15/4} (c+d x)^{5/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{15/4}\,{\left (c+d\,x\right )}^{5/4}} \,d x \]
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