Integrand size = 19, antiderivative size = 136 \[ \int \frac {1}{(a+b x)^{17/4} (c+d x)^{3/4}} \, dx=-\frac {4 \sqrt [4]{c+d x}}{13 (b c-a d) (a+b x)^{13/4}}+\frac {16 d \sqrt [4]{c+d x}}{39 (b c-a d)^2 (a+b x)^{9/4}}-\frac {128 d^2 \sqrt [4]{c+d x}}{195 (b c-a d)^3 (a+b x)^{5/4}}+\frac {512 d^3 \sqrt [4]{c+d x}}{195 (b c-a d)^4 \sqrt [4]{a+b x}} \]
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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} \[ \int \frac {1}{(a+b x)^{17/4} (c+d x)^{3/4}} \, dx=\frac {512 d^3 \sqrt [4]{c+d x}}{195 \sqrt [4]{a+b x} (b c-a d)^4}-\frac {128 d^2 \sqrt [4]{c+d x}}{195 (a+b x)^{5/4} (b c-a d)^3}+\frac {16 d \sqrt [4]{c+d x}}{39 (a+b x)^{9/4} (b c-a d)^2}-\frac {4 \sqrt [4]{c+d x}}{13 (a+b x)^{13/4} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \sqrt [4]{c+d x}}{13 (b c-a d) (a+b x)^{13/4}}-\frac {(12 d) \int \frac {1}{(a+b x)^{13/4} (c+d x)^{3/4}} \, dx}{13 (b c-a d)} \\ & = -\frac {4 \sqrt [4]{c+d x}}{13 (b c-a d) (a+b x)^{13/4}}+\frac {16 d \sqrt [4]{c+d x}}{39 (b c-a d)^2 (a+b x)^{9/4}}+\frac {\left (32 d^2\right ) \int \frac {1}{(a+b x)^{9/4} (c+d x)^{3/4}} \, dx}{39 (b c-a d)^2} \\ & = -\frac {4 \sqrt [4]{c+d x}}{13 (b c-a d) (a+b x)^{13/4}}+\frac {16 d \sqrt [4]{c+d x}}{39 (b c-a d)^2 (a+b x)^{9/4}}-\frac {128 d^2 \sqrt [4]{c+d x}}{195 (b c-a d)^3 (a+b x)^{5/4}}-\frac {\left (128 d^3\right ) \int \frac {1}{(a+b x)^{5/4} (c+d x)^{3/4}} \, dx}{195 (b c-a d)^3} \\ & = -\frac {4 \sqrt [4]{c+d x}}{13 (b c-a d) (a+b x)^{13/4}}+\frac {16 d \sqrt [4]{c+d x}}{39 (b c-a d)^2 (a+b x)^{9/4}}-\frac {128 d^2 \sqrt [4]{c+d x}}{195 (b c-a d)^3 (a+b x)^{5/4}}+\frac {512 d^3 \sqrt [4]{c+d x}}{195 (b c-a d)^4 \sqrt [4]{a+b x}} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a+b x)^{17/4} (c+d x)^{3/4}} \, dx=\frac {4 \sqrt [4]{c+d x} \left (195 a^3 d^3-117 a^2 b d^2 (c-4 d x)+13 a b^2 d \left (5 c^2-8 c d x+32 d^2 x^2\right )+b^3 \left (-15 c^3+20 c^2 d x-32 c d^2 x^2+128 d^3 x^3\right )\right )}{195 (b c-a d)^4 (a+b x)^{13/4}} \]
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Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
method | result | size |
gosper | \(\frac {4 \left (d x +c \right )^{\frac {1}{4}} \left (128 d^{3} x^{3} b^{3}+416 x^{2} a \,b^{2} d^{3}-32 x^{2} b^{3} c \,d^{2}+468 x \,a^{2} b \,d^{3}-104 x a \,b^{2} c \,d^{2}+20 x \,b^{3} c^{2} d +195 a^{3} d^{3}-117 a^{2} b c \,d^{2}+65 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right )}{195 \left (b x +a \right )^{\frac {13}{4}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (112) = 224\).
Time = 0.24 (sec) , antiderivative size = 419, normalized size of antiderivative = 3.08 \[ \int \frac {1}{(a+b x)^{17/4} (c+d x)^{3/4}} \, dx=\frac {4 \, {\left (128 \, b^{3} d^{3} x^{3} - 15 \, b^{3} c^{3} + 65 \, a b^{2} c^{2} d - 117 \, a^{2} b c d^{2} + 195 \, a^{3} d^{3} - 32 \, {\left (b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (5 \, b^{3} c^{2} d - 26 \, a b^{2} c d^{2} + 117 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{195 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \]
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Timed out. \[ \int \frac {1}{(a+b x)^{17/4} (c+d x)^{3/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+b x)^{17/4} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {17}{4}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{17/4} (c+d x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {17}{4}} {\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
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Time = 1.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.54 \[ \int \frac {1}{(a+b x)^{17/4} (c+d x)^{3/4}} \, dx=\frac {{\left (c+d\,x\right )}^{1/4}\,\left (\frac {512\,d^3\,x^3}{195\,{\left (a\,d-b\,c\right )}^4}+\frac {780\,a^3\,d^3-468\,a^2\,b\,c\,d^2+260\,a\,b^2\,c^2\,d-60\,b^3\,c^3}{195\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d\,x\,\left (117\,a^2\,d^2-26\,a\,b\,c\,d+5\,b^2\,c^2\right )}{195\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {128\,d^2\,x^2\,\left (13\,a\,d-b\,c\right )}{195\,b\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,{\left (a+b\,x\right )}^{1/4}+\frac {a^3\,{\left (a+b\,x\right )}^{1/4}}{b^3}+\frac {3\,a\,x^2\,{\left (a+b\,x\right )}^{1/4}}{b}+\frac {3\,a^2\,x\,{\left (a+b\,x\right )}^{1/4}}{b^2}} \]
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