Integrand size = 19, antiderivative size = 180 \[ \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{9/2}} \, dx=\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}+\frac {512 b \sqrt {a x+b x^3}}{35 a^6 x^{7/2}}-\frac {1024 b^2 \sqrt {a x+b x^3}}{35 a^7 x^{3/2}} \]
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Time = 0.19 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2040, 2041, 2039} \[ \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{9/2}} \, dx=-\frac {1024 b^2 \sqrt {a x+b x^3}}{35 a^7 x^{3/2}}+\frac {512 b \sqrt {a x+b x^3}}{35 a^6 x^{7/2}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}} \]
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Rule 2039
Rule 2040
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12 \int \frac {1}{x^{5/2} \left (a x+b x^3\right )^{7/2}} \, dx}{7 a} \\ & = \frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {24 \int \frac {1}{x^{7/2} \left (a x+b x^3\right )^{5/2}} \, dx}{7 a^2} \\ & = \frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64 \int \frac {1}{x^{9/2} \left (a x+b x^3\right )^{3/2}} \, dx}{7 a^3} \\ & = \frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}+\frac {384 \int \frac {1}{x^{11/2} \sqrt {a x+b x^3}} \, dx}{7 a^4} \\ & = \frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}-\frac {(1536 b) \int \frac {1}{x^{7/2} \sqrt {a x+b x^3}} \, dx}{35 a^5} \\ & = \frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}+\frac {512 b \sqrt {a x+b x^3}}{35 a^6 x^{7/2}}+\frac {\left (1024 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {a x+b x^3}} \, dx}{35 a^6} \\ & = \frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}+\frac {512 b \sqrt {a x+b x^3}}{35 a^6 x^{7/2}}-\frac {1024 b^2 \sqrt {a x+b x^3}}{35 a^7 x^{3/2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{9/2}} \, dx=\frac {-7 a^6+28 a^5 b x^2-280 a^4 b^2 x^4-2240 a^3 b^3 x^6-4480 a^2 b^4 x^8-3584 a b^5 x^{10}-1024 b^6 x^{12}}{35 a^7 x^{3/2} \left (x \left (a+b x^2\right )\right )^{7/2}} \]
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Time = 2.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right ) \left (1024 b^{6} x^{12}+3584 a \,b^{5} x^{10}+4480 a^{2} b^{4} x^{8}+2240 a^{3} b^{3} x^{6}+280 a^{4} b^{2} x^{4}-28 a^{5} b \,x^{2}+7 a^{6}\right )}{35 \sqrt {x}\, a^{7} \left (b \,x^{3}+a x \right )^{\frac {9}{2}}}\) | \(92\) |
default | \(-\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (1024 b^{6} x^{12}+3584 a \,b^{5} x^{10}+4480 a^{2} b^{4} x^{8}+2240 a^{3} b^{3} x^{6}+280 a^{4} b^{2} x^{4}-28 a^{5} b \,x^{2}+7 a^{6}\right )}{35 x^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{4} a^{7}}\) | \(94\) |
risch | \(-\frac {\left (b \,x^{2}+a \right ) \left (66 b^{2} x^{4}-8 a b \,x^{2}+a^{2}\right )}{5 a^{7} x^{\frac {9}{2}} \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {\left (b \,x^{2}+a \right ) x^{\frac {3}{2}} \left (562 b^{3} x^{6}+1792 a \,b^{2} x^{4}+1925 a^{2} b \,x^{2}+700 a^{3}\right ) b^{3}}{35 a^{7} \left (x^{8} b^{4}+4 a \,b^{3} x^{6}+6 a^{2} x^{4} b^{2}+4 a^{3} b \,x^{2}+a^{4}\right ) \sqrt {x \left (b \,x^{2}+a \right )}}\) | \(150\) |
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Time = 0.37 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{9/2}} \, dx=-\frac {{\left (1024 \, b^{6} x^{12} + 3584 \, a b^{5} x^{10} + 4480 \, a^{2} b^{4} x^{8} + 2240 \, a^{3} b^{3} x^{6} + 280 \, a^{4} b^{2} x^{4} - 28 \, a^{5} b x^{2} + 7 \, a^{6}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{35 \, {\left (a^{7} b^{4} x^{14} + 4 \, a^{8} b^{3} x^{12} + 6 \, a^{9} b^{2} x^{10} + 4 \, a^{10} b x^{8} + a^{11} x^{6}\right )}} \]
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\[ \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {1}{x^{\frac {3}{2}} \left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \]
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\[ \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{9/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}} x^{\frac {3}{2}}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{9/2}} \, dx=-\frac {{\left ({\left (2 \, x^{2} {\left (\frac {281 \, b^{6} x^{2}}{a^{7}} + \frac {896 \, b^{5}}{a^{6}}\right )} + \frac {1925 \, b^{4}}{a^{5}}\right )} x^{2} + \frac {700 \, b^{3}}{a^{4}}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {4 \, {\left (25 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {5}{2}} - 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {5}{2}} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {5}{2}} - 140 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {5}{2}} + 33 \, a^{4} b^{\frac {5}{2}}\right )}}{5 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{6}} \]
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Timed out. \[ \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {1}{x^{3/2}\,{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]
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