Integrand size = 26, antiderivative size = 247 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{12}} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {2 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}+\frac {5 a b^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b^5 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]
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Time = 0.04 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 276} \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{12}} \, dx=\frac {b^5 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {5 a b^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {2 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )} \]
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Rule 276
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^5}{x^{12}} \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (5 a b^9+\frac {a^5 b^5}{x^{12}}+\frac {5 a^4 b^6}{x^9}+\frac {10 a^3 b^7}{x^6}+\frac {10 a^2 b^8}{x^3}+b^{10} x^3\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = -\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {2 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}+\frac {5 a b^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b^5 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.34 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{12}} \, dx=-\frac {\sqrt {\left (a+b x^3\right )^2} \left (8 a^5+55 a^4 b x^3+176 a^3 b^2 x^6+440 a^2 b^3 x^9-440 a b^4 x^{12}-22 b^5 x^{15}\right )}{88 x^{11} \left (a+b x^3\right )} \]
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Time = 12.87 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.32
method | result | size |
gosper | \(-\frac {\left (-22 b^{5} x^{15}-440 a \,b^{4} x^{12}+440 a^{2} b^{3} x^{9}+176 a^{3} b^{2} x^{6}+55 a^{4} b \,x^{3}+8 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{88 \left (b \,x^{3}+a \right )^{5} x^{11}}\) | \(80\) |
default | \(-\frac {\left (-22 b^{5} x^{15}-440 a \,b^{4} x^{12}+440 a^{2} b^{3} x^{9}+176 a^{3} b^{2} x^{6}+55 a^{4} b \,x^{3}+8 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{88 \left (b \,x^{3}+a \right )^{5} x^{11}}\) | \(80\) |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{4} \left (\frac {1}{4} b \,x^{4}+5 a x \right )}{b \,x^{3}+a}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-5 a^{2} b^{3} x^{9}-2 a^{3} b^{2} x^{6}-\frac {5}{8} a^{4} b \,x^{3}-\frac {1}{11} a^{5}\right )}{\left (b \,x^{3}+a \right ) x^{11}}\) | \(98\) |
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.24 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{12}} \, dx=\frac {22 \, b^{5} x^{15} + 440 \, a b^{4} x^{12} - 440 \, a^{2} b^{3} x^{9} - 176 \, a^{3} b^{2} x^{6} - 55 \, a^{4} b x^{3} - 8 \, a^{5}}{88 \, x^{11}} \]
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\[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{12}} \, dx=\int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{12}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.24 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{12}} \, dx=\frac {22 \, b^{5} x^{15} + 440 \, a b^{4} x^{12} - 440 \, a^{2} b^{3} x^{9} - 176 \, a^{3} b^{2} x^{6} - 55 \, a^{4} b x^{3} - 8 \, a^{5}}{88 \, x^{11}} \]
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Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.43 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{12}} \, dx=\frac {1}{4} \, b^{5} x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + 5 \, a b^{4} x \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {440 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 176 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 55 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 8 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{88 \, x^{11}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{12}} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x^{12}} \,d x \]
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