Integrand size = 26, antiderivative size = 251 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx=\frac {5 a^4 b x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {5 a^3 b^2 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {5 a b^4 x^{12} \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac {b^5 x^{15} \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3} \]
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Time = 0.05 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1369, 272, 45} \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx=\frac {b^5 x^{15} \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac {5 a b^4 x^{12} \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {a^5 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a^4 b x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {5 a^3 b^2 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )} \]
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Rule 45
Rule 272
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} \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x} \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \text {Subst}\left (\int \left (5 a^4 b^6+\frac {a^5 b^5}{x}+10 a^3 b^7 x+10 a^2 b^8 x^2+5 a b^9 x^3+b^{10} x^4\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {5 a^4 b x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {5 a^3 b^2 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {5 a b^4 x^{12} \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac {b^5 x^{15} \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.33 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx=\frac {\sqrt {\left (a+b x^3\right )^2} \left (b x^3 \left (300 a^4+300 a^3 b x^3+200 a^2 b^2 x^6+75 a b^3 x^9+12 b^4 x^{12}\right )+180 a^5 \log (x)\right )}{180 \left (a+b x^3\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.30
method | result | size |
pseudoelliptic | \(\frac {\operatorname {csgn}\left (b \,x^{3}+a \right ) \left (\frac {b^{5} x^{15}}{5}+\frac {5 a \,b^{4} x^{12}}{4}+\frac {10 a^{2} b^{3} x^{9}}{3}+5 a^{3} b^{2} x^{6}+5 a^{4} b \,x^{3}+a^{5} \ln \left (b \,x^{3}\right )+\frac {137 a^{5}}{60}\right )}{3}\) | \(75\) |
default | \(\frac {{\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}} \left (12 b^{5} x^{15}+75 a \,b^{4} x^{12}+200 a^{2} b^{3} x^{9}+300 a^{3} b^{2} x^{6}+300 a^{4} b \,x^{3}+180 a^{5} \ln \left (x \right )\right )}{180 \left (b \,x^{3}+a \right )^{5}}\) | \(79\) |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b \left (\frac {1}{15} b^{4} x^{15}+\frac {5}{12} a \,b^{3} x^{12}+\frac {10}{9} a^{2} b^{2} x^{9}+\frac {5}{3} a^{3} b \,x^{6}+\frac {5}{3} a^{4} x^{3}\right )}{b \,x^{3}+a}+\frac {a^{5} \ln \left (x \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{b \,x^{3}+a}\) | \(96\) |
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Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx=\frac {1}{15} \, b^{5} x^{15} + \frac {5}{12} \, a b^{4} x^{12} + \frac {10}{9} \, a^{2} b^{3} x^{9} + \frac {5}{3} \, a^{3} b^{2} x^{6} + \frac {5}{3} \, a^{4} b x^{3} + a^{5} \log \left (x\right ) \]
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\[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx=\int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx=\frac {1}{6} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a^{3} b x^{3} + \frac {1}{3} \, \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} a^{5} \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \frac {1}{3} \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} a^{5} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) + \frac {1}{12} \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a b x^{3} + \frac {1}{2} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a^{4} + \frac {7}{36} \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a^{2} + \frac {1}{15} \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.41 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx=\frac {1}{15} \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{12} \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{9} \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{3} \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{3} \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{5} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x} \,d x \]
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