Integrand size = 28, antiderivative size = 196 \[ \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x} \, dx=\frac {3 a^2 b^2 x^n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac {3 a b^3 x^{2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a b+b^2 x^n\right )}+\frac {b^4 x^{3 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )}+\frac {a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \log (x)}{a+b x^n} \]
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Time = 0.03 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1369, 272, 45} \[ \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x} \, dx=\frac {3 a^2 b^2 x^n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac {b^4 x^{3 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )}+\frac {3 a b^3 x^{2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a b+b^2 x^n\right )}+\frac {a^3 \log (x) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
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Rule 45
Rule 272
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \frac {\left (a b+b^2 x^n\right )^3}{x} \, dx}{b^2 \left (a b+b^2 x^n\right )} \\ & = \frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^3}{x} \, dx,x,x^n\right )}{b^2 n \left (a b+b^2 x^n\right )} \\ & = \frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \text {Subst}\left (\int \left (3 a^2 b^4+\frac {a^3 b^3}{x}+3 a b^5 x+b^6 x^2\right ) \, dx,x,x^n\right )}{b^2 n \left (a b+b^2 x^n\right )} \\ & = \frac {3 a^2 b^2 x^n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac {3 a b^3 x^{2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a b+b^2 x^n\right )}+\frac {b^4 x^{3 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )}+\frac {a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \log (x)}{a+b x^n} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.34 \[ \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x} \, dx=\frac {\sqrt {\left (a+b x^n\right )^2} \left (b x^n \left (18 a^2+9 a b x^n+2 b^2 x^{2 n}\right )+6 a^3 \log \left (x^n\right )\right )}{6 n \left (a+b x^n\right )} \]
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Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{3} \ln \left (x \right )}{a +b \,x^{n}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{3} x^{3 n}}{3 \left (a +b \,x^{n}\right ) n}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{2} a \,x^{2 n}}{2 \left (a +b \,x^{n}\right ) n}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{2} b \,x^{n}}{\left (a +b \,x^{n}\right ) n}\) | \(127\) |
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Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x} \, dx=\frac {6 \, a^{3} n \log \left (x\right ) + 2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, n} \]
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\[ \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x} \, dx=\int \frac {\left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}}{x}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x} \, dx=a^{3} \log \left (x\right ) + \frac {2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, n} \]
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\[ \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x} \, dx=\int { \frac {{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x} \, dx=\int \frac {{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}}{x} \,d x \]
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