Integrand size = 30, antiderivative size = 108 \[ \int (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac {b^2 x^{1+n} (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1+m+n) \left (a b+b^2 x^n\right )} \]
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Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1369, 14, 20, 30} \[ \int (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {b^2 x^{n+1} (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(m+n+1) \left (a b+b^2 x^n\right )}+\frac {a (d x)^{m+1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (m+1) \left (a+b x^n\right )} \]
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Rule 14
Rule 20
Rule 30
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int (d x)^m \left (a b+b^2 x^n\right ) \, dx}{a b+b^2 x^n} \\ & = \frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \left (a b (d x)^m+b^2 x^n (d x)^m\right ) \, dx}{a b+b^2 x^n} \\ & = \frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac {\left (b^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^n (d x)^m \, dx}{a b+b^2 x^n} \\ & = \frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac {\left (b^2 x^{-m} (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^{m+n} \, dx}{a b+b^2 x^n} \\ & = \frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac {b^2 x^{1+n} (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1+m+n) \left (a b+b^2 x^n\right )} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.51 \[ \int (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {x (d x)^m \sqrt {\left (a+b x^n\right )^2} \left (a (1+m+n)+b (1+m) x^n\right )}{(1+m) (1+m+n) \left (a+b x^n\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, x \left (m b \,x^{n}+a m +a n +b \,x^{n}+a \right ) d^{m} x^{m} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i d x \right ) m \left (\operatorname {csgn}\left (i d x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i d \right )\right )}{2}}}{\left (a +b \,x^{n}\right ) \left (1+m \right ) \left (1+m +n \right )}\) | \(99\) |
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Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.53 \[ \int (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {{\left (b m + b\right )} x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + {\left (a m + a n + a\right )} x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{2} + {\left (m + 1\right )} n + 2 \, m + 1} \]
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\[ \int (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\int \left (d x\right )^{m} \sqrt {\left (a + b x^{n}\right )^{2}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.44 \[ \int (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {a d^{m} {\left (m + n + 1\right )} x x^{m} + b d^{m} {\left (m + 1\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m^{2} + m {\left (n + 2\right )} + n + 1} \]
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Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.60 \[ \int (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {b m x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + a m x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + b m x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + a n x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + b x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + a x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + b x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} \mathrm {sgn}\left (b x^{n} + a\right )}{m^{2} + m n + 2 \, m + n + 1} \]
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Timed out. \[ \int (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n} \,d x \]
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