Integrand size = 24, antiderivative size = 206 \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {a^3 x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{\left (a+b x^n\right )^3}+\frac {3 a^2 b^4 x^{1+n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+n) \left (a b+b^2 x^n\right )^3}+\frac {3 a b^5 x^{1+2 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+2 n) \left (a b+b^2 x^n\right )^3}+\frac {b^6 x^{1+3 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+3 n) \left (a b+b^2 x^n\right )^3} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1357, 250} \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {b^6 x^{3 n+1} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(3 n+1) \left (a b+b^2 x^n\right )^3}+\frac {3 a b^5 x^{2 n+1} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(2 n+1) \left (a b+b^2 x^n\right )^3}+\frac {3 a^2 b^4 x^{n+1} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(n+1) \left (a b+b^2 x^n\right )^3}+\frac {a^3 x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{\left (a+b x^n\right )^3} \]
[In]
[Out]
Rule 250
Rule 1357
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \int \left (2 a b+2 b^2 x^n\right )^3 \, dx}{\left (2 a b+2 b^2 x^n\right )^3} \\ & = \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \int \left (8 a^3 b^3+24 a^2 b^4 x^n+24 a b^5 x^{2 n}+8 b^6 x^{3 n}\right ) \, dx}{\left (2 a b+2 b^2 x^n\right )^3} \\ & = \frac {a^3 x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{\left (a+b x^n\right )^3}+\frac {3 a^2 b^4 x^{1+n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+n) \left (a b+b^2 x^n\right )^3}+\frac {3 a b^5 x^{1+2 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+2 n) \left (a b+b^2 x^n\right )^3}+\frac {b^6 x^{1+3 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+3 n) \left (a b+b^2 x^n\right )^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.59 \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {x \sqrt {\left (a+b x^n\right )^2} \left (a^3 \left (1+6 n+11 n^2+6 n^3\right )+3 a^2 b \left (1+5 n+6 n^2\right ) x^n+3 a b^2 \left (1+4 n+3 n^2\right ) x^{2 n}+b^3 \left (1+3 n+2 n^2\right ) x^{3 n}\right )}{(1+n) (1+2 n) (1+3 n) \left (a+b x^n\right )} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{3} x}{a +b \,x^{n}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{3} x \,x^{3 n}}{\left (a +b \,x^{n}\right ) \left (1+3 n \right )}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{2} a x \,x^{2 n}}{\left (a +b \,x^{n}\right ) \left (1+2 n \right )}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{2} b x \,x^{n}}{\left (a +b \,x^{n}\right ) \left (1+n \right )}\) | \(138\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.63 \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {{\left (2 \, b^{3} n^{2} + 3 \, b^{3} n + b^{3}\right )} x x^{3 \, n} + 3 \, {\left (3 \, a b^{2} n^{2} + 4 \, a b^{2} n + a b^{2}\right )} x x^{2 \, n} + 3 \, {\left (6 \, a^{2} b n^{2} + 5 \, a^{2} b n + a^{2} b\right )} x x^{n} + {\left (6 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 6 \, a^{3} n + a^{3}\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
[In]
[Out]
\[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\int \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac {3}{2}}\, dx \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.49 \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {{\left (2 \, n^{2} + 3 \, n + 1\right )} b^{3} x x^{3 \, n} + 3 \, {\left (3 \, n^{2} + 4 \, n + 1\right )} a b^{2} x x^{2 \, n} + 3 \, {\left (6 \, n^{2} + 5 \, n + 1\right )} a^{2} b x x^{n} + {\left (6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1\right )} a^{3} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.28 \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {6 \, a^{3} n^{3} x \mathrm {sgn}\left (b x^{n} + a\right ) + 2 \, b^{3} n^{2} x x^{3 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 9 \, a b^{2} n^{2} x x^{2 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 18 \, a^{2} b n^{2} x x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + 11 \, a^{3} n^{2} x \mathrm {sgn}\left (b x^{n} + a\right ) + 3 \, b^{3} n x x^{3 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 12 \, a b^{2} n x x^{2 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 15 \, a^{2} b n x x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + 6 \, a^{3} n x \mathrm {sgn}\left (b x^{n} + a\right ) + b^{3} x x^{3 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 3 \, a b^{2} x x^{2 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 3 \, a^{2} b x x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + a^{3} x \mathrm {sgn}\left (b x^{n} + a\right )}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
[In]
[Out]
Timed out. \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\int {\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2} \,d x \]
[In]
[Out]