Integrand size = 23, antiderivative size = 176 \[ \int \left (a+x^2\right )^2 \left (x-\sqrt {a+x^2}\right )^n \, dx=-\frac {a^5 \left (x-\sqrt {a+x^2}\right )^{-5+n}}{32 (5-n)}-\frac {5 a^4 \left (x-\sqrt {a+x^2}\right )^{-3+n}}{32 (3-n)}-\frac {5 a^3 \left (x-\sqrt {a+x^2}\right )^{-1+n}}{16 (1-n)}+\frac {5 a^2 \left (x-\sqrt {a+x^2}\right )^{1+n}}{16 (1+n)}+\frac {5 a \left (x-\sqrt {a+x^2}\right )^{3+n}}{32 (3+n)}+\frac {\left (x-\sqrt {a+x^2}\right )^{5+n}}{32 (5+n)} \]
-1/32*a^5*(x-(x^2+a)^(1/2))^(-5+n)/(5-n)-5/32*a^4*(x-(x^2+a)^(1/2))^(-3+n) /(3-n)-5/16*a^3*(x-(x^2+a)^(1/2))^(-1+n)/(1-n)+5/16*a^2*(x-(x^2+a)^(1/2))^ (1+n)/(1+n)+5/32*a*(x-(x^2+a)^(1/2))^(3+n)/(3+n)+1/32*(x-(x^2+a)^(1/2))^(5 +n)/(5+n)
Time = 0.41 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.85 \[ \int \left (a+x^2\right )^2 \left (x-\sqrt {a+x^2}\right )^n \, dx=\frac {1}{32} \left (x-\sqrt {a+x^2}\right )^{-5+n} \left (\frac {a^5}{-5+n}+\frac {5 a^4 \left (x-\sqrt {a+x^2}\right )^2}{-3+n}+\frac {10 a^3 \left (x-\sqrt {a+x^2}\right )^4}{-1+n}+\frac {10 a^2 \left (x-\sqrt {a+x^2}\right )^6}{1+n}+\frac {5 a \left (x-\sqrt {a+x^2}\right )^8}{3+n}+\frac {\left (x-\sqrt {a+x^2}\right )^{10}}{5+n}\right ) \]
((x - Sqrt[a + x^2])^(-5 + n)*(a^5/(-5 + n) + (5*a^4*(x - Sqrt[a + x^2])^2 )/(-3 + n) + (10*a^3*(x - Sqrt[a + x^2])^4)/(-1 + n) + (10*a^2*(x - Sqrt[a + x^2])^6)/(1 + n) + (5*a*(x - Sqrt[a + x^2])^8)/(3 + n) + (x - Sqrt[a + x^2])^10/(5 + n)))/32
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2547, 244, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+x^2\right )^2 \left (x-\sqrt {a+x^2}\right )^n \, dx\) |
\(\Big \downarrow \) 2547 |
\(\displaystyle \frac {1}{32} \int \left (x-\sqrt {x^2+a}\right )^{n-6} \left (\left (x-\sqrt {x^2+a}\right )^2+a\right )^5d\left (x-\sqrt {x^2+a}\right )\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {1}{32} \int \left (a^5 \left (x-\sqrt {x^2+a}\right )^{n-6}+5 a^4 \left (x-\sqrt {x^2+a}\right )^{n-4}+10 a^3 \left (x-\sqrt {x^2+a}\right )^{n-2}+10 a^2 \left (x-\sqrt {x^2+a}\right )^n+5 a \left (x-\sqrt {x^2+a}\right )^{n+2}+\left (x-\sqrt {x^2+a}\right )^{n+4}\right )d\left (x-\sqrt {x^2+a}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{32} \left (-\frac {a^5 \left (x-\sqrt {a+x^2}\right )^{n-5}}{5-n}-\frac {5 a^4 \left (x-\sqrt {a+x^2}\right )^{n-3}}{3-n}-\frac {10 a^3 \left (x-\sqrt {a+x^2}\right )^{n-1}}{1-n}+\frac {10 a^2 \left (x-\sqrt {a+x^2}\right )^{n+1}}{n+1}+\frac {5 a \left (x-\sqrt {a+x^2}\right )^{n+3}}{n+3}+\frac {\left (x-\sqrt {a+x^2}\right )^{n+5}}{n+5}\right )\) |
(-((a^5*(x - Sqrt[a + x^2])^(-5 + n))/(5 - n)) - (5*a^4*(x - Sqrt[a + x^2] )^(-3 + n))/(3 - n) - (10*a^3*(x - Sqrt[a + x^2])^(-1 + n))/(1 - n) + (10* a^2*(x - Sqrt[a + x^2])^(1 + n))/(1 + n) + (5*a*(x - Sqrt[a + x^2])^(3 + n ))/(3 + n) + (x - Sqrt[a + x^2])^(5 + n)/(5 + n))/32
3.5.90.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_ .)*(x_)^2])^(n_.), x_Symbol] :> Simp[(1/(2^(2*m + 1)*e*f^(2*m)))*(i/c)^m Subst[Int[x^n*((d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1)/(-d + x)^(2*(m + 1))), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IntegerQ[2*m] && (Intege rQ[m] || GtQ[i/c, 0])
\[\int \left (x^{2}+a \right )^{2} \left (x -\sqrt {x^{2}+a}\right )^{n}d x\]
Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.90 \[ \int \left (a+x^2\right )^2 \left (x-\sqrt {a+x^2}\right )^n \, dx=-\frac {{\left (5 \, {\left (n^{4} - 10 \, n^{2} + 9\right )} x^{5} + 10 \, {\left (a n^{4} - 16 \, a n^{2} + 15 \, a\right )} x^{3} + 5 \, {\left (a^{2} n^{4} - 22 \, a^{2} n^{2} + 45 \, a^{2}\right )} x + {\left (a^{2} n^{5} - 30 \, a^{2} n^{3} + {\left (n^{5} - 10 \, n^{3} + 9 \, n\right )} x^{4} + 149 \, a^{2} n + 2 \, {\left (a n^{5} - 20 \, a n^{3} + 19 \, a n\right )} x^{2}\right )} \sqrt {x^{2} + a}\right )} {\left (x - \sqrt {x^{2} + a}\right )}^{n}}{n^{6} - 35 \, n^{4} + 259 \, n^{2} - 225} \]
-(5*(n^4 - 10*n^2 + 9)*x^5 + 10*(a*n^4 - 16*a*n^2 + 15*a)*x^3 + 5*(a^2*n^4 - 22*a^2*n^2 + 45*a^2)*x + (a^2*n^5 - 30*a^2*n^3 + (n^5 - 10*n^3 + 9*n)*x ^4 + 149*a^2*n + 2*(a*n^5 - 20*a*n^3 + 19*a*n)*x^2)*sqrt(x^2 + a))*(x - sq rt(x^2 + a))^n/(n^6 - 35*n^4 + 259*n^2 - 225)
\[ \int \left (a+x^2\right )^2 \left (x-\sqrt {a+x^2}\right )^n \, dx=\int \left (a + x^{2}\right )^{2} \left (x - \sqrt {a + x^{2}}\right )^{n}\, dx \]
\[ \int \left (a+x^2\right )^2 \left (x-\sqrt {a+x^2}\right )^n \, dx=\int { {\left (x^{2} + a\right )}^{2} {\left (x - \sqrt {x^{2} + a}\right )}^{n} \,d x } \]
\[ \int \left (a+x^2\right )^2 \left (x-\sqrt {a+x^2}\right )^n \, dx=\int { {\left (x^{2} + a\right )}^{2} {\left (x - \sqrt {x^{2} + a}\right )}^{n} \,d x } \]
Timed out. \[ \int \left (a+x^2\right )^2 \left (x-\sqrt {a+x^2}\right )^n \, dx=\int {\left (x-\sqrt {x^2+a}\right )}^n\,{\left (x^2+a\right )}^2 \,d x \]