Integrand size = 25, antiderivative size = 201 \[ \int \left (a+x^2\right )^{5/2} \left (x-\sqrt {a+x^2}\right )^n \, dx=\frac {a^6 \left (x-\sqrt {a+x^2}\right )^{-6+n}}{64 (6-n)}+\frac {3 a^5 \left (x-\sqrt {a+x^2}\right )^{-4+n}}{32 (4-n)}+\frac {15 a^4 \left (x-\sqrt {a+x^2}\right )^{-2+n}}{64 (2-n)}-\frac {5 a^3 \left (x-\sqrt {a+x^2}\right )^n}{16 n}-\frac {15 a^2 \left (x-\sqrt {a+x^2}\right )^{2+n}}{64 (2+n)}-\frac {3 a \left (x-\sqrt {a+x^2}\right )^{4+n}}{32 (4+n)}-\frac {\left (x-\sqrt {a+x^2}\right )^{6+n}}{64 (6+n)} \]
1/64*a^6*(x-(x^2+a)^(1/2))^(-6+n)/(6-n)+3/32*a^5*(x-(x^2+a)^(1/2))^(-4+n)/ (4-n)+15/64*a^4*(x-(x^2+a)^(1/2))^(-2+n)/(2-n)-5/16*a^3*(x-(x^2+a)^(1/2))^ n/n-15/64*a^2*(x-(x^2+a)^(1/2))^(2+n)/(2+n)-3/32*a*(x-(x^2+a)^(1/2))^(4+n) /(4+n)-1/64*(x-(x^2+a)^(1/2))^(6+n)/(6+n)
Time = 0.41 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.86 \[ \int \left (a+x^2\right )^{5/2} \left (x-\sqrt {a+x^2}\right )^n \, dx=\frac {1}{64} \left (x-\sqrt {a+x^2}\right )^n \left (-\frac {20 a^3}{n}-\frac {a^6}{(-6+n) \left (x-\sqrt {a+x^2}\right )^6}-\frac {6 a^5}{(-4+n) \left (x-\sqrt {a+x^2}\right )^4}-\frac {15 a^4}{(-2+n) \left (x-\sqrt {a+x^2}\right )^2}-\frac {15 a^2 \left (x-\sqrt {a+x^2}\right )^2}{2+n}-\frac {6 a \left (x-\sqrt {a+x^2}\right )^4}{4+n}-\frac {\left (x-\sqrt {a+x^2}\right )^6}{6+n}\right ) \]
((x - Sqrt[a + x^2])^n*((-20*a^3)/n - a^6/((-6 + n)*(x - Sqrt[a + x^2])^6) - (6*a^5)/((-4 + n)*(x - Sqrt[a + x^2])^4) - (15*a^4)/((-2 + n)*(x - Sqrt [a + x^2])^2) - (15*a^2*(x - Sqrt[a + x^2])^2)/(2 + n) - (6*a*(x - Sqrt[a + x^2])^4)/(4 + n) - (x - Sqrt[a + x^2])^6/(6 + n)))/64
Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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.
\(\displaystyle \int \left (a+x^2\right )^{5/2} \left (x-\sqrt {a+x^2}\right )^n \, dx\) |
\(\Big \downarrow \) 2547 |
\(\displaystyle -\frac {1}{64} \int \left (x-\sqrt {x^2+a}\right )^{n-7} \left (\left (x-\sqrt {x^2+a}\right )^2+a\right )^6d\left (x-\sqrt {x^2+a}\right )\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {1}{64} \int \left (a^6 \left (x-\sqrt {x^2+a}\right )^{n-7}+6 a^5 \left (x-\sqrt {x^2+a}\right )^{n-5}+15 a^4 \left (x-\sqrt {x^2+a}\right )^{n-3}+20 a^3 \left (x-\sqrt {x^2+a}\right )^{n-1}+15 a^2 \left (x-\sqrt {x^2+a}\right )^{n+1}+6 a \left (x-\sqrt {x^2+a}\right )^{n+3}+\left (x-\sqrt {x^2+a}\right )^{n+5}\right )d\left (x-\sqrt {x^2+a}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{64} \left (\frac {a^6 \left (x-\sqrt {a+x^2}\right )^{n-6}}{6-n}+\frac {6 a^5 \left (x-\sqrt {a+x^2}\right )^{n-4}}{4-n}+\frac {15 a^4 \left (x-\sqrt {a+x^2}\right )^{n-2}}{2-n}-\frac {20 a^3 \left (x-\sqrt {a+x^2}\right )^n}{n}-\frac {15 a^2 \left (x-\sqrt {a+x^2}\right )^{n+2}}{n+2}-\frac {6 a \left (x-\sqrt {a+x^2}\right )^{n+4}}{n+4}-\frac {\left (x-\sqrt {a+x^2}\right )^{n+6}}{n+6}\right )\) |
((a^6*(x - Sqrt[a + x^2])^(-6 + n))/(6 - n) + (6*a^5*(x - Sqrt[a + x^2])^( -4 + n))/(4 - n) + (15*a^4*(x - Sqrt[a + x^2])^(-2 + n))/(2 - n) - (20*a^3 *(x - Sqrt[a + x^2])^n)/n - (15*a^2*(x - Sqrt[a + x^2])^(2 + n))/(2 + n) - (6*a*(x - Sqrt[a + x^2])^(4 + n))/(4 + n) - (x - Sqrt[a + x^2])^(6 + n)/( 6 + n))/64
3.6.1.3.1 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])
\[\int \left (x^{2}+a \right )^{\frac {5}{2}} \left (x -\sqrt {x^{2}+a}\right )^{n}d x\]
Time = 0.28 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.01 \[ \int \left (a+x^2\right )^{5/2} \left (x-\sqrt {a+x^2}\right )^n \, dx=-\frac {{\left (a^{3} n^{6} - 50 \, a^{3} n^{4} + {\left (n^{6} - 20 \, n^{4} + 64 \, n^{2}\right )} x^{6} + 544 \, a^{3} n^{2} + 3 \, {\left (a n^{6} - 30 \, a n^{4} + 104 \, a n^{2}\right )} x^{4} - 720 \, a^{3} + 3 \, {\left (a^{2} n^{6} - 40 \, a^{2} n^{4} + 264 \, a^{2} n^{2}\right )} x^{2} + 6 \, {\left ({\left (n^{5} - 20 \, n^{3} + 64 \, n\right )} x^{5} + 2 \, {\left (a n^{5} - 30 \, a n^{3} + 104 \, a n\right )} x^{3} + {\left (a^{2} n^{5} - 40 \, a^{2} n^{3} + 264 \, a^{2} n\right )} x\right )} \sqrt {x^{2} + a}\right )} {\left (x - \sqrt {x^{2} + a}\right )}^{n}}{n^{7} - 56 \, n^{5} + 784 \, n^{3} - 2304 \, n} \]
-(a^3*n^6 - 50*a^3*n^4 + (n^6 - 20*n^4 + 64*n^2)*x^6 + 544*a^3*n^2 + 3*(a* n^6 - 30*a*n^4 + 104*a*n^2)*x^4 - 720*a^3 + 3*(a^2*n^6 - 40*a^2*n^4 + 264* a^2*n^2)*x^2 + 6*((n^5 - 20*n^3 + 64*n)*x^5 + 2*(a*n^5 - 30*a*n^3 + 104*a* n)*x^3 + (a^2*n^5 - 40*a^2*n^3 + 264*a^2*n)*x)*sqrt(x^2 + a))*(x - sqrt(x^ 2 + a))^n/(n^7 - 56*n^5 + 784*n^3 - 2304*n)
Exception generated. \[ \int \left (a+x^2\right )^{5/2} \left (x-\sqrt {a+x^2}\right )^n \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int \left (a+x^2\right )^{5/2} \left (x-\sqrt {a+x^2}\right )^n \, dx=\int { {\left (x^{2} + a\right )}^{\frac {5}{2}} {\left (x - \sqrt {x^{2} + a}\right )}^{n} \,d x } \]
\[ \int \left (a+x^2\right )^{5/2} \left (x-\sqrt {a+x^2}\right )^n \, dx=\int { {\left (x^{2} + a\right )}^{\frac {5}{2}} {\left (x - \sqrt {x^{2} + a}\right )}^{n} \,d x } \]
Timed out. \[ \int \left (a+x^2\right )^{5/2} \left (x-\sqrt {a+x^2}\right )^n \, dx=\int {\left (x-\sqrt {x^2+a}\right )}^n\,{\left (x^2+a\right )}^{5/2} \,d x \]