Integrand size = 26, antiderivative size = 255 \[ \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {a^5 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac {5 a^4 b x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {10 a^3 b^2 x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {10 a^2 b^3 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {a b^4 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {b^5 x^{17} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )} \] Output:
a^5*x^7*((b*x^2+a)^2)^(1/2)/(7*b*x^2+7*a)+5*a^4*b*x^9*((b*x^2+a)^2)^(1/2)/ (9*b*x^2+9*a)+10*a^3*b^2*x^11*((b*x^2+a)^2)^(1/2)/(11*b*x^2+11*a)+10*a^2*b ^3*x^13*((b*x^2+a)^2)^(1/2)/(13*b*x^2+13*a)+a*b^4*x^15*((b*x^2+a)^2)^(1/2) /(3*b*x^2+3*a)+b^5*x^17*((b*x^2+a)^2)^(1/2)/(17*b*x^2+17*a)
Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.33 \[ \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {x^7 \sqrt {\left (a+b x^2\right )^2} \left (21879 a^5+85085 a^4 b x^2+139230 a^3 b^2 x^4+117810 a^2 b^3 x^6+51051 a b^4 x^8+9009 b^5 x^{10}\right )}{153153 \left (a+b x^2\right )} \] Input:
Integrate[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
Output:
(x^7*Sqrt[(a + b*x^2)^2]*(21879*a^5 + 85085*a^4*b*x^2 + 139230*a^3*b^2*x^4 + 117810*a^2*b^3*x^6 + 51051*a*b^4*x^8 + 9009*b^5*x^10))/(153153*(a + b*x ^2))
Time = 0.39 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.40, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1384, 27, 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.
\(\displaystyle \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int b^5 x^6 \left (b x^2+a\right )^5dx}{b^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int x^6 \left (b x^2+a\right )^5dx}{a+b x^2}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (b^5 x^{16}+5 a b^4 x^{14}+10 a^2 b^3 x^{12}+10 a^3 b^2 x^{10}+5 a^4 b x^8+a^5 x^6\right )dx}{a+b x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (\frac {a^5 x^7}{7}+\frac {5}{9} a^4 b x^9+\frac {10}{11} a^3 b^2 x^{11}+\frac {10}{13} a^2 b^3 x^{13}+\frac {1}{3} a b^4 x^{15}+\frac {b^5 x^{17}}{17}\right )}{a+b x^2}\) |
Input:
Int[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*((a^5*x^7)/7 + (5*a^4*b*x^9)/9 + (10*a^3* b^2*x^11)/11 + (10*a^2*b^3*x^13)/13 + (a*b^4*x^15)/3 + (b^5*x^17)/17))/(a + b*x^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Time = 2.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.31
method | result | size |
gosper | \(\frac {x^{7} \left (9009 x^{10} b^{5}+51051 a \,x^{8} b^{4}+117810 a^{2} x^{6} b^{3}+139230 a^{3} x^{4} b^{2}+85085 x^{2} a^{4} b +21879 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{153153 \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
default | \(\frac {x^{7} \left (9009 x^{10} b^{5}+51051 a \,x^{8} b^{4}+117810 a^{2} x^{6} b^{3}+139230 a^{3} x^{4} b^{2}+85085 x^{2} a^{4} b +21879 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{153153 \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
orering | \(\frac {x^{7} \left (9009 x^{10} b^{5}+51051 a \,x^{8} b^{4}+117810 a^{2} x^{6} b^{3}+139230 a^{3} x^{4} b^{2}+85085 x^{2} a^{4} b +21879 a^{5}\right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{2}}}{153153 \left (b \,x^{2}+a \right )^{5}}\) | \(89\) |
risch | \(\frac {a^{5} x^{7} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{7 b \,x^{2}+7 a}+\frac {5 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b \,a^{4} x^{9}}{9 \left (b \,x^{2}+a \right )}+\frac {10 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, a^{3} b^{2} x^{11}}{11 \left (b \,x^{2}+a \right )}+\frac {10 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, a^{2} b^{3} x^{13}}{13 \left (b \,x^{2}+a \right )}+\frac {a \,b^{4} x^{15} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 b \,x^{2}+3 a}+\frac {b^{5} x^{17} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{17 b \,x^{2}+17 a}\) | \(178\) |
Input:
int(x^6*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/153153*x^7*(9009*b^5*x^10+51051*a*b^4*x^8+117810*a^2*b^3*x^6+139230*a^3* b^2*x^4+85085*a^4*b*x^2+21879*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.22 \[ \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {1}{17} \, b^{5} x^{17} + \frac {1}{3} \, a b^{4} x^{15} + \frac {10}{13} \, a^{2} b^{3} x^{13} + \frac {10}{11} \, a^{3} b^{2} x^{11} + \frac {5}{9} \, a^{4} b x^{9} + \frac {1}{7} \, a^{5} x^{7} \] Input:
integrate(x^6*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")
Output:
1/17*b^5*x^17 + 1/3*a*b^4*x^15 + 10/13*a^2*b^3*x^13 + 10/11*a^3*b^2*x^11 + 5/9*a^4*b*x^9 + 1/7*a^5*x^7
\[ \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\int x^{6} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \] Input:
integrate(x**6*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
Output:
Integral(x**6*((a + b*x**2)**2)**(5/2), x)
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.22 \[ \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {1}{17} \, b^{5} x^{17} + \frac {1}{3} \, a b^{4} x^{15} + \frac {10}{13} \, a^{2} b^{3} x^{13} + \frac {10}{11} \, a^{3} b^{2} x^{11} + \frac {5}{9} \, a^{4} b x^{9} + \frac {1}{7} \, a^{5} x^{7} \] Input:
integrate(x^6*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")
Output:
1/17*b^5*x^17 + 1/3*a*b^4*x^15 + 10/13*a^2*b^3*x^13 + 10/11*a^3*b^2*x^11 + 5/9*a^4*b*x^9 + 1/7*a^5*x^7
Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.41 \[ \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {1}{17} \, b^{5} x^{17} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{3} \, a b^{4} x^{15} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{13} \, a^{2} b^{3} x^{13} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{11} \, a^{3} b^{2} x^{11} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{9} \, a^{4} b x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{7} \, a^{5} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) \] Input:
integrate(x^6*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Output:
1/17*b^5*x^17*sgn(b*x^2 + a) + 1/3*a*b^4*x^15*sgn(b*x^2 + a) + 10/13*a^2*b ^3*x^13*sgn(b*x^2 + a) + 10/11*a^3*b^2*x^11*sgn(b*x^2 + a) + 5/9*a^4*b*x^9 *sgn(b*x^2 + a) + 1/7*a^5*x^7*sgn(b*x^2 + a)
Timed out. \[ \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\int x^6\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \] Input:
int(x^6*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
Output:
int(x^6*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.23 \[ \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {x^{7} \left (9009 b^{5} x^{10}+51051 a \,b^{4} x^{8}+117810 a^{2} b^{3} x^{6}+139230 a^{3} b^{2} x^{4}+85085 a^{4} b \,x^{2}+21879 a^{5}\right )}{153153} \] Input:
int(x^6*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
Output:
(x**7*(21879*a**5 + 85085*a**4*b*x**2 + 139230*a**3*b**2*x**4 + 117810*a** 2*b**3*x**6 + 51051*a*b**4*x**8 + 9009*b**5*x**10))/153153