\(\int \frac {x^8}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [602]

Optimal result

Integrand size = 26, antiderivative size = 210 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {35 x}{128 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {x^7}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 x^5}{48 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {35 x^3}{192 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {35 \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 \sqrt {a} b^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:

-35/128*x/b^4/((b*x^2+a)^2)^(1/2)-1/8*x^7/b/(b*x^2+a)^3/((b*x^2+a)^2)^(1/2 
)-7/48*x^5/b^2/(b*x^2+a)^2/((b*x^2+a)^2)^(1/2)-35/192*x^3/b^3/(b*x^2+a)/(( 
b*x^2+a)^2)^(1/2)+35/128*(b*x^2+a)*arctan(b^(1/2)*x/a^(1/2))/a^(1/2)/b^(9/ 
2)/((b*x^2+a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.50 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {-\sqrt {a} \sqrt {b} x \left (105 a^3+385 a^2 b x^2+511 a b^2 x^4+279 b^3 x^6\right )+105 \left (a+b x^2\right )^4 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{384 \sqrt {a} b^{9/2} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \] Input:

Integrate[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

(-(Sqrt[a]*Sqrt[b]*x*(105*a^3 + 385*a^2*b*x^2 + 511*a*b^2*x^4 + 279*b^3*x^ 
6)) + 105*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(384*Sqrt[a]*b^(9/2)* 
(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.74, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1384, 27, 252, 252, 252, 252, 218}

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.

\(\displaystyle \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1384

\(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {x^8}{b^5 \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {x^8}{\left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {7 \int \frac {x^6}{\left (b x^2+a\right )^4}dx}{8 b}-\frac {x^7}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {7 \left (\frac {5 \int \frac {x^4}{\left (b x^2+a\right )^3}dx}{6 b}-\frac {x^5}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^7}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {x^2}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x^3}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^7}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{b x^2+a}dx}{2 b}-\frac {x}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^3}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^7}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {x}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^3}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^7}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

Input:

Int[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

((a + b*x^2)*(-1/8*x^7/(b*(a + b*x^2)^4) + (7*(-1/6*x^5/(b*(a + b*x^2)^3) 
+ (5*(-1/4*x^3/(b*(a + b*x^2)^2) + (3*(-1/2*x/(b*(a + b*x^2)) + ArcTan[(Sq 
rt[b]*x)/Sqrt[a]]/(2*Sqrt[a]*b^(3/2))))/(4*b)))/(6*b)))/(8*b)))/Sqrt[a^2 + 
 2*a*b*x^2 + b^2*x^4]
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 1384

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)])
 

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70

method	result	size
risch	\(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {93 x^{7}}{128 b}-\frac {511 a \,x^{5}}{384 b^{2}}-\frac {385 a^{2} x^{3}}{384 b^{3}}-\frac {35 a^{3} x}{128 b^{4}}\right )}{\left (b \,x^{2}+a \right )^{5}}-\frac {35 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (b x +\sqrt {-a b}\right )}{256 \left (b \,x^{2}+a \right ) \sqrt {-a b}\, b^{4}}+\frac {35 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (-b x +\sqrt {-a b}\right )}{256 \left (b \,x^{2}+a \right ) \sqrt {-a b}\, b^{4}}\)	\(146\)
default	\(-\frac {\left (-105 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) b^{4} x^{8}+279 \sqrt {a b}\, b^{3} x^{7}-420 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a \,b^{3} x^{6}+511 \sqrt {a b}\, a \,b^{2} x^{5}-630 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{2} b^{2} x^{4}+385 \sqrt {a b}\, a^{2} b \,x^{3}-420 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{3} b \,x^{2}+105 \sqrt {a b}\, a^{3} x -105 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{4}\right ) \left (b \,x^{2}+a \right )}{384 \sqrt {a b}\, b^{4} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}\)	\(169\)

Input:

int(x^8/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

((b*x^2+a)^2)^(1/2)/(b*x^2+a)^5*(-93/128/b*x^7-511/384*a/b^2*x^5-385/384*a 
^2/b^3*x^3-35/128*a^3/b^4*x)-35/256*((b*x^2+a)^2)^(1/2)/(b*x^2+a)/(-a*b)^( 
1/2)/b^4*ln(b*x+(-a*b)^(1/2))+35/256*((b*x^2+a)^2)^(1/2)/(b*x^2+a)/(-a*b)^ 
(1/2)/b^4*ln(-b*x+(-a*b)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.52 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\left [-\frac {558 \, a b^{4} x^{7} + 1022 \, a^{2} b^{3} x^{5} + 770 \, a^{3} b^{2} x^{3} + 210 \, a^{4} b x + 105 \, {\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{768 \, {\left (a b^{9} x^{8} + 4 \, a^{2} b^{8} x^{6} + 6 \, a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}, -\frac {279 \, a b^{4} x^{7} + 511 \, a^{2} b^{3} x^{5} + 385 \, a^{3} b^{2} x^{3} + 105 \, a^{4} b x - 105 \, {\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{384 \, {\left (a b^{9} x^{8} + 4 \, a^{2} b^{8} x^{6} + 6 \, a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}\right ] \] Input:

integrate(x^8/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")
 

Output:

[-1/768*(558*a*b^4*x^7 + 1022*a^2*b^3*x^5 + 770*a^3*b^2*x^3 + 210*a^4*b*x 
+ 105*(b^4*x^8 + 4*a*b^3*x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4)*sqrt(-a* 
b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a*b^9*x^8 + 4*a^2*b^8*x 
^6 + 6*a^3*b^7*x^4 + 4*a^4*b^6*x^2 + a^5*b^5), -1/384*(279*a*b^4*x^7 + 511 
*a^2*b^3*x^5 + 385*a^3*b^2*x^3 + 105*a^4*b*x - 105*(b^4*x^8 + 4*a*b^3*x^6 
+ 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a*b 
^9*x^8 + 4*a^2*b^8*x^6 + 6*a^3*b^7*x^4 + 4*a^4*b^6*x^2 + a^5*b^5)]
 

Sympy [F]

\[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {x^{8}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**8/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
 

Output:

Integral(x**8/((a + b*x**2)**2)**(5/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.49 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {279 \, b^{3} x^{7} + 511 \, a b^{2} x^{5} + 385 \, a^{2} b x^{3} + 105 \, a^{3} x}{384 \, {\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} + \frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} b^{4}} \] Input:

integrate(x^8/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")
 

Output:

-1/384*(279*b^3*x^7 + 511*a*b^2*x^5 + 385*a^2*b*x^3 + 105*a^3*x)/(b^8*x^8 
+ 4*a*b^7*x^6 + 6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4) + 35/128*arctan(b 
*x/sqrt(a*b))/(sqrt(a*b)*b^4)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.41 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} b^{4} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {279 \, b^{3} x^{7} + 511 \, a b^{2} x^{5} + 385 \, a^{2} b x^{3} + 105 \, a^{3} x}{384 \, {\left (b x^{2} + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x^{2} + a\right )} \] Input:

integrate(x^8/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
 

Output:

35/128*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4*sgn(b*x^2 + a)) - 1/384*(279*b 
^3*x^7 + 511*a*b^2*x^5 + 385*a^2*b*x^3 + 105*a^3*x)/((b*x^2 + a)^4*b^4*sgn 
(b*x^2 + a))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {x^8}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:

int(x^8/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
 

Output:

int(x^8/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4}+420 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b \,x^{2}+630 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} x^{4}+420 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} x^{6}+105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{4} x^{8}-105 a^{4} b x -385 a^{3} b^{2} x^{3}-511 a^{2} b^{3} x^{5}-279 a \,b^{4} x^{7}}{384 a \,b^{5} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:

int(x^8/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
 

Output:

(105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4 + 420*sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*x**2 + 630*sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*x**4 + 420*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a*b**3*x**6 + 105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqr 
t(a)))*b**4*x**8 - 105*a**4*b*x - 385*a**3*b**2*x**3 - 511*a**2*b**3*x**5 
- 279*a*b**4*x**7)/(384*a*b**5*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 
4*a*b**3*x**6 + b**4*x**8))